The third wheel: How red squirrels affect the
dynamics of the lynx-snowshoe hare relationship
A Thesis Submitted to the Committee on Graduate Studies in Partial Fulfillment of the
Requirements for the Degree of a Master of Science in the Faculty of Arts and Science
TRENT UNIVERSITY
Peterborough, Ontario, Canada
© Copyright by Kevin W. Chan 2017
Environmental and Life Sciences M.Sc. graduate program
January 2017
Abstract
The third wheel: How red squirrels affect the dynamics of the lynx-snowshoe hare
relationship
Kevin W. Chan
Population cycles are regular fluctuations in population densities, however, in recent
years many cycles have begun to disappear. With Canada lynx this dampening has also
been seen with decreasing latitude corresponding to an increase in prey diversity. My
study investigates the role of alternate prey on the stability of the lynx-hare cycle by first
comparing the functional responses of two sympatric but ecologically distinct predators
on a primary and alternate prey. I then populated a three species predator-prey model to
investigate the role of alternate prey on population stability. My results showed that
alternate prey can promote stability, though they are unlikely to “stop the cycle”.
Furthermore, stability offered by alternate prey is contingent on its ability to increase
intraspecific competition. My study highlights that population cycles are not governed by
a single factor and that future research needs to be cognizant of interactions between
alternate prey and intraspecific competition.
Keywords: Canada Lynx (Lynx Canadensis), Coyote (Canis latrans), Red squirrel
(Tamiasciuris hudsonicus), Snowshoe hare (Lupus americanus), functional responses,
population cycles, population modeling, alternate prey, interference competition, density
dependence.
ii
Acknowledgements
First and foremost, I would like to express my gratitude to my supervisor, Dennis
Murray, who took a chance on me when no other supervisor would. Through his
supervision, I have learned more about ecology than I thought possible, and more
importantly, what it means to be a member of the scientific community. I have seen
through him the value of maintaining a curious attitude but also the importance of being
able to focus on a single topic. In this acknowledgement I must also apologize for the
many, many pages of horrible writing that he must have endured to help me produce this
manuscript. I would also like to thank members of my committee members, Wenying
Feng and Jeff Row, for their time and willingness to adapt to the sometimes sudden
changes to my project.
I owe a debt of gratitude to Stan Boutin, Charles Krebs and Mark O‟Donoghue,
without whom this document would not exist. It is through their foresight, audacity, and
years of painstaking work that an ecological record of this magnitude exists. I feel
particularly grateful for their insight and their permission to partake in this project even if
only through the virtual reality of historic data.
In addition I need to extend my thanks to the members of the Murray lab, who
greatly enhanced my graduate experience, and provided me with exceptional role models.
In particular, Tom Hossie, an exceptional scientist on whom I heavily relied on not only
for insight on understanding functional responses but also as a mentor and guide through
my career as a masters student. Special thanks also goes to Catarina Ferreira, a great
researcher and even better friend, and to Adrian Borlestean and Guillame Baptiste-
iii
Rousseau, whose expertise in R gave me a great foundation on which to rise to the ranks
of middling R guru.
I am deeply indebted to Ian Fife, Ayden Sherrit, Tim Irvin, Luis Pavon, Mike
Peers, Yasmine Majchrzak, and Adam Marques for their invaluable friendship in helping
me keep my sanity through the years spent starring at a computer screen. To my parents,
who encouraged a strange child with bizarre interests and continue to support my
unconventional life wandering the backwoods of the world. And finally, to Anna Cho,
whose patience and support means more to me than she realizes.
iv
Dedication
To my parents, who nurtured (and tolerated) my curiosity and sense of adventure.
To my friends, who have challenged my assumptions and patience and made me better for it.
And to you, yes you, because honestly who reads this?!
v
Co-authorship Statement
I have written my dissertation in manuscript format, as each of my chapters are being
prepared for submission in peer-reviewed literature. Chapter 2 is being submitted to
Ecology, while Chapter 3 is being submitted to Oikos. These manuscripts contain four to
five authors in addition to myself. I devised the general outline and objectives of this
thesis, and was the primary contributor to the analysis and manuscript preparation. S.
Boutin, C.J.Krebs, and M. O‟Donoghue graciously contributed data collected over the
span of their long term study and provided useful criticisms to all manuscripts. D. Murray
provided supervision to all components of the thesis. Additionally co-author T. Hossie
(Chapter 2) gave additional supervision made important contributions to the manuscript
as a whole.
vi
Table of Contents
Abstract ...................................................................................................................................... ii
Acknowledgements ................................................................................................................... iii
Co-authorship Statement ............................................................................................................ vi
Table of Contents ..................................................................................................................... vii
List of Figures ........................................................................................................................... ix
List of Tables ............................................................................................................................. xi
List of Equations....................................................................................................................... xii
Chapter 1: General Introduction .................................................................................................. 1
Chapter 2 .................................................................................................................................. 11
The role of alternate prey and intraspecific interactions on predator functional responses
in a natural vertebrate system .................................................................................................... 11
Abstract ..................................................................................................................... 12
Introduction ............................................................................................................... 14
Methods ..................................................................................................................... 18
Results ....................................................................................................................... 23
Discussion.................................................................................................................. 25
Tables and Figures ..................................................................................................... 31
Chapter 3 .................................................................................................................................. 37
vii
The role of alternate prey in the maintenance of population cycles: A case study using
Canada lynx .............................................................................................................................. 37
Abstract ..................................................................................................................... 38
Introduction ............................................................................................................... 39
Methods ..................................................................................................................... 42
Results ....................................................................................................................... 51
Discussion.................................................................................................................. 55
Tables and Figures .................................................................................................................... 63
Chapter 4: General discussion ................................................................................................... 71
References ................................................................................................................................ 74
Appendix I: Multispecies functional response derivation .......................................................... 87
Appendix II: Methods Testing .................................................................................................. 91
Appendix III. Parameter estimates from functional response model fitting ................................ 95
Appendix IV: Comparison of single species and multi-species functional responses in a
model selection framework ....................................................................................................... 99
Appendix V: Results of predator density-independent with predator density-dependent
mortality model fitting and stability analysis. .......................................................................... 103
Appendix VI: Raw Data.......................................................................................................... 105
viii
List of Figures
Figure 2.1. Comparison of prey-dependent (left), ratio-dependent (middle) and inverse
(right) responses of lynx to primary (top) and alternate (bottom) prey. The best-fit models
were hyperbolic (AICc weight = 0.47, pseudo R2 = 0.67; solid) and sigmoidal preydependent (AICc weight = 0.40, pseudo R2 = 0.68; dashed) for primary prey and linear
ratio-dependent response (pseudo R2 = 0.55, AICc weight = 0.33) and prey-dependent
inverse (θ = 2) for alternate prey (pseudo R2 = 0.66, AICc weight = 0.31). Best fit models
are shown for all inverse graphs with hare predation best represented by an inverse ratiodependence (θ = 2, pseudo R2 = 0.66, AICc weight = 0.01) in contrast to prey-dependent
inverse predation in squirrels. ................................................................................................... 35
Figure 2.2. Comparison of prey-dependent (left) and ratio-dependent (right) functional
responses of coyote to primary (top) and alternate (bottom) prey. The best-fit models
were linear ratio-dependent for both primary prey (pseudo R2 = 0.49, AICc weight =
0.28) and alternate prey (pseudo R2 = 0.78, AICc weight = 0.38). Best fit models are
shown for all inverse graphs with hare predation best represented by an inverse ratiodependent model (θ = 2, pseudo R2 = 0.66, AICc weight = 0.01) in contrast to squirrel
predation being inverse prey-dependent (θ = 2, pseudo R2 = 0.49, AICc weight = 0.03). ........... 36
Figure 3.1. a) The observed average winter densities of snowshoe hares (black), red
squirrels (grey), and lynx (dashed) in the Kluane study area, Yukon from 1987-2011. b)
Representative biomass of snowshoe hares (average biomass = 1.5kg) and red squirrels
(average biomass =0.25kg) in the Kluane lake study area from 1988-2001. .............................. 65
ix
Figure 3.2. Comparison of the estimated densities (solid) and observed densites (dotted)
for a) Hares and b) Lynx using unweighted residuals in the Levenburg-Marquardt
algorithm. ................................................................................................................................. 66
Figure 3.3. Simulation of hare cycle at 3 different densities of squirrels, no squirrels (0
per 100 km2), average observed densities (23,000 per 100 km2), and maximum observed
densities (39,500 per 100 km2) showing a reduction in the amplitude and period of the
cycle as squirrel densities increase. ........................................................................................... 67
Figure 3.4. Elasticity analysis of system stability to proportional changes in parameter.
The real part of the eigenvalues of the jacobian matrix are plotted against values of the
parameter being shifted (i.e. 1 = best fit model estimate). The horizontal black line
indicates an eigenvalue of zero. A positive real eigenvalue represents an unstable system
(i.e. a source) while a negative eigenvalue represents a stable system (i.e. a sink). .................... 68
Figure 3.5. Elasticity analysis of the stability of the jacobian matrix in respect to
proportional changes in squirrel density (i.e. 1 = 23,000 squirrels per 100km2). ........................ 69
Figure 3.6. Elasticity analysis of system stability to proportional changes in parameters
and changes in squirrel density: no squirrels (light grey), average squirrel densities
(23,000 squirrels per 100 km2; medium grey) and 3 times average squirrel density (69,000
squirrels per 100 km2; black). The real part of the eigenvalues of the jacobian matrix are
plotted against values of the parameter being shifted (i.e. 1 = best fit model estimate)............... 70
x
List of Tables
Table 2.1. Performance of lynx (P) functional response models relative to primary prey
(snowshoe hare; N1) density. Squirrel density is denoted by N2. ................................................ 31
Table 2.2. Performance of lynx (P) functional response models relative to alternate prey
(red squirrel; N2) density. Hare density is denoted by N1. .......................................................... 32
Table 2.3. Performance of coyote (P) functional response models relative to primary prey
(snowshoe hare; N1) density. Squirrel density is denoted by N2. ................................................ 33
Table 2.4. Performance of coyote (P) functional response models relative to alternate prey
(red squirrel; N2) density. Hare density is denoted by N1. .......................................................... 34
Table 3.1. Model selection of for the numerical response of hare and lynx. ............................... 63
Table 3.2. Parameter estimates of the best fit numerical response model estimated using
with the Levenburg-Marquardt method. .................................................................................... 64
xi
List of Equations
Model
Constant
Form
Prey-Dependent
Holling- Zero handling time (Type I)
Hyperbolic Holling disc equation (Type II)
Sigmoidal Holling disc equation (Type III)
Inverse
Ratio-Dependent
Arditi-Ginzburg-Zero handling time(Type I)
Hyperbolic Arditi-Ginzburg (Type II)
Sigmoidal Arditi-Ginzburg (Type II)
Inverse
Predator-Dependent
Hassell-Varley (Type I)
Hyperbolic Arditi-Akҫakaya (Type II)
Sigmoidal Arditi-Akҫakaya (Type III)
Inverse
*Variables: N = prey density, P = predator density. Parameters: a = attack rate, h =
handling time, c = maximum killing rate, θ = shape parameter, m = interference
coefficient
xii
Chapter 1: General Introduction
One of the central tenets of population ecology is that stability is a natural consequence of
complexity; in other words the more diverse a community the less likely it is to fluctuate
and be upset by disturbances and invasions (May 1973a, McCann 2000). This tenant has
garnered support from repeated observations of natural and experimental systems that
simple communities are much more likely to undergo large fluctuations in population and
or collapse, than complex ones (Odum 1953, May 1973a, Elton 2000). In addition, it is
noted that systems undergoing regular population cycles tend to lose their cycles with a
north south gradient as numbers of generalist predators increase (Hanski et al. 1991, Roth
et al. 2007). Similarly, species richness and alternate prey use increases along this
gradient and seemingly can increase stability (Roth et al. 2007, Shier and Boyce 2009).
Thus it can be inferred that the cyclic propensity of a population can be diminished by
increasing alternate prey reliance and availability. However, these observations remain
circumstantial and the alternate prey hypothesis remains difficult to test experimentally or
in the field due to the spatial scale and time needed to collect usable field data (Tilman
and Downing 1994, Tilman 1996, Krebs et al. 2001b). Consequently, population ecology
has relied heavily on mathematics to simulate experimental changes in simple single prey,
single predator systems rather than testing real ecosystems (May 1976, Turchin 2003).
While mathematical models are used to approximate population dynamics in nature,
many researchers question the validity of such models in characterizing real world
dynamics as they can produce seemingly unrealistic predictions. For example, some
models may give rise to the „paradox of enrichment‟, whereby the addition of resources to
a predator-prey system destabilizes population dynamics (Rosenzweig 1971, Turchin
2003); an outcome seemingly counter-intuitive in light of the observation that system
1
complexity should promote system stability. Likewise, simple models can be destabilized
by adding a single trophic interaction (May 1973a). Expanding on this theme, Robert
May challenged the longstanding notion that stability and complexity increase in unison
by illustrating that by increasing the number of species or the number of interactions, the
parameter space in which a modeled community can remain stable becomes increasingly
small (May 1972, 1973a). May‟s work further revealed that species interactions such as
amensalism, commensalism, and predator-prey interactions generally lead to system
stability, while mutualism and interspecific competition generally do not, and that the
probability of a modeled system being stable declines with increasingly strong
interactions (May 1973a, 1973b, Jansen and Kokkoris 2003).
While these results are surprising, May‟s early models were criticized as being too
simplistic, for example, by not specifically addressing intraspecific competition, which
has been shown to increase the stability of a community (Yodzis 1981, Roy and
Chattopadhyay 2007). Increasingly, it seems that current mathematical models do support
observations that stability is generally found in more complex systems (Yodzis 1981,
Jansen and Kokkoris 2003, Allesina and Tang 2012), yet there remains a lack of
understanding of the mechanisms contributing to stability (de Mazancourt et al. 2013,
Loreau and de Mazancourt 2013).Therefore, to examine the mechanisms driving stability
in a community, we reduce the community to its essential components, a focal predator, a
primary prey, and an alternate prey, to more easily determine the effect of their individual
interactions. From this point, we can begin to test the alternate prey hypothesis and
determine how alternate prey affect the stability of the community.
2
The population model
In order to assess the effect of alternate prey on population stability we first need to
develop a multi-prey population model. Traditionally the framework for any type of
population ecology model has been to quantify the rate of change in a population using
the following structure (Turchin 2003):
[
[
]
]
*
+
*
+
Exploring the individual components in the above frame work, the prey‟s rate of increase
in the absence of predators is analogous to a single species model for population growth.
Barring exceptions such as the Allee effect, most single species systems can be modeled
can be using a logistic equation (eq. 1.1) (Hassell 1978, Turchin 2003, Ginzburg and
Arditi 2012).
(
)
(1.1)
Here population growth of species N is moderated by two parameters: r, the intrinsic rate
of population growth, and k, the carrying capacity. This allows for the population growth
rate to decrease monotonically as the population approaches carrying capacity and
accelerating growth at low densities that decelerate to equilibrium at the carrying capacity
(Turchin 2003).
Predator growth, on the other hand, is defined by how many prey can be converted to new
predators (Rosenzweig and MacArthur 1963, Turchin 2003) and can be represented by
3
the number of predators (P) multiplied by the summation of products of the prey specific
conversion rate (χi) and the prey specific functional response (Fi )(eq.1.2).
(∑(
(1.2)
The functional response is defined as the relationship between the rate at which a predator
can kill a prey item and the density of the prey. Therefore, this relationship also defines
the mortality of prey accounted for by the predator and will be discussed in greater detail
below. The total prey mortality is then accounted as the combination of the functional
response with the product of the linear non-specific mortality parameter ( 𝛿 ) and prey
density. This parameter includes predation from predators not included in the model as
well as natural mortality. Similarly, 𝛿 is a linear term represents the total mortality of
predators irrespective of cause.
The above framework can then be represented mathematically by combining these
components to form the following coupled equations expanded to accommodate multiple
prey (Rosenzweig and MacArthur 1963, Turchin 2003) where
change in the population of prey i and
represents the rate of
represents the rate of change in the predator
population. .
(
)
𝛿
((∑
𝛿
(1.3a)
(1.3b)
The functional response
From the above equations we see that the direct link between the prey equation and the
predator equation is the functional response, Fi, which is the relationship between the
predator killing rate on prey i and the prey density. Thus, the proper characterization of
4
the functional response is critical for determining the dynamics of the population.
Originally, three functional responses shapes were characterized by C.S. Holling (Holling
1959a, Turchin 2003); a linear, hyperbolic, and a sigmoidal response and can all be
described by the Holling disc equation (eq. 1.4).
(
(1.4)
The Holling equation provides a mechanistic explanation of the killing rate by defining it
as the function of attack rate (a) (i.e. the proportion of prey encountered and killed during
one time step), and the handing time per prey item (h). θ is an additional parameter used
to define the shape of the response between hyperbolic (θ = 1) and sigmoidal (θ = 2). The
simplest response, the linear or type I response, is characterized by a predator killing rate
that is unaffected by the time it takes to physically manipulate and digest prey (h = 0).
The most basic Lotka-Volterra model utilized a linear functional response where the
killing rate is constantly increasing with prey density. This model, however, is seen as
unrealistic as linear functional responses are thought to only be well represented by filter
feeders (Turchin 2003, Jeschke et al. 2004), where handling time is seen to be negligible
or predators are able to capture and handle prey simultaneously.
More commonly the functional response takes the shape of a hyperbolic curve (Jeschke et
al. 2004), and is distinguished from the linear response by a saturation of the killing rate
at high densities of prey. In the hyperbolic response, handing time limits the time
available to the predator for searching resulting in a response with a continually
decreasing rate (Hassell 1978). Similarly, predator efficiency in the sigmoidal response is
limited by handling time, but this response is differentiated by an accelerating kill rate at
low prey densities and a decelerating kill rate at intermediate to high prey densities.
5
Mathematically this can be described using an attack rate that increases with increasing
prey density resulting in low active searching by predators at low prey densities but high
active searching at high densities. This switch in foraging behavior can be seen as a result
of predator learning (Holling 1959a, Hassell 1978), prey refuges from predation
(Rosenzweig and MacArthur 1963, Hossie and Murray 2010), and prey switching
(Murdoch 1969, Turchin 2003). Consequently, the hyperbolic is widely thought to be
represented by specialist predators while the sigmoidal by generalist.
Because the Holling equation is a declining function resulting from the difference
between the total time and the time spent on active searching of prey, it follows that
alternate prey should decrease the amount of time available for active searching of the
primary prey. However, to date, the focus of most studies has been on single predator,
single prey interactions with very few studies explicitly testing predator interactions with
alternate prey (Koen-Alonso 2007, Smout et al. 2010). Moreover, the Holling responses
were characterized only as a function of prey density, known as prey-dependence, and did
not consider the effect of predator density on the killing rate. However, many proponents
such as Hassel and Varley (1969) , Beddington (1975) and DeAngelis (1975), and most
recently Arditi and Ginzburg (1989, 2012) have demonstrated the importance of predator
density as intraspecific competition (herein referred to as interference), which can lead to
reductions in hunting efficiency. This interference should reduce the amount of time
available for searching, thus weakening the direct relationship between the killing rate
and prey density. Commonly predator interference is incorporated as either a parameter in
the functional response (i.e., “predator-dependent” functional response; Hassell and
Varley 1969b, DeAngelis et al. 1975b) but can also be represented as the proportion of
6
prey available to each predator (i.e. “ratio-dependent” functional response; Arditi &
Ginzburg 1989).
Prey dependent models continue to dominate predator-prey theory (Abrams and Ginzburg
2000, Turchin 2003, Ginzburg and Arditi 2012) and while there is increasing acceptance
of mechanistic predator-dependent models (Abrams 2015), ratio-dependent models draw
persistent criticism due to their lack of a mechanistic explanation and oversimplification
of complex behavioral interactions between predator and prey (Abrams and Ginzburg
2000, Ginzburg and Arditi 2012, Abrams 2015). It is important to note that preydependence is not free of criticism and has long been known to create anomalous and
apparently unrealistic dynamics (e.g. paradox of enrichment; Jensen & Ginzburg, 2005).
Consequently, neither prey-dependence, ratio-dependence or predator-dependence should
be taken dogmatically but seen as a range of possible responses along a continuum,
starting with zero interference (prey-dependence; F(N)), and incorporating interference
either as a separate parameter (predator-dependence; F(N,P)) or as the proportion of prey
available to each predator (ratio-dependence; F(N/P)).
The model system
To provide an empirical test of the above theory, I used data from an extensive study of
the boreal forest ecosystem, near Kluane lake, Yukon. This project included
documenting functional responses of lynx (Lynx canadensis) and coyotes (Canis latrans)
on their primary prey, snowshoe hares (Lepus americanus), and primary alternate prey,
red squirrels (Tamiasciurus hudsonicus). Lynx and coyotes experience lagged densitydependent population cycles with a period of approximately 9-11 years with the
snowshoe hare, however the red squirrel population dynamics are largely reflective of the
cone crop (Krebs et al. 2001b). The Kluane study specifically examined ecosystem
7
dynamics and spanned a complete snowshoe hare cycle. While this research project
officially ended in 1996, some data collection has continued to this day. The Kluane
dataset represents an especially rich source of information on population dynamics and
predator-prey relationships in the boreal forest ecosystem, making it well suited for
investigating the role of primary and alternate prey in predator population dynamics.
It is important to provide greater background detail about the predator-prey system under
investigation. Firstly, this system is dominated by two similar sized, sympatric predators,
the lynx (Lynx canadensis) and the coyote (Canis latrans), that share a common primary
prey the, snowshoe hare (Lepus americanus). Snowshoe hares dominate much of the
energy flow in the Kluane ecosystem and while avian predators (e.g. great horned owls
and goshawks) also contribute to this system, coyotes and lynx are responsible for
approximately 60% of adult snowshoe hare deaths over the course of the cycle (Krebs et
al. 2001b). Wolves and other large mammalian predators are known to take snowshoe
hares but only incidentally supporting the idea that these dynamics are largely driven by
the interactions of these 3 species (Krebs et al. 2001b).
Second, due to the large energy flow from hares, during periods of low hare availability,
both lynx and coyotes require an increased consumption of alternate prey to fulfill their
energetic needs and consume an increased number of red squirrels (Tamiasciurus
hudsonicus) and other prey (Peromyscus, Microtus, and Myodes spp.)(O‟Donoghue et al.
1998b, Krebs et al. 2001b). Red squirrels are an important alternate prey, particularly for
lynx, and can constitute up to 37% of the diet of lynx and up to 13% in coyotes during
years of low hare density (O‟Donoghue et al. 1998a, Krebs et al. 2001b). Moreover, both
predators undergo population cycles with a period of 8-11 years and experience a
8
dampening of these cycles at southern latitudes corresponding with an increased
consumption of red squirrels and alternate prey (Roth et al. 2007).
Finally, while these predators share the same prey, they represent vastly different foraging
behaviors, coyotes in the Yukon being more reliant on hares and having a narrower
dietary breadth than lynx, allowing us to compare the relative importance of alternate
prey to 2 ecologically distinct predators in a natural ecosystem (Murray et al. 1995,
O‟Donoghue et al. 1998a).
Goal of thesis
I first developed a generalized functional response incorporating multiple prey and
interference to use as a basis for examining predator-prey interactions. This response will
be used to investigate the differences in the functional responses between predator with a
narrow dietary breadth (coyotes) and one with a wider dietary breadth (lynx) and their use
of a common primary prey (snowshoe hares) and alternate prey (red squirrels) (Chapter
2). I predict that lynx, due to their wider dietary breadth and solitary nature (see Poole
1995, Krebs et al. 2001), should be more affected by alternate prey density and less by
intraspecific interference than coyotes which are more social and reliant on hares
(Kleiman and Eisenberg 1973, O‟Donoghue et al. 1998b). Following this, I developed a
generalized multi-prey population model for lynx to assess the stabilizing or destabilizing
effect of alternate prey on a community population (Chapter 3). Because lynx-hare
populations at the southern periphery of the species range undergo a decrease in cyclicity
correlating with alternate prey use (Roth et al. 2007), we predict that increased alternate
prey density should result in stability.
By integrating predator-prey theory with an assessment of long term behavioral and
population data sets, I will demonstrate that alternate prey and predator interference
9
play a critical role in the characterization of the functional response and in assessing
the stability of a predator-prey community. General conclusions and future directions
will be addressed in Chapter 4. In summary, I will address the following research
questions in greater detail in the following chapters:
1) How do alternate prey and intraspecific interference affect the functional responses of
2 similarly sized predators that differ greatly in their dietary breadth? [Chapter 2]
2) Do alternate prey serve to stabilize or destabilize population dynamics? [Chapter 3]
10
Chapter 2
The role of alternate prey and intraspecific interactions on predator functional
responses in a natural vertebrate system
Chan, K., Boutin, S., Hossie, T, Krebs, C.J., O‟Donoghue, M., Murray, D.L.
11
Abstract
To better understand the complex and variable patterns of predator foraging behaviour in
natural systems, it is critical to determine how density-dependent predation and predator
hunting success are mediated by alternate prey or predator interference. Despite
considerable theory and debate seeking to place predator-prey interactions in a more
realistic context, few empirical studies have quantified the role of alternate prey or
intraspecific interactions on predator-prey dynamics. Using snowtracking data, we
assessed the functional responses of 2 similarly-sized, sympatric carnivores, lynx (Lynx
canadensis) and coyotes (Canis latrans), foraging on common primary (snowshoe hares;
Lepus americanus) and alternate (red squirrels; Tamiasciurus hudsonicus) prey within a
natural system. Lynx exhibited a hyperbolic prey-dependent response to changes in hare
density, which is characteristic of predators relying primarily on a single prey species. In
contrast, the lynx-squirrel response was found to be linear ratio-dependent, or inversely
dependent on hare density. The coyote-hare and coyote-squirrel interactions also were
linear and influenced by predator density. We explain these novel results by the apparent
use of spatial and temporal refuges by prey, and the likelihood that predators commonly
experience interference and lack of satiation when foraging. Our study provides empirical
support from a natural predator-prey system that: 1) the predation rate may not be limited
at high prey densities when prey are small or rarely captured; 2) interference competition
may influence the predator functional response; and 3) predator interference has a
variable role across different prey types. Ultimately, distinct functional responses of
predators to different prey types illustrates the complexity associated with predator-prey
interactions in natural systems and highlights the need to investigate predator behavior
12
and predation rate in relation to the broader ecological community comprised of
competitors and alternate prey.
13
Introduction
Understanding how predators influence prey population dynamics is central to
predator-prey ecology, but our ability to rigorously quantify predation effects remains
challenged by the difficulty in estimating predator kill rates across space and time
(Berryman 1992, Turchin 2003). The functional response is an important component of
the predation rate, establishing the density–dependent relationship between predator
killing rate and prey abundance; the functional response can vary dynamically
depending on a variety of ecological and environmental factors (Berryman 1992,
Abrams and Ginzburg 2000, Turchin 2003). Indeed, predator foraging and predation
success is affected by variation in predator foraging tactics (Fryxell et al. 2007), prey
handling and digestion time (Jeschke et al. 2004), prey escape or avoidance behavior
(Jeschke et al. 2004), alternate food resources (Smout et al. 2010), habitat structure
(Poggiale et al. 1998, Hossie and Murray 2010), and environmental conditions (Stenseth
and Shabbar 2004). Notably, these interactions have been revealed almost exclusively
through controlled lab experiments involving single predator-prey dyads. Although this
work has greatly elucidated mechanisms shaping the predator functional response, the
role of the above factors in a real world context remains largely unknown. Indeed, it is
unclear whether our current understanding of the predator functional response, derived
from highly simplified systems, robustly explains real world predator-prey population
dynamics (Berryman 1992, Ginzburg and Arditi 2012).
The shape of the functional response varies widely across systems, and is
determined by factors that limit the predator‟s attack rate and maximum consumption
rate. For example, a linear (Type I) functional response represents a killing rate that
increases proportionally with prey density, and is most often seen in filter feeders where
14
prey handling time is negligible and predators capture and handle prey simultaneously
(Turchin 2003, Jeschke et al. 2004). More commonly, limits set by prey handling and
digestion result in a hyperbolic (Type II) or sigmoidal (Type III) functional response,
reflecting a constantly declining predator efficiency as prey density increases (Berryman
1992, Turchin 2003, Jeschke et al. 2004). Hyperbolic functional responses have been
reported across a broad range of taxa and are the predominant functional form of the
predator killing rate (Jeschke et al. 2004). Alternatively, sigmoidal (Type III) functional
responses involve a killing rate that increases across a range of intermediate prey
densities and decelerates at high densities (Hassell 1978). Such responses often are
attributed to predators that can switch between primary and alternate prey depending on
the density of primary prey (Murdoch 1969, Turchin 2003), however, sigmoidal
functional responses can also result from predator learning (Holling 1959a, Hassell
1978) and prey refuges from predation (Rosenzweig and MacArthur 1963, Hossie and
Murray 2010).
To date, theoretical and empirical research on functional responses has largely
centered on single-predator, single-prey systems, often in controlled lab situations (e.g.
Sarnelle & Wilson 2008; Hossie & Murray 2010; Ginzburg & Arditi 2012). While the
simplicity of these systems is useful for revealing basic interactions, the complexity
associated with natural predator-prey relationships, where multiple prey species are
available to predators, warrants additional attention. Specifically, there is a need to
consider not only the density of the primary prey, but also that of alternate prey. For
example, one might expect that variation in the reliance on primary and alternate prey
should affect the shape of individual functional responses and that predator species with
wider dietary breadths will be more strongly influenced by changes in alternate prey
15
density than those with narrow diets (Holling 1959a). This may be especially impactful
if there is a large differential between the energy returns of primary and alternate prey
(Holling 1959a, Hassell 1978).
An additional uncertainty concerns the role of predator interactions (herein
referred to as interference) on the functional response (Abrams and Ginzburg 2000,
Abrams 2015). Traditional (prey-dependent) functional response models assume that
predator kill rates depend solely on prey density though, more recently, there is growing
appreciation that conspecific predators can impact predator foraging through processes
such as exploitative competition and strife (Turchin 2003, Ginzburg and Arditi 2012).
While prey-dependent functional response models form a useful conceptual starting
point, they can generate unrealistic population dynamics, for example, when additional
food destabilizes predator-prey dynamics (i.e. paradox of enrichment), or when stable
populations cannot be maintained at low prey densities (i.e. paradox of biological
control) (Rosenzweig 1971, Vucetich et al. 2002, Roy and Chattopadhyay 2007).
However, the main criticism of prey-dependent functional response models is that they
ignore how interference can weaken the direct link between prey density and the
predator‟s per capita kill rate (Abrams and Ginzburg 2000, Ginzburg and Arditi 2012).
Alternate forms of the functional response incorporate interference either as a
component of the functional response itself (i.e., “predator-dependent”; Hassell and
Varley 1969, Beddington 1975a, DeAngelis et al. 1975b) or via the relative abundance
or prey per predator (i.e. “ratio-dependent”; Arditi & Ginzburg 1989).
Interestingly, the extent to which predator- or ratio-dependent functional
responses outperform prey-dependent models appears to vary across systems (Ginzburg
and Arditi 2012, Abrams 2015). Of the few empirical tests assessing the variety of
16
functional responses, it seems that, depending on the system, prey-dependent (Arditi and
Saïah 1992), ratio-dependent (Vucetich et al. 2002, Ginzburg and Arditi 2012) and
predator-dependent responses (Skalski and Gilliam 2001, Kratina et al. 2009) each can
appropriately characterize the predator killing rate. Since it remains unknown to what
degree functional responses are driven by primary prey density, interference, or the
availability of alternate prey, it seems appropriate to consider these models as a gradient
of potential responses driven by a range of ecological or environmental factors
(Ginzburg and Arditi 2012).
Our study compares the shape and form of the functional response between
contrasting predators in a natural multi-prey community. In the Yukon, Canada lynx
(Lynx canadensis) and coyotes (Canis latrans) rely on common primary (snowshoe
hares, Lepus americanus) and alternate (red squirrel, Tamiasciurus hudsonicus) prey.
Both predators rely extensively on snowshoe hares, but kill more red squirrels when
hare densities are low. While lynx and coyotes are similarly-sized and sympatric in the
boreal forest, they are functionally distinct predators with lynx in the Kluane region
having a marginally wider dietary breadth and exhibiting stronger prey switching at low
hare densities (O‟Donoghue et al. 1998b, Mowat et al. 2000). Both predators, however,
are strongly reliant on snowshoe hares and accordingly, we predict that lynx, even with
their wider diet, should have a hyperbolic or weakly sigmoidal prey-dependent
functional response to hares, reflecting strong handling and digestive constraints and
prey switching (O‟Donoghue et al. 1998b). In contrast, lynx functional response to
squirrels should either follow a sigmoidal response (i.e., reflecting an active switch to
squirrels) or one that is inverse to hare density when feeding on squirrels. Because
coyotes in our study area are strongly dependent on hares and show little evidence of
17
prey switching (O‟Donoghue et al. 1998a, Krebs et al. 2001b), they should also follow a
hyperbolic prey-dependent functional response when feeding on hares and either a
constant response (i.e. reflecting opportunistic predation not dependent on prey density)
or one that is weakly inverse to hare density. We also expect that coyotes will
experience stronger interference due to their hierarchical social structure and
territoriality (Kleiman and Eisenberg 1973, Gese 2001, Neale and Sacks 2001); this
means that their functional responses are more likely to be predator-dependent or ratiodependent.
Methods
We studied lynx and coyote functional responses relative to their primary
(snowshoe hare) and alternate (red squirrel) prey by snowtracking predators in
southwest Yukon during November-March of 1987-1988 to 1996-1997 (O‟Donoghue et
al. 1998a, 1998b). We note, however, that both species, consume additional prey
(Peromyscus, Microtus, and Clethrionomys spp.) that can comprise a substantive portion
of their diet during hare lows (Coyotes = 18-44%, Lynx =23-25%; O‟Donoghue et al.
1998a). Kills of each prey species were recorded as they were encountered on fresh
trails, providing a continuous record of prey encounters, habitat use, and other activities
related to hunting behavior in winter (Murray et al. 1995, O‟Donoghue et al. 1998a). In
total, lynx and coyotes were snowtracked 3568 km and 2752km, respectively and killed
a total of 313 and 99 hares, and 183 and 30 red squirrels, respectively. Recognizing that
we had low numbers of kills to support our model fitting, we provide details on
additional analyses conducted to ascertain confidence in our results in Appendix II. This
study was part of a larger effort assessing the community-level dynamics of the boreal
forest ecosystem (Krebs et al. 2001b), and the present paper expands on our previous
18
work (O‟Donoghue et al. 1998a) by considering multiple prey and interference as
potential determinants of the predator response. We include 2 additional years of
predation data not reported in the original O‟Donoghue et al. (1998a) study, spanning
increasing predator and hare numbers in the study area. Of particular note, this paper
extends the previous analysis of lynx and coyote predation to include a gamut of
candidate functional response models that may better reflect the ecological complexity
and diversity in our study area.
Instantaneous prey kill rates were estimated by comparing distances travelled by
snowtracked lynx and coyotes to movement rates of radio-collared predators,
calculating kill rate per individual predator, per day (O‟Donoghue et al. 1998a). To
estimate winter abundance, we used updated Efford‟s maximum-likelihood estimates of
prey abundance from mark-recapture grids averaged between fall and spring rather than
the original jack-knife estimations (O‟Donoghue et al. 1998a, Efford et al. 2009, Krebs
et al. 2011). Predator densities were estimated from collared individuals and
snowtracking. Both predator and prey densities are available online (see
http://www.zoology.ubc.ca/~krebs/kluane.html). Density estimates for prey and
predators were standardized as number of individuals per 100 km2. Our observed kill
rates are approximations of instantaneous kill rates, as both prey species produce young
in spring-summer pulses and abundances decline continually over the course of the year.
However, the intra-annual decline in prey abundance is assumed to have a negligible
impact on approximation in a natural system of this scale (Jost et al. 2005).
To explore factors influencing each predator‟s per capita kill rate, we developed
a multispecies functional response model incorporating effects of both primary and
alternate prey density. It is important that since we sought to assess not only the effect
19
of red squirrels on the killing rate of hares (alternate on primary), but also the effect of
hares on the killing rate of red squirrels (primary on alternate), that we distinguish
„primary prey‟ as the prey killed at the highest rate relative to its overall availability
(snowshoe hare) and „alternate prey‟ as prey with a killing rate closer to its availability
(red squirrel). Our additional nomenclature refers to „focal prey‟ as the prey type whose
functional response is under immediate review, whereas „non-focal prey‟ refers to the
prey type not under current review.
The multispecies functional response model was based on the formulation
originally proposed by Murdoch (1973), where the denominator is the summation of the
actual searching time for all prey items. Because the original model assumes each
response is hyperbolic and prey-dependent, this model was further developed to allow
the shape (linear to sigmoidal; Smout et al. 2010) and strength of interference to vary
separately for each prey item (eq. 2.1, Appendix I for full derivation).
(
Here,
(
∑
(
(2.1)
is the kill rate of the predator on the focal prey species (i), P is the
density of the predator, and N is the density of the prey (i,j). Parameter a is the attack
rate or search efficiency of the predator and can be defined as the proportion of prey
encountered per predator per unit of searching time, whereas h is the handling time for
prey type i, accounting for both physical manipulation and digestion of prey. Parameter
θ controls the shape of the curve and reflects an attack rate that is either constant
(hyperbolic response; θ = 1) or increases as prey densities increase (sigmoidal response;
θ = 2). A linear response can be elicited when handling time is negligible (h ≈ 0).
Parameter m is a continuous parameter indicating the degree to which interference plays
20
a role in the functional response, with m = 0 indicating no interference and larger
numbers indicating stronger interference (Turchin 2003, Ginzburg and Arditi 2012). We
note a similar derivation of this multispecies equation has been independently obtained
by Koen-Alonso (2007) using the Beddington-DeAngelis models to account for waste
time as opposed to our model which employs a Hassell-Varley interference parameter
(Hassell and Varley 1969).
Our goal was to examine three key aspects of the functional response: 1) shape
of the response curve; 2) prey-, predator-, or ratio-dependency of the response; and 3)
whether the response was dependent upon focal or non-focal prey density. To do this,
we simplified the multispecies model into its single-species components, which were
then fit to the data and compared using likelihood methods. Specifically, by letting the
density of prey species j (Nj) equal 0, we effectively removed the second prey species
and derived the single-species predator-dependent equation (f(Ni,P), eq. 2.2).
(
(2.2)
Likewise, if m = 0, the predator-dependent equation (eq. 2.2) reduces to the
prey-dependent functional response (f(Ni ), eq. 2.3).
(
(2.3)
Indeed, equation 2.2 is mathematically equivalent to the Hassell-Varley equation
(Hassell and Varley 1969), whereas equation 2.3 is equivalent to the Holling disc
equation (Holling 1959b). Equations 2.1 and 2.2 do not explicitly consider ratiodependence, however, if m = 1 we obtain the Arditi-Ginzburg ratio-dependent model
(Arditi and Ginzburg 1989).
21
To examine whether killing rate for alternate prey is strongly influenced by time
spent hunting primary prey, we consider an additional set of models where non-focal
prey species (j) drives kill rate of the focal species i (f(Nj ,P), eq. 2.4).
(
(
)
(2.4)
The above relationship allowed us to model the functional response as an inverse
function corresponding to a nonlinear decline in the killing rate of species i as the
density of species j increases (see Appendix I for full derivation). Here, parameter θ is
linked to the shape of the functional response of species j using the Holling equation,
where larger values represent a steeper decline in the use of focal prey as non-focal prey
density increases. This model is largely phenomenological as parameter c2 represents
the maximum killing rate of squirrels and d1 is a constant that is the product of the hare
attack rate (a1) and handling time (h1). Though we expected that the killing rate of
squirrels might best be explained by an inverse function of hare density, we did not
expect the reciprocal relationship to be supported empirically.
We fit the models using the nls2 function in R (Grothendieck 2013, R
Development Core Team 2016) following the methodology outlined by Vucetich et al.
(2002). For each predator-prey combination we fit 16 competing functional response
models: a constant model (model 1), and 5 models for each of the prey-dependent
(models 2-6; linear, hyperbolic, sigmoidal, and 2 inverse ( θ = 1, θ = 2)), ratio-dependent
(models 7-11), and predator-dependent models (models 12-16) (see Table S1). The most
parsimonious model with strong explanatory power was identified from other candidate
models using the lowest ΔAICc, with models ΔAICc < 2 being considered as statistically
22
indistinguishable (Burnham and Anderson 2002). Pseudo R2 values were calculated as
the squared coefficient of determination of the predicted and observed values.
Results
When considering the 16 candidate models involving lynx-hare interactions, we
found that the top model was hyperbolic and prey-dependent (Table 1, Fig. 1, AICc
weight = 0.47, pseudo R2 = 0.67). The lynx predation response on hares indicated that
the maximum killing rate (h-1) was between 1.3-1.7 hares killed per lynx per day (fitted
parameters ± 95% CI; a = 0.0004± 0.0001 km2 ·d-1, h = 0.6610 ± 0.0820 d; see
Appendix III for parameter estimates of all 16 models). Model weights also revealed the
sigmoidal prey-dependent model as statistically indistinguishable from the top model
(AICc weight = 0.40, pseudo R2 = 0.68; a = 1.59e-07 ± 6.30e-08 km2 ·d-1, h = 0.810 ±
0.0566 d-1·N-1). The third ranked model (hyperbolic, predator-dependent response),
indicated low contribution from interference in lynx predation on hares (AICc weight =
0.02, pseudo R2 = 0.67, parameter m = 0.0475 ± 0.37) but had a ΔAICc > 2.
For the lynx-red squirrel interaction, the linear ratio-dependent functional
response (AICc weight = 0.33, pseudo R2 = 0.50) had the best fit though the inverse (θ =
2) response (AICc weight = 0.30, pseudo R2 = 0.66) was statistically indistinguishable
(ΔAICc = 0.11, Table 2, Fig. 1). Accordingly, both interference and hare density
appeared to contribute greatly to lynx predation of squirrels. The estimated attack rate
for the linear model was 0.00009-0.0001 P·km2 ·d-1 (a = 0.0001 ± 1.76 e-05),
representing a constantly increasing mortality rate imposed on red squirrels by lynx.
Because of the lack of red squirrel handling time in either the linear or inverse model
(see Appendix I), we infer from our results that time required for lynx to handle
squirrels was negligible compared to active search time. The estimates from the inverse
23
(θ = 2) response correspond to a theoretical maximum killing rate of 1.2-1.7 squirrels
per day (c = 1.48 ± 0.25) though because the linear model lacks predator satiation, these
estimates should be treated with caution. Regardless, we affirm that the lynx-squirrel
relationship may be governed by the ratio of lynx to squirrels or hare density, with
squirrel handling time playing a minor role in the shape of the response.
The best-fit model describing coyote predation on hares was a linear ratiodependent response (AICc weight = 0.40, pseudo R2 = 0.49, fitted parameter a = 0.16 ±
0.06 P·km2 ·d-1; Table 3). The linear nature of this relationship implied that coyote
predation on hares was not limited by handling time, but strong support for the both the
ratio- and predator-dependent models indicated that coyote per capita kill rate of hares
was to some degree influenced by interference (m = 0.57 ± 0.17; see Appendix III).
Interestingly, the linear nature of the coyote-hare response seems unequivocal, as the
linear predator-dependent (Fig. 2; AICc weight =0.26, pseudo R2 = 0.64), and preydependent model (AICc weight = 0.20, pseudo R2 = 0.64) were statistically
indistinguishable from the linear ratio-dependent response (Δ AICc <2, Table 3).
Notably, non-linear responses did not describe the coyote-hare relationship particularly
well.
The coyote-red squirrel functional response was best described by the linear
ratio-dependent model (Table 4; AICc weight = 0.58, pseudo R2 = 0.58), with the attack
rate estimated between 2.0 e-05 and 2.5 e-05 km2·P-1·d-1 (a = 2.27 e-05 ± 2.48 e-06). No
other models had ΔAICc < 2. The second ( AICc weight = 0.13, pseudo R2 = 0.66,) and
third (AICc weight = 0.11, pseudo R2 = 0.68) best models were the linear predatordependent and sigmoidal ratio-dependent models respectively (Table 4). Therefore, our
24
prediction that the functional response of coyotes to squirrels would be ratio-dependent
was supported, but this relationship was not strongly dependent on hare density.
Discussion
Consistent with our predictions, lynx kill rates on primary prey (hares) were
strongly constrained by handling and digestive time at high hare densities, as indicated
by either a hyperbolic or sigmoidal prey-dependent functional response to changes in
primary prey (hares). In addition, the lynx response to alternate prey (squirrels) was
found to be either ratio-dependent and linear or inverse to primary prey densities,
diverging from our prediction that a switching relationship would be best characterized
by a sigmoidal response. Coyote predation was linear with evidence favoring
interference models, which suggest that, broadly speaking, coyote foraging in our study
area appear to be limited more by interference than by prey handling constraints.
Accordingly, our findings diverge from the traditional view that in natural systems
predators consistently become satiated at high prey densities (Jeschke et al. 2004), and
that the expression of interference is a consistent factor influencing a predator‟s
functional response across its full range of food items (Koen-Alonso 2007, Williams
2007). Collectively, our findings reflect the complexity of predator-prey relationships in
natural environments and highlight the need to examine functional responses and their
effects on prey populations in a context reflecting real-world complexities involving
multispecies interactions.
It is well understood that lynx rely extensively on hares as prey across a wide
range of hare densities (O‟Donoghue et al. 1998b). Therefore, the hyperbolic functional
response to hares is not surprising (O‟Donoghue et al. 1998a), and reinforces the
prevalence of similar responses across the majority of predator species (Jeschke et al.
25
2004). Yet the lynx-hare response had a degree of support for the sigmoidal response
model, indicating that our predation data provided equivocal support for active diet
switching by lynx. Parameter estimates for both models of the lynx-hare functional
response clearly show that prey handling and digestion constraints strongly limit lynx
ability to capture additional hares as hare densities increase (see Turchin 2003, Jeschke
et al. 2004), which is consistent with theory showing that specialist predators experience
constraints in processing prey at high densities (Hassell 1978, Turchin 2003).
In contrast, the lynx-squirrel response was either linear or inverse; both models
are not limited by handling and digestion time of the focal prey. Interestingly, linear
functional responses are largely attributed to filter feeders, where small prey can be
ingested and processed while search for new prey occurs simultaneously (Hassell 1978,
Jeschke et al. 2004). Obviously, squirrels impose some handling and digestive
constraints on lynx, implying that the linear functional response does not reflect
limitless prey capture and processing. The linear functional response of lynx to squirrels
is likely attributable to the small body mass and thus lower handling time of squirrels
(i.e. squirrel biomass is approximately 15% hare biomass; O‟Donoghue et al. 1998a). In
addition, red squirrels have an arboreal activity refuge from terrestrial predators,
especially during winter (Krebs et al. 2001b). Thus, it follows that lynx may be unable
to capture squirrels at sufficiently high rates to experience satiation.
While evidence for active switching in the form of a sigmoidal response was
only detected in the lynx-hare relationship, evidence of the inverse response (Table 2)
was found in the lynx-squirrel response. This was supported by a strong influence of
primary prey density on the consumption rate of alternate prey. Because support for the
inverse model was found only in the lynx-squirrel relationship (ΔAIC = 0.11), we
26
contend that lynx are largely reliant on hares and likely switch to alternate prey either
gradually or as a last resort as hare numbers decline, in order to fulfill basic energetic
needs. We are mindful, however, that our data may not provide sufficient statistical
power to fully discriminate among the functional forms of the predator response
(Trexler et al. 1988, Marshal and Boutin 1999, see Appendix II for additional
analysese). Moreover, given the temporal correlation between predator and hare
densities in this system, it is likely that alternate prey consumption by predators is
governed by multiple factors and that squirrel predation by lynx is strongly influenced
by both interference and hare consumption.
The coyote responses were also found to be linear, though the conditions
promoting a linear functional response likely differ from those influencing the lynxsquirrel relationship. Coyotes, more so than lynx, are known to kill snowshoe hares in
excess of their nutritional needs (i.e. surplus killing; see O‟Donoghue et al. 1998a,
Krebs et al. 2001), perhaps to help meet their greater energy requirements due to high
activity and extensive movements (Pekins and Mautz 1990, O‟Donoghue et al. 1998a,
Laundré and Hernández 2003). In many cases, surplus hares are also cached, resulting in
a lower handling time than consumption (O‟Donoghue et al. 1998a). In the case of
coyotes and squirrels, the prey comprise a relatively small component of coyote diet
(O‟Donoghue et al. 1998a), thereby signifying that the functional response to squirrels
should not be constrained by handling and digestion. Again we are mindful that our
results are constrained by small sample sizes (see Appendix II), but given the surprising
degree to which the linear response best fit both coyotes and lynx predation, our results
underline the need for closer attention to the ecological context in which linear,
sigmoidal, and inverse functional responses are examined.
27
Presuming the lynx-squirrel response is governed, at least in part, by predator
density, a central question raised by our results concerns how lynx can exhibit preydependence in hares but ratio-dependence in squirrels. In theory, predator-dependence
can arise from: 1) group hunting; 2) social interactions; 3) aggressive interactions; 4)
anti-predator defences which are more strongly expressed as predator density increases;
or 5) a limited number of high quality sites where predators capture prey rapidly
(Abrams and Ginzburg 2000, Turchin 2003). Cases 1-3 describe scenarios involving
direct interference between predators, where conspecifics interactions reduce hunting
efficiency (Beddington 1975, DeAngelis et al. 1975). In contrast, cases 4 and 5 describe
scenarios where interference is indirect. Direct interference is typically modeled as
„waste time‟, where time lost on predator-predator interactions reduces active searching
time for prey. However, we dismiss that lynx are strongly influenced by group hunting
or aggressive interactions (cases 1-3), as „waste time‟ should be consistent across
functional responses for that predator. Among squirrels, alarm calls in the presence of
predators may be considered an inducible defense and thus predators may experience
reduced foraging efficiency on squirrels when predator abundance is high (case 4). In
addition, squirrel predation is largely restricted to areas where squirrels are vulnerable to
terrestrial predation (i.e., winter food caches „middens‟; case 5), implying that
interference in the lynx-squirrel response likely relates to indirect interference that is
imposed through low encounter or capture rates by lynx. We recognize, however, that
ratio-dependent functional responses remain an area of contention in ecology, with
critics cautioning that use of relatively simple models necessarily ignores the influence
of factors such as movement, landscape heterogeneity, and prey behavior, on predation
rates. Such real-world complexities may lead to indirect interference (i.e., pseudo28
interference) even in the absence of direct interactions between predators (See Free et al.
1977, Abrams and Walters 1996, Abrams 2014). Indeed, because lynx hunting success
on hares in our study area is not strongly influenced by habitat type (Murray et al.
1994), lynx predation pressure could be largely homogenous across the landscape,
making it unlikely that indirect interference strongly influences the lynx-hare
relationship.
While the coyote-hare response was determined to be linear, we were unable to
distinguish between prey-, ratio- and predator-dependent models. By inference, it seems
that predator interference may not play an overwhelmingly dominant role in the coyotehare relationship. Yet, the fact that the top 2 models were ratio- and predator-dependent,
respectively, gives credence to the perception that interference does contribute to coyote
functional responses. Here, we infer that direct interference may play a role in coyote
foraging given that ratio-dependence was observed in both hares and squirrels. Indirect
interference may also be implicated given the presence of hare refuges from coyote
predation in deep snow and cover (Murray and Boutin 1991, Murray et al. 1995), in
addition to the aforementioned arboreal and activity refuge of red squirrels.
We conclude that future work on predator functional responses will benefit from
a more holistic approach that considers predators in an ecosystem context rather than as
a solitary actor in a binary relationship with prey. Our results add to a growing body of
evidence supporting that predator kill rates can be influenced by a complex suite of
extrinsic factors that have not been the focus of close attention in traditional predatorprey research (Vucetich et al. 2002, Vos et al. 2004, Hossie and Murray 2010). In our
study, consideration of predator-dependent functional responses and alternate prey
allowed us to better understand the role of the two predator species in the boreal
29
ecosystem. Our results also highlight new areas that should be prioritized for further
investigation. For example, the extent to which predator- or ratio-dependent responses
truly reflect limitations imposed by intraspecific rather than interspecific competition
among predators continues to be a novel avenue for further investigation. In our study,
lynx and coyote numbers were strongly correlated (R2 = 0.92), demonstrating the
importance of determining whether interactions across predator guilds can also
influence predator dependency. Likewise, it is important to remember that while our
results strongly suggest that predator density is a key factor influencing predator kill
rates, limited sample sizes, co-varying predator and prey numbers, and the tendency for
models to oversimplify complex interactions, may obscure our ability to determine the
true drivers of predation. Consequently, the mechanisms underlying the observed levels
of interference in our system remain unclear and it follows that additional insight into
the effect of relative prey abundances, spatial refuges, and predator energy demands,
will help reveal mechanisms governing predator responses to prey limitation.
Ultimately, our study reinforces the notion that processes governing predator-prey
interactions and the role of predators on prey populations will benefit from more
extensive research into how and when multispecies interactions affect predator
functional responses.
30
Tables and Figures
Table 2.1. Performance of lynx (P) functional response models relative to primary prey
(snowshoe hare; N1) density. Squirrel density is denoted by N2.
Δ
Weight
Rank
Pseudo R2
C
6.93
0.01
7
0.00
Linear
a1N1
7.12
0.01
8
0.47
Hyperbolic (θ = 1)
a1N1/(1+a1h1N1)
0.00
0.47
1
0.67
Sigmoidal (θ = 2)
a1N12/(1+a1h1 N12)
0.30
0.40
2
0.69
Inverse (θ = 1)
c1/(1+ d2N2)
11.21
0.00
13.5
0.00
Inverse (θ = 2)
c1/(1+ d2N22)
11.21
0.00
13.5
0.00
Linear
a1(N1/P)
10.88
0.00
12
0.24
Hyperbolic (θ = 1)
a1N/(P+ a1h1N1)
6.67
0.02
5
0.37
Sigmoidal (θ = 2)
a1N2/(P2+ a1h1 N12)
6.71
0.02
6
0.37
Inverse (θ = 1)
c1/(1+ d2N2/P)
9.06
0.01
10
0.19
Inverse (θ = 2)
c1/((1+ d2 (N2/P)2)
8.94
0.01
9
0.20
Linear
a1N1P-m
10.24
0.00
11
0.48
Hyperbolic (θ = 1)
a1N1/(Pm+ a1h1 N1)
5.99
0.02
3
0.67
Sigmoidal (θ = 2)
a1N2/(P2m+ a1h1 N12)
6.30
0.02
4
0.69
Inverse (θ = 1)
c1/(1+ d2N2/Pm)
14.26
0.00
15
0.27
Inverse (θ = 2)
c1/((1+ d2(N2/ Pm)2)
14.47
0.00
16
0.24
Model
Constant
Form
Prey-Dependent
Ratio-Dependent
Predator-Dependent
31
Table 2.2. Performance of lynx (P) functional response models relative to alternate prey
(red squirrel; N2) density. Hare density is denoted by N1.
Δ
Weight
Rank
Pseudo R2
C
5.96
0.02
9
0.00
Linear
a2N2
5.21
0.02
8
0.18
Hyperbolic (θ = 1)
a2N2/(1+a2h2N2)
9.50
0.00
16
0.18
Sigmoidal (θ = 2)
a2N22/(1+a2h2 N22)
4.89
0.03
7
0.18
Inverse (θ = 1)
c2/(1+d1N1)
2.83
0.08
3
0.56
Inverse (θ = 2)
c2/(1+ d1N12)
0.11
0.31
2
0.66
Linear
a2 (N2/P)
0.00
0.33
1
0.50
Hyperbolic (θ = 1)
a2N/(P+ a2h2N2)
4.29
0.04
6
0.50
Sigmoidal (θ = 2)
a2N2/(P2+ a2h2 N22)
3.09
0.07
4
0.51
Inverse (θ = 1)
c2/(1+ d1N1/P)
8.37
0.00
12
0.18
Inverse (θ = 2)
c2/((1+ d1(N1/P)2)
7.98
0.01
11
0.21
Linear
a2N1P-m
3.13
0.07
5
0.51
Hyperbolic (θ = 1)
a2N1/(Pm+ a2h2 N2)
9.01
0.00
15
0.51
Sigmoidal (θ = 2)
a2N2/(P2m+a2h2N22)
8.92
0.00
14
0.52
Inverse (θ = 1)
c2/(1+ d1N1/Pm)
8.83
0.00
13
0.56
Inverse (θ = 2)
c2/((1+ d1(N1/ Pm)2)
6.11
0.02
10
0.66
Model
Constant
Form
Prey-Dependent
Ratio-Dependent
Predator-Dependent
32
Table 2.3. Performance of coyote (P) functional response models relative to primary prey
(snowshoe hare; N1) density. Squirrel density is denoted by N2.
Δ
Weight
Rank
Pseudo R2
C
6.71
0.01
7
0.00
Linear
a1N1
1.38
0.20
3
0.48
Hyperbolic (θ = 1)
a1N1/(1+a1h1N1)
5.31
0.03
5
0.44
Sigmoidal (θ = 2)
a1N12/(1+a1h1 N12)
6.82
0.01
8
0.36
Inverse (θ = 1)
c1/(1+ d2N2)
10.81
0.00
14
0.02
Inverse (θ = 2)
c1/(1+ d2N22)
10.75
0.00
13
0.02
Linear
a1(N1/P)
0.00
0.40
1
0.50
Hyperbolic (θ = 1)
a1N/(P+ a1h1N1)
5.67
0.02
6
0.48
Sigmoidal (θ = 2)
a1N2/(P2+ a1h1 N12)
4.91
0.03
4
0.46
Inverse (θ = 1)
c1/(1+ d2N2/P)
9.88
0.00
12
0.11
Inverse (θ = 2)
c1/((1+ d2 (N2/P)2)
9.16
0.00
11
0.17
Linear
a1N1P-m
0.93
0.25
2
0.64
Hyperbolic (θ = 1)
a1N1/(Pm+ a1h1 N1)
6.93
0.01
9
0.64
Sigmoidal (θ = 2)
a1N2/(P2m+ a1h1 N12)
9.05
0.04
10
0.64
Inverse (θ = 1)
c1/(1+ d2N2/Pm)
14.39
0.00
16
0.23
Inverse (θ = 2)
c1/((1+ d2(N2/ Pm)2)
14.36
0.00
15
0.23
Model
Constant
Form
Prey-Dependent
Ratio-Dependent
Predator-Dependent
33
Table 2.4. Performance of coyote (P) functional response models relative to alternate prey
(red squirrel; N2) density. Hare density is denoted by N1.
Δ
Weight
Rank
Pseudo R2
C
8.57
0.01
8
0.00
Linear
a2N2
7.34
0.01
7
0.20
Hyperbolic (θ = 1)
a2N2/(1+a2h2N2)
11.63
0.00
14
0.20
Sigmoidal (θ = 2)
a2N22/(1+a2h2 N22)
10.72
0.00
13
0.22
Inverse (θ = 1)
c2/(1+d1N1)
6.19
0.03
6
0.49
Inverse (θ = 2)
c2/(1+ d1N12)
6.07
0.03
5
0.49
Linear
a2 (N2/P)
0.00
0.58
1
0.58
Hyperbolic (θ = 1)
a2N/(P+ a2h2N2)
4.29
0.07
4
0.58
Sigmoidal (θ = 2)
a2N2/(P2+ a2h2 N22)
3.27
0.11
3
0.68
Inverse (θ = 1)
c2/(1+ d1N1/P)
10.09
0.00
12
0.24
Inverse (θ = 2)
c2/((1+ d1(N1/P)2)
9.86
0.00
11
0.26
Linear
a2N1P-m
3.02
0.13
2
0.66
Hyperbolic (θ = 1)
a2N1/(Pm+ a2h2 N2)
9.02
0.01
9
0.66
Sigmoidal (θ = 2)
a2N2/(P2m+a2h2N22)
9.14
0.01
10
0.66
Inverse (θ = 1)
c2/(1+ d1N1/Pm)
12.19
0.00
16
0.49
Inverse (θ = 2)
c2/((1+ d1(N1/ Pm)2)
12.07
0.00
15
0.49
Model
Constant
Form
Prey-Dependent
Ratio-Dependent
Predator-Dependent
34
35
Figure 2.1. Comparison of prey-dependent (left), ratio-dependent (middle) and inverse (right) responses of lynx to primary (top) and
alternate (bottom) prey. The best-fit models were hyperbolic (AIC c weight = 0.47, pseudo R2 = 0.67; solid) and sigmoidal preydependent (AICc weight = 0.40, pseudo R2 = 0.68; dashed) for primary prey and linear ratio-dependent response (pseudo R2 = 0.55,
AICc weight = 0.33) and prey-dependent inverse (θ = 2) for alternate prey (pseudo R2 = 0.66, AICc weight = 0.31). Best fit models are
shown for all inverse graphs with hare predation best represented by an inverse ratio-dependence (θ = 2, pseudo R2 = 0.66, AICc
weight = 0.01) in contrast to prey-dependent inverse predation in squirrels.
36
Figure 2.2. Comparison of prey-dependent (left) and ratio-dependent (right) functional responses of coyote to primary (top) and
alternate (bottom) prey. The best-fit models were linear ratio-dependent for both primary prey (pseudo R2 = 0.49, AICc weight = 0.28)
and alternate prey (pseudo R2 = 0.78, AICc weight = 0.38). Best fit models are shown for all inverse graphs with hare predation best
represented by an inverse ratio-dependent model (θ = 2, pseudo R2 = 0.66, AICc weight = 0.01) in contrast to squirrel predation being
inverse prey-dependent (θ = 2, pseudo R2 = 0.49, AICc weight = 0.03).
Chapter 3
The role of alternate prey in the maintenance of population cycles: A case study
using Canada lynx
Authorship: Chan, K., Boutin, S., Krebs, C.J., O‟Donoghue, M., Murray, D.L.
37
Abstract
Canada lynx (Lynx canadensis) populations, as well as those of their primary prey,
snowshoe hares (Lepus americanus), exhibit regular fluctuations across the boreal forest
of North America. The absence of pronounced lynx and hare cycles in their southern
range corresponds with broader lynx diet breadth, giving rise to the prediction that
alternate prey reduce predator reliance on primary prey and contribute to increased
numerical stability. We assessed the effect of alternate prey, specifically red squirrels
(Tamiasciurus hudsonicus), on the lynx-snowshoe hare cycle, by parameterizing a simple
2-prey, 1-predator model using predation rate and population density estimates from a
longstanding field study in southwest Yukon (1987-2001). In our modeled system, we
show that although inclusion of squirrels in the lynx diet increases system stability. When
squirrel numbers are considered across a realistic range of population densities, their
stabilizing role is minor on lynx dynamics and does not lead to full cyclic attenuation.
Further, density-independent mortality rates of the predator and capture efficiency of the
prey were strongly stabilizing to both lynx and hare populations, though it is likely that
spatial variability in multiple factors work to increase numerical stability in the southern
lynx range. These results are consistent when stability analyses are conducted in a
multivariate context. Our results highlight that multiple factors may contribute to
increased numerical stability in natural predator-prey systems, while emphasizing that
further insight into species interactions will provide a more robust framework for
explaining patterns of spatio-temporal variability in predator population cycles.
38
Introduction
Animal population cycles involve regular, periodic fluctuations in abundance
that are driven by time-delayed density-dependent processes such as food shortage or
predation (Krebs et al. 2001b, Turchin 2003). Population cycles occur across a variety of
landscapes in Europe and North America and are widely documented among insects,
birds and mammals that often serve as keystone species in temperate ecosystems (Kendall
et al. 1998, 1999, Turchin 2003). Current interest regarding population cycles extends
past the mere detection of cycles and factors by which they are driven, to explaining
patterns of spatio-temporal variability in cyclic dynamics. Indeed, a variety of species
exhibit variability in cyclic magnitude and intensity, usually spanning latitudinal or even
longitudinal gradients (Erb et al. 2000, Williams et al. 2004, Row et al. 2014). There is
also recognition that some cyclic populations exhibit novel temporal variability in their
dynamics, including recent cyclic attenuation and population stability (Ims et al. 2008,
Cornulier et al. 2013). While the underlying causes of this variability are not well
understood, ultimately, changes in species and community interactions are likely culprits
(Klemola et al. 2002, Saitoh et al. 2006, Guillaumet et al. 2015).
Victim-exploiter relationships (i.e., predation, herbivory, parasitism) are
regarded as the driving interactions at the root of delayed density dependence and
population cycles (Turchin 2003). This implies that fundamental changes in the nature of
such interactions, such as the intensity and magnitude of the effects of parasites or
predators, should contribute strongly to spatio-temporal variability in cyclic propensity
(Turchin 2003, Ginzburg and Arditi 2012). For example, a dominant hypothesis offered
to explain cyclic persistence and attenuation, known as the generalist predator hypothesis
(Hanski et al. 1991, Turchin and Hanski 1997), suggests that increased numbers of
39
generalist predators at lower latitudes increases competition and reduces the importance
of specialist predator-prey relationships and thereby contributes to cyclic attenuation
(Klemola et al. 2002, Ims et al. 2008). By extension, it follows that increased prey
diversity along a spatial gradient could also allow predators to exploit a larger number of
alternate prey and thereby decouple the close relationship between a specialist predator
and its prey (Johnson et al. 2000, Shier and Boyce 2009). Additionally, declines in the
relative abundance of primary prey species may force predators to find adequate biomass
through stronger reliance on alternate prey (Kjellander and Nordström 2003, Leeuwen
and Jansen 2013). Ultimately, these processes support an extension of the generalist
predator hypothesis, namely the alternate prey hypothesis, that invokes spatial variability
in cyclic propensity of populations depending on the availability of, and reliance on,
alternate prey by the predator (Roth et al. 2007, Shier and Boyce 2009). It follows that the
recent attenuation of cyclic phenomena in several northern species may be related to the
alternate prey hypothesis and how local changes in diversity and consumptive patterns of
resident predators affects the availability of primary and alternate prey (Johnson et al.
2000, Ims et al. 2008).
Birds and mammals of the boreal forest of North America provide important
examples of spatio-temporal variability in cyclic population dynamics. Among the bestknown cases are Canada lynx (Lynx canadensis) and their primary prey, snowshoe hare
(Lepus americanus), where lynx and hare populations cycle dramatically in the core of
their range but exhibit dampened oscillations at lower latitudes (Roth et al. 2007, Murray
et al. 2008). While lynx reliance on snowshoe hares is well documented (Krebs et al.
2001b, Turchin 2003), research shows that lynx diet encompasses a variety of prey
species including red squirrel (Tamiasciurus hudsonicus), arctic ground squirrel
40
(Spermophilus parryii), grouse (Bonasa sp.; Lagopus sp.), and small rodents
(Clethrionomys sp.; Microtus sp.; Peromyscus maniculatus) that are used increasingly
during periods of low hare abundance (O‟Donoghue et al. 1998a). Similarly, in the
southern part of their range, lynx rely less on snowshoe hares as prey and their lower
consumption of hares correlates with attenuated numerical cyclicity (Roth et al. 2007). It
follows that if alternate prey are important in sustaining individual lynx during periods of
low hare abundance in northern latitudes, then the increased prevalence of alternate prey
in the lynx diet should stabilize their dynamics.
Ecological systems are governed by complex factors, and when investigating
how variability in victim-exploiter relationships can affect population dynamics, it is
important to consider that factors like prey abundance or prey capture rate may not only
have direct but also synergistic effects on populations (Ives and Murray 1997, Vos et al.
2004). For example, if primary prey are seasonally abundant and predators rely on
alternate prey to survive through periods of primary prey scarcity, then predator
numerical stability will be governed not only by direct effects of prey abundance and
capture probability, but also by relative changes in abundance of different prey species.
Likewise, how the predator responds to relative changes in prey, either through variable
feeding efficiency or altered recruitment or survival, can have profound effects on
population dynamics. Accordingly, population stability properties may be a consequence
of interactions between multiple ecological or environmental factors.
Here we examine how alternate prey affect population cycles of predators and
their primary prey, either directly or through interactions with other factors. As model
system, we use populations of Canada lynx, snowshoe hares, and red squirrels. First, we
develop a general predator-prey model that incorporates both primary and alternate prey
41
species using a multi-species functional response. We then parameterize a RosenzweigMacArthur population dynamics model using lynx, snowshoe hare, and red squirrel data
from the long-term study of the boreal ecosystem at Kluane Lake, Yukon (Krebs et al.
2001b, Krebs 2016). In the study area, populations of the three species have been
monitored for > 25 years, and the 10-year cyclic patterns in lynx and hares are especially
well-documented (Fig. 3.1a). On the other hand, red squirrel numbers do not follow
regular oscillations in the boreal forest but rather are closely associated with spruce
masting events (Krebs et al. 2001b)(Fig. 3.1a). While cone masts have been linked to
several exogenous factors (Krebs et al. 2012), the limited predictive ability of these
models suggest that masting events are generally stochastic. However, unlike hares, red
squirrels provide a small but relatively consistent source of energy for predators in the
Kluane lake study area (Fig 3.1b), implying that they may have a previously unexplored
influence on lynx-hare population dynamics. Therefore, we predict that changes in the
availability of alternate prey (red squirrels) for lynx will contribute to changes in stability
in lynx numbers, either directly via additional biomass, or through interactions with other
primary population parameters such as lynx mortality or recruitment. In other words, we
expect that relative importance of alternate prey in the predator diet influences the
population dynamics of predators and their primary prey.
Methods
Data collection
Lynx were snowtracked during the winters of 1987-1988 to 1996-1997, and
hunting behavior, activity, habitat use, and prey kills were recorded (O‟Donoghue et al.
1998a). Tracking sessions were conducted from late October through late March, and we
42
estimated kill rates for hares and squirrels by recording kills along tracks and calculating
distances traveled by snowtracked lynx from measured movement rates and activity rates
of radio-collared. Prey kill rates were adjusted for group size to obtain per capita kill rates
(O‟Donoghue et al. 1998, Ch.2). The lynx numerical responses to changes in prey
abundance were parameterized using population density estimates from the Kluane
project (see http://www.zoology.ubc.ca/~krebs/kluane.html). These data were obtained as
part of a larger project assessing the dynamics of the boreal forest ecosystem (Krebs et al.
2001b), with snowshoe hare and red squirrel densities being estimated by averaging the
winter and spring population estimates from livetrapping sessions using Efford‟s
maximum likelihood estimator (Efford et al. 2009, Krebs et al. 2011). Lynx populations
were estimated during the winters of 1987-1988 to 2000-2001 (10 years) using radiotelemetry of radio-collared animals to estimate home range sizes, supplemented with
snow transect data to interpolate the number of uncollared animals in the study area
(O‟Donoghue et al. 1997, Krebs et al. 2001b). Radio collaring of lynx and coyotes ended
in the winter of 1996-1997, though estimation using visual encounters and snowtracking
continued until winter 2000-2001 (14 years total; M. O‟Donoghue, pers. com.). While
density estimates of hares and squirrels in our study area comprise a longer population
time series (hares: 39 years; squirrels: 27 years) than that available for lynx, concurrent
estimates of both prey species and lynx are required for population parameter estimation,
thereby limiting the amount of usable data to years with lynx density estimates (n=14).
Monitoring of lynx populations continued in Kluane after 2001, however, later estimates
could not be used because after 2001 lynx numbers were indexed using a different
methodology and direct population estimates were not available. Density estimates for all
species were expressed as number of individuals per 100 km2.
43
Lynx functional response
O‟Donoghue et al. (1998a) found that the lynx-hare functional response to be
hyperbolic prey-dependent. There is rarely sufficient data to distinguish between forms of
the functional response (Marshal and Boutin 1999, Vucetich et al. 2002), and more recent
work found equivocal support for the sigmoidal prey-dependent response for the
sigmoidal prey-dependent response (Ch. 2). Here, the relationship of the killing rate is
related to prey density as a function of searching efficiency (a) and handling time (h),
where h accounts for the time required to capture, process, and digest prey items. It
follows that in the hyperbolic functional response as handling time increases, there are
limits to the total amount of time left for searching for prey. To incorporate alternate prey
in the lynx functional response, we used a multispecies expansion of the Holling
functional response (Murdoch 1973) that explicitly incorporates non-independence
between searching time for each prey type:
(
(
Where
∑
(
(3.1)
is the kill rate of the predator on the focal prey species (i; species
whose killing rate is being defined by the functional response), P is the density of the
predator, and N is the density of the prey (i, j). Parameter 𝜃 was incorporated in more
recent iterations of the multispecies model as a binary shape parameter to represent both
hyperbolic (𝜃
and sigmoidal (𝜃
2 functional responses; this flexibility is akin to
that offered by the related functional response model based on the Michaelis-Menten
equation for enzyme kinetics (Turchin 2003). Notably, Murdoch‟s work was limited to
scenarios where functional responses were driven by prey density and did not consider
44
intraspecific interference, which can lead to killing rates that are more driven by predator
numbers or predator-to-prey ratio (Murdoch 1973, Abrams and Ginzburg 2000).
More recent work has also highlighted the importance of intraspecific
interference (ratio- and predator-dependence) in reducing predator capture efficiency.
Predator-dependence, of which ratio-dependence is a special form, is a process whereby
conspecific predators can negatively impact predator foraging through processes such as
exploitative competition, weakening the direct link between prey density and the
predator‟s per capita kill rate. This interference can manifest either through direct (e.g.,
social interactions and territoriality; Beddington 1975, DeAngelis et al. 1975) or indirect
(e.g., spatial heterogeneity in predation pressure and inducible antipredator defenses;
Beddington and Lawton 1977, Abrams and Walters 1996) factors. Therefore, we
incorporated intraspecific interference as the coefficient parameter m (eq. 3.1) (Hassell
and Varley 1969, Ch.2). A functional response having m = 0 indicates no interference
(prey dependence), whereas positive values indicate an increasing degree of intraspecific
competition. These generalities allow us to explore a broader range of potential nonlinear
functional responses that can more realistically reflect the complexity of real-world
predator-prey interactions inherent in the lynx-hare-squirrel system.
Our previous work (Ch. 2) also highlighted that the multi-species functional
response can allow for killing rate of the focal prey species (i) to be more strongly
influenced by the density of another prey species (j) and consequently, the killing rate of
species i declines inversely as species j density increases. This relationship can arise when
there is a high differential between prey preferences or when the handling time of species
j is disproportionately high in comparison to time spent hunting the focal species. It
45
follows that the killing rate of alternate prey are more likely to be affected by primary
prey density than the reverse.
Our recent efforts show that the lynx-red squirrel functional response is best
represented by either a linear ratio-dependent or an inverse (θ = 2) prey-dependent
response (Ch. 2). In the previous study, we simplified the full multispecies model into its
nested single species components to more closely document individual lynx-prey
interactions (Ch. 2). The present study compares the performance of the multispecies
model (eq. 3.1) alongside the nested single species models, specifically to validate
whether the multispecies model can outperform the single species models. However,
fitting all possible combinations of the multispecies model (81 models) would be
inappropriate as many models are not biologically relevant to our study species. Thus, to
conserve as many biologically relevant models as possible, we chose to restrict testing of
the multispecies models a priori to combinations of models with a ΔAICc < 4 in the
single species analysis. These models were the hyperbolic and sigmoidal response preydependent responses for hares and the linear ratio-dependent, sigmoidal ratio-dependent,
and inverse (θ = 1 and θ = 2) responses for squirrels (see Appendix IV for detailed
methods and results). The most parsimonious model for the combined single-species and
multispecies model selection was identified from other candidate models using the lowest
ΔAICc, with models Δ<2 being considered as statistically indistinguishable (Burnham
and Anderson 2002). The fit of hare response was not improved through the use of the
multispecies functional response. In contrast, we found that the lynx functional response
on red squirrels was best represented by the multispecies equation below:
(
46
(3.2)
Where N1 represents hare density, N2 represents squirrel density and d1
represents the product of the attack rate and handling time of hares and can collectively
be referred to as “hare” time. Although it is understood that alternate prey other than red
squirrels are consumed by lynx in our study area, these comprise a relatively small
proportion (< 25% of biomass during cyclic low) of the number of prey killed and
biomass consumed (Krebs et al. 2001b).
Numerical responses
The cyclic relationship between predator and prey can be represented by the
Lotka-Volterra class of equations, which can be further extended to include non-linear
functional responses using the Rosenzweig-MacArthur (RM) model (Turchin 2003). The
RM model was initially developed as a 2-species predator-prey system, but the model can
be expanded to the multi-prey context by incorporating a multispecies functional response
to account for prey preference based on relative abundance of different prey types
(Rosenzweig and MacArthur 1963, Post et al. 2000). Given that the lynx-snowshoe hare
functional response is hyperbolic and prey-dependent and the lynx-squirrel response is
multispecies response reliant on hare density, the population dynamics of our 3-species
system are described as:
(
)
(
)
𝛿
(3.3a)
𝛿
(
(3.3b)
𝛿
47
)
(3.3c)
Where N1 represents density of the primary prey (hares), N2 represents density
of the alternate prey (red squirrels), and P represents density of the predator (lynx), all
expressed in absolute numbers per unit area. θ is a shape parameter defining the hare
functional response as either hyperbolic (θ = 1) or sigmoidal (θ = 2). Parameter ri is the
intrinsic growth rate of species i, ki is the predator-independent carrying capacity per unit
area, and 𝛿 is the mortality rate or species i in the absence of predator P (i.e. predator
independent mortality). For the predator equation, χi is the conversion rate of prey species
i into new predators, 𝛿 is the density-dependent mortality of predators, and H represents
density-independent (e.g. harvest) mortality. Traditionally, the RM model does not
include a parameter for density-dependent predator mortality, however, given that prey
growth is self-limited using a logistic growth curve, it follows that predator growth should
be limited not just by biomass conversion (Messier 1994, Serrouya 2013). It is likely that
for lynx, prey biomass is the most limiting factor for population growth), although to an
uncertain degree density-dependent mortality also may be implicated (Steury and Murray
2004). Thus, we included both density-independent and density-dependent sources of
death as parameters to be tested in our model.
Examining the alternate prey equation (3.3b), we see that the linear ratiodependent response produces a mortality rate that is equal to aN2/(1+d1N12), thereby
resulting in a unique form of ratio dependence known as „donor control‟ (Poggiale et al.
1998, Abrams 2015). Donor control allows bottom-up control where the density of the
prey determines the density of its predators but the predators do not offer a reciprocal
control. As a consequence, predator population dynamics are proportional to the prey
population density, but prey population dynamics do not explicitly depend on predators
48
(Ginzburg and Arditi 2012). In addition, we note that red squirrel densities do not
fluctuate according to predator-prey interactions (as they do for hares) but primarily due
to masting of coniferous trees (Klenner and Krebs 1991, Krebs et al. 2001b, Fletcher et
al. 2013). Here, we use a stochastic function, , to adjust squirrel abundance to reflect the
reproductive potential associated with a stochastic mast crop (equation 3.2b).
Model selection and parameter estimation
We tested three competing numerical response models to determine the
importance of the extra density-dependent mortality parameter: a model with only
density-independent predator mortality (H; traditional RM model), a model with only
density-dependent predator mortality (𝛿 ), and a model with both density-dependent and
density-independent mortality (𝛿
; eq. 3.2). These were also tested with both a
hyperbolic and sigmoidal hare response, resulting in a total of 6 competing models (2
functional responses x 3 mortality models). The most parsimonious model was identified
from other candidate models using the lowest ΔAICc, with models Δ < 2 being
considered as indistinguishable (Burnham and Anderson 2002).
The models were fit using a Levenberg-Marquardt method for finding the
minimum sum of squares of a non-linear function combined with simulated annealing.
The Levenberg-Marquardt algorithm was carried out using the nls.lm function in R
(Elzhov et al. 2013, R Development Core Team 2016). Simulated annealing with an
initial starting temperature of 100 was then used to reduce the time to find optimal
solutions. A literature review was used to find the biologically relevant starting points for
the nls.lm function, as well as maximum values to serve as upper bounds for each
parameter (Tyson et al. 2009). The lower bounds for all parameters were set at zero.
49
Because red squirrel population dynamics are more directly affected by seasonal masting
than by predation, we treated the observed red squirrel densities as a parameter with
known values that changed at each time step rather than attempting to fit a deterministic
equation to a stochastic variable. In essence, this approach mimics a perfect fit for red
squirrel population densities, while reducing computational difficulty. However, because
we used this approach, none of the parameters specific to equation 3.2b (e.g. r2, k2, δ2)
can be estimated and as a result are they not included in the stability analysis described
below. This is justified by the fact that these equations and parameters do not account for
variability in reproductive output and mortality in red squirrels specifically as a result of
tree mast (Fletcher et al. 2013). Accordingly, estimates produced from fitting red squirrel
density observations with equation 3.2b should have limited biological relevance.
Stability analysis
Model stability was assessed by evaluating eigenvalues of the jacobian matrix
of the model at equilibrium. While a 3-species model (eq. 3.2) is represented by a 3 x 3
jacobian matrix, because squirrel densities are stochastic and relatively constant (Krebs et
al. 2001b), squirrel density was set at the mean observed value for the study area
(~23,000 red squirrels per 100 km2, S. Boutin, unpubl.). This allowed us to reduce the
model by one equation to a 2 x 2 matrix without losing the effect of squirrel density on
the system. Every n x n matrix produces n eigenvalues and thus our 2 x 2 system has 2
eigenvalues. The eigenvalues of the matrix are constructed with both a real and imaginary
part; a negative real part in all eigenvalues indicates that the system is a sink and stable,
whereas a positive real part any eigenvalue indicates an unstable system (i.e., a source;
see May 1973a, Fryxell et al. 2007). Here we define stability as when the equilibrium of
50
the system is non-oscillatory (no positive real-parts). The imaginary part also indicates
whether the system is oscillatory (i.e. limit cycles), however, we did not consider the
imaginary part as we did not distinguish between oscillatory and chaotic instability, and in
instances when both real eigenvalues where negative, the imaginary part was equal to 0
(i.e. no oscillation). The analysis was performed by changing a single parameter while
holding other parameters constant and examining variability in the real part of the
eigenvalues. Parameter estimates from the best fit model were used as a baseline and
changes in stability were quantified as proportional changes to the baseline estimate (i.e.
1 = baseline estimate). Stability of the community matrix to variability in alternate prey
(squirrel) density was examined by treating squirrel density as a variable parameter in a
separate sensitivity analysis. By excluding factors considered as constants across the
range (i.e. conversion rates, handling times), we focused on factors with the greatest
effect on system stability across a range of squirrel densities.
Results
Model selection and parameter estimation
The best-fit predator model was the hyperbolic hare response with densityindependent mortality of the predator (RSS = 46696016, AICc weight = 0.76; Table 3.1);
parameter estimates for this model are presented in Table 3.2. The second best model was
the hyperbolic hare response with combined density-independent and density-dependent
predator mortality, but this model had ΔAICc > 2 (ΔAICc = 2.7, AICc weight = 0.22).
Because this model was so close to the cutoff, we included parameters and additional
stability analysis in Appendix V. While the explained variance is greater (RSS =
44651558), these results should be taken with caution as the confidence intervals are
51
much larger than those of the best fit model ( see Appendix V). The remaining models
were all had a ΔAICc > 9 with the density-dependent mortality models providing
virtually no explanatory power.
We note that for hares, our modeled population abundance had a close fit to
estimated hare abundance in the study area (pseudo R2 = 0.94, Fig. 3.2a). Broadly
speaking, our simulations captured the period, amplitude, and phase observed in the hare
population survey data. However, although the model roughly captured the periodicity of
the lynx population cycle, we note poor fit with the observed lynx population time series
in the Kluane study area (pseudo R2 = 0.07, Fig. 3.2b). Below, we discuss the likely
reasons for this disparity. Nevertheless, we conclude that our model qualitatively captured
the properties found in the lynx-hare cycle, allowing us to assess the role of alternate prey
(squirrels) on system stability.
We ran simulations on the best fit model using parameter estimates using three
different densities of red squirrels; i) no squirrels (0 squirrels∙100km2); ii) average
observed density in the study area (23,000 squirrels∙100km2) and; iii) highest squirrel
density observed in the study area (39,500 squirrels∙100km2) (Fig. 3.3). With increasing
squirrel density, there is a qualitative change in cyclic properties for both lynx and hare
populations, as evidenced by a reduced peak density of the hare cycle, and an increased
low density of the lynx cycle (Fig. 3.3). Thus, the presence and abundance of alternate
prey in our system led to reduced cyclic amplitude in both the predator and primary prey.
Likewise, as squirrel densities increased in the model system, there was a notable
reduction in the dominant Lomb-Scargle cycle period from 9.2 years (no squirrels) to 7.9
years (39,500 squirrels).
52
Stability analysis
Our stability analysis revealed that a number of parameters influenced
cyclic propensity in our modeled system (Fig. 4).The real parts of the 2 eigenvalues
produced showed that higher hare growth rates (r1), higher hare carrying capacity (k1),
higher attack rates on hares (a), and higher hare-to-lynx conversion rates (χ1) resulted in
qualitative reduction in system stability. Higher hare handling times (h), higher lynx
attack rates on red squirrels (a2), higher hare mortality rates (𝛿 ), higher red squirrel to
lynx conversion rates (χ2), and higher density-independent lynx mortality (H) lead to
increased system stability. Surprisingly, “hare” time (d1) in the inverse function had no
effect on the stability of the system. Similarly, the effect of density-dependent mortality
appears to have little to no effect on system stability (see Appendix V). The real part of
the eigenvalues for hare handling time (h1) and lynx mortality (H) also decreased below
zero when parameter values were low (Fig. 4), indicating a return to a stable system.
Notably, this increased stability was associated with a stable equilibrium that occurred
when the system reached extinction. Overall, most parameters indicated a stable
equilibrium within 0.3-1.4 times the estimated parameter at baseline squirrel densities
(23,000 squirrels per 100 km2), with the exception of squirrel attack rate (a2), squirrel
conversion rate (χ2), and hare mortality rate (𝛿 ), which require values > 8 times the
baseline estimate to produce a stable equilibrium. However, only hare mortality (𝛿 )
generated a stable equilibrium within parameter confidence limits.
To summarize our results, an increase in lynx capture or conversion efficiency
from red squirrels or an increase in lynx mortality can stabilize cyclic properties of our
model system. In contrast, higher lynx conversion efficiency for hares was destabilizing
53
and increased cyclic propensity (Fig 4). This pattern was further evidenced by treating red
squirrel density as a parameter in the stability analysis. When red squirrel density was
increased, the model system became increasingly stable (Fig. 5). Thus, our stability
analysis confirmed that inclusion of alternate prey in our modeled system led to reduced
cyclic propensity for both predators and their primary prey. However, notwithstanding
these qualitative findings, we note that the effect of squirrel density on cyclic properties
was negligible, as densities of at least 159,000 squirrels per 100 km2 (~ 7 times the
average observed density in our study area) would be required for the system to reach a
stable equilibrium (eigenvalues < 0). By comparison, squirrel attack rates (a2) would also
need to be >7 times the estimated value to fully attenuate the lynx-hare cycle. These
results highlight that, overall, lynx and hare cyclic properties are relatively insensitive to
changes in single parameters like squirrel density.
For the multivariate stability analysis, we assessed the influence of squirrel
density in tandem with each of the most stabilizing model parameters (intrinsic rate of
increase of hares (r1); hare carrying capacity (k1); hare attack rate (a1); lynx mortality
(H)), on model properties. Intrinsic rates of increase in hares and carrying capacity had
the strongest interaction with squirrel density and elicited the largest differences between
equilibrium points (Fig. 3.6). Interestingly, lynx mortality and hare attack rate responded
by forming 2 stable regions at the extremes when squirrel densities were high. Overall,
although select population parameters revealed interactions with squirrel density to
further increase system stability, these effects were limited as squirrel densities had to be
5 times greater than average to see any appreciable effect.
54
Discussion
Our 3-species predator-prey model revealed that higher red squirrel consumption
increased stability in the lynx-hare system, offering support, in principle, for the alternate
prey hypothesis. Accordingly, we infer that as a general concept, increasing dietary
breadth of predators may contribute to lower cyclic propensity among predators and their
primary prey (Hanski et al. 1991, Roth et al. 2007). However, although alternate prey
consumption increased relative stability in our system, we found that absolute changes in
squirrel density alone were insufficient to fully stabilize lynx-hare population dynamics.
Even when considered in tandem with variation among other demographic features of the
lynx-hare population cycle, the influence of alternate prey on system stability was
relatively modest. This implies that the alternate prey hypothesis may only partially
explain spatial variability in cyclic propensity of lynx and hare populations, and that other
factors, including those that may interact with alternate prey abundance and predator
capture rates, contribute to regional variation in predator-prey population dynamics.
Our stability analysis considered 10 population parameters as potentially affecting
the lynx-hare cycle; 6 of these (intrinsic rate of increase of hares (r1); hare carrying
capacity (k1); hare attack rate (a1); hare handling time (h1); hare conversion rate (χ1); lynx
mortality (H)) had a strong influence on cyclic propensity. Of these, only lynx mortality,
hare carrying capacity, and hare attack rate are likely to exhibit strong variability across
the geographical range of lynx and hares. For example, hare densities appear to be
considerably lower in the southern range (Aubry et al. 2000, Hodges 2000) indicating a
lower carrying capacity or population growth rate, though evidence for large geographic
variability in productivity in hares output remains equivocal (Murray 2000). Likewise,
hunting efficiency of lynx on hares is largely linked to snow and environmental
55
conditions, indicating a probable geographic gradient in hunting ability (Stenseth and
Shabbar 2004, Peers et al. 2014). Lynx mortality is substantially higher in the southern
boreal forest, with annual survival rates being comparable to those in the northern boreal
during low hare densities (Aubry et al. 2000, Steury and Murray 2004). Therefore, it is
likely that regional variability in lynx population dynamics is influenced to some degree
by local conditions, and that the role of alternate prey on system stability is regulated by
other phenomena that vary through space and time.
Squirrel densities across the distribution of lynx and hares are not well known, but
they are more likely to vary due to locally available resources than over a broad
geographic scale (Klenner and Krebs 1991, Bayne and Hobson 2000). We are not aware
of reports of red squirrels ever reaching the population density required to generate a
stable equilibrium in our model system (~ 16 per ha), meaning that the stabilizing effect
of squirrels on lynx and hares should be considered only as qualitative support for the
alternate prey hypothesis. Furthermore, when squirrel density is considered concurrently
with changes in hare parameters, there is little influence on system stability and only hare
growth rate and hare carrying capacity contribute to stability in the direction consistent
with observed hare demographic parameters in the southern boreal forest (Fig. 3.6)(
Murray 2000). It seems more probable that geographic variation in lynx and hare
demographics (e.g., regional hare carrying capacity or density-independent mortality for
lynx) contribute more forcefully to stabilizing lynx-hare population dynamics. However,
it is important to emphasize that parameter and density estimates used in our model are
those required to achieve numerical stability rather than actually mimic latitudinal
reduction in cyclic amplitude (e.g., see Ives and Murray 1997), and therefore our failure
56
to „stop the cycle‟ by incorporating alternate prey in lynx-hare dynamics should not be a
focal point of interest (see Turchin 2003, Ch. 13).
It is understood that in general the diversity of alternate prey types increases at
lower latitudes (Ims et al. 2008), and our use of a single alternate prey type represents an
over-simplification of a fundamentally complex predator-prey system. In our Yukon
study area (O‟Donoghue et al. 1998a, 1998b) and elsewhere in the lynx range (Aubry et
al. 2000, Buskirk et al. 2000) alternate prey are strongly consumed by lynx only when
hares are scarce and alternate prey other than red squirrels (e.g., grouse (Bonasa spp.) and
small mammals (Microtus spp.; Clethryonomys spp; Peromyscus spp.)) are of lower
preference and used in smaller amounts than squirrels. Therefore, given the minor effect
of squirrel density on stability in our modeled system, it is unlikely that other alternate
prey would contribute strongly to system dynamics, individually or combined.
Furthermore, our model suggests that alternate prey do not stabilize cyclic population
dynamics via the provision of additional biomass to predators but rather through
increased intraspecific competition between predators. This is highlighted by the lack of
predator interference in the primary prey response where enrichment is destabilizing (e.g.
higher carrying capacity; Fig. 3.4), in contrast to the stabilizing effect of increasing
alternate prey density (Fig. 3.5).This interference can arise either from direct predator
interactions such as group hunting and increased agonistic interactions associated with
these, or indirectly via behaviors resulting in higher aggregation of individual predators
specifically within areas where high quality hunting sites are limited (see Hassell and
May 1974, Abrams and Walters 1996). For example, Arditi and Saiah (1992) were able to
induce or mitigate presence of interference within functional responses of Daphnia and
Simocephalus via modifications of predator distribution. The resulting spatial aggregation
57
of predators can lead to a density-dependent reduction in predator search efficiency
(Fryxell et al. 2007) and accordingly, we can infer that interference is more likely to be
found in predators where: 1) predator movements cause higher than random encounter
rates with conspecifics (Gurarie and Ovaskainen 2013); 2) non-random movement and
searching leads to higher prey encounter rates (Free et al. 1977); and 3) predators
disproportionately select prey habitat (Vos et al. 2004).
Previous work illustrated the importance of intraspecific competition for stability
in food webs, as system stability cannot arise unless no populations in the community
experience destabilizing positive feedback in intraspecific interactions (i.e. all m ≤ 0), and
at least one population in the community exhibits self-stabilization (i.e. m ≠ 0) (May
1973a, Yodzis 1981). That being said, because our model was an initial attempt to test the
alternate prey hypothesis on cyclic propensity of predators and prey, we acknowledge that
the parameter representing intraspecific competition associated with both hare and
squirrel predation is over-simplified by using purely prey-dependent (m = 0) and ratiodependent (m = 1) responses (Abrams 2015). Perfect prey- or ratio-dependence is rare and
we infer that in natural predator-prey systems there may be a higher degree of stability
when interference is present in predator-prey relationships with strong interaction
strengths (i.e., primary prey).
The fact that intraspecific competition appears to strongly influence stability
suggests a more likely explanation can be found in the generalist predator hypothesis,
where competition between generalist and specialist predators could be seen to reduce the
importance of specialist predation. Similar latitudinal gradients have been observed in
Fennoscandian microtines (Microtus spp., Clethrionomys spp., Lemmus lemmus;
Bjornstad et al. 1995, Hanski and Korpimäki 1995) and previous experimental work
58
(Korpimäki and Norrdahl 1998) has shown that removal of specialist predators can reduce
declines seen in fenoscandian voles (Microtus spp. and Clethrionomys glareolus). Further
theoretical work resulting from this experiment (Turchin and Hanski 2001, Korpimäki et
al. 2002) also revealed the strong driving effect of specialist predation in driving the
decline phase of voles. Other work has also indicated that clines in generalist predation
can offer a common explanation in the geographic decline of both small mammal and
insect population cycles in Fennoscandia (Klemola et al. 2002). However, predation of
field voles in Northern England has a lower degree of support for the generalist predator
hypothesis (Lambin et al. 2000) suggesting the presence of population cycles may not
have a single overarching explanation.
While it is especially difficult to test the alternate prey hypothesis in a field
context involving large mammals, and our compromise was to use empirical field data to
estimate parameters supporting a population model and stability analysis, experimental
studies have tested the ability of alternate prey to affect stability in more controlled
environments. For example, Holyoak and Sachdev (1998)were able to show that
generalist protists that switched to omnivorous or cannibalistic diets, had longer
population persistence and more stationary population dynamics. In another study
(Flaherty 1969) the presence of alternate prey created a more homogenous distribution of
predatory mites and population stability increased according to habitat complexity. On the
other hand, Luckinbill (1979) found that alternate prey destabilized populations of
Didinium and Paramecium, which is in direct contrast with the above results from
Flaherty (1969). Similarly, in a field setting Tilman (1996) showed that stability of
grassland community biomass was shaped by interspecific plant competition, although
59
species richness actually had little effect on system stability. In sum, empirical support for
the alternate prey hypothesis, even in controlled systems, remains equivocal.
Overall, our model was able to adequately capture snowshoe hare cyclic
dynamics, however, there were notable inconsistencies as predicted lynx densities that
were 3 times greater those that have been observed in our area (O‟Donoghue et al. 1997).
In addition, there was a longer time lag (approx. 2 years) between population changes in
lynx numbers following those of hares (Boutin et al. 1995). We explain these deviations
primarily through: 1) field estimation of lynx and hare densities at different spatial scales,
and 2) omission of seasonality in demographic rate estimation. Specifically, predator
densities were estimated as lynx per100 km2 scale whereas hare densities were scaled to
100 km2 using estimates derived from 60 ha mark-recapture study areas (Krebs et al.
2011). We acknowledge that predator estimates are subject a larger uncertainty
(O‟Donoghue et al. 1997) and that hare estimates may be located in above-average hare
habitat in our region (D. Murray, pers. obs.) and subsequently, estimation of predator and
prey densities at incompatible spatial scales would result in a greater inconsistency
between estimated and predicted values.
Additionally, our functional responses were estimated using winter predation
rates determined from snowtracking (O‟Donoghue et al. 1998a) and the inability of our
model to realistically capture the numerical properties of the lynx cycle was likely
influenced by the model‟s omission of seasonal changes in demographic rates of predator
and prey. Other work has sought to incorporate seasonal dynamics in predator prey
models (Hanski and Korpimäki 1995, Klausmeier and Litchman 2012) and there is
substantial evidence indicating the increased use of alternate prey by lynx and, more
importantly, a decrease in hare predation and mortality during summer months (Mowat et
60
al. 2000, Feierabend and Kielland 2015). Because our model predicts a substantial
increase in stability with decreased primary prey use, predators with seasonal variation in
diet to select for lower preference prey would experience a seasonal increase in stability.
However, as seasonal kill rates and prey switching of predators in general remain poorly
understood we were unable to incorporate these factors (Sand et al. 2008, Metz et al.
2011).
In conclusion, our results show that alternate prey can stabilize predator-prey
population dynamics, although these effects had relatively minor influence on stability
and cyclic propensity in our particular system. While our results indicate that reliance on
squirrels alone may not be sufficient to provide numerical stability to the lynx-hare
system, our goal was to illustrate that increased reliance upon alternate prey was
functionally stabilizing. To that end, our efforts offer an important first step in developing
a robust understanding of predator-prey population determinants by showing that
inclusion of alternate prey in the predator diet may contribute to stable population
dynamics. It also appears that the stabilizing effects of alternate prey are more largely
influenced by interference competition. As a result, this study complements a growing
body of literature highlighting the role of intraspecific interactions and alternate prey in
community dynamics (Post et al. 2000, Berlow et al. 2004, Le Bourlot et al. 2014) and
suggests that our understanding of predator-predator interactions may be wellcomplemented by future investigations focusing on individual predator movements in
heterogeneous landscapes, encounter rates of predators, interspecies competition, and
seasonal variability in prey choice and predation. Such efforts to link landscape,
behavioral and population ecology of predators and prey will be necessary to establish a
61
more realistic understanding of community interactions and predict their ultimate
biological consequences.
62
Tables and Figures
Table 3.1. Model selection of for the numerical response of hare and lynx.
Pseudo R2
Hare
Model
RSS
AICc
Δ
Weight
response
Hyperbolic
Sigmoidal
(hare, lynx)
DI
46696016
500.2
0
0.76
0.94, 0.07
DD
603516529
571.9
71.7
0.00
0.004, 0.004
DI+DD
44651558
502.9
2.7
0.22
0.94, 0.14
DI
65620322
509.7
9.5
0.01
0.60, 0.004
DD
626000000
572.9
72.5
0.00
0.004, 0.004
DI+DD
174443385
541.1
40.9
0.00
0.42, 0.04
63
Table 3.2. Parameter estimates of the best fit numerical response model estimated using
with the Levenburg-Marquardt method.
Parameter
Value ± 95%CI
1.99 ± 0.50 y-1
r1
Intrinsic growth rate of hares
k1
Carrying capacity of hares
a
Attack rate (Hares)
h
Handling time (Hares)
a2
Attack rate (Rsq)
f
“Hare” time
χ1
Conversion rate of hares to lynx
0.006 ± 0.000002 N1-1
χ2
Conversion rate of red squirrels
0.001 ± 0.001 N2-1
22,792 ± 2860 N1
0.15 ± 0.04 km2·y-1 *
0.0018 ± 0.0002 y·N1-1 *
0.062 ± 0.009 P-1·y-1 *
4.02 e-08 ± 2.57 e-08 km2·N1-1 *
to lynx
δ1
Hare mortality rate independent
0.012 ± 0.35 y-1
of lynx
H
1.87 ± 0.03 y-1
Predator mortality rate
* These values were estimated from snowtracking data (see Appendix IV for full
methodology)
64
20
40000
18
35000
16
14
30000
12
25000
10
20000
8
15000
6
10000
4
5000
2
0
1987
1992
1997
2002
2007
Lynx per 100 km2
Hares and red squirrels per 100 km2
45000
0
2012
Year
45000
Red squirrel
40000
Biomass (kg)
35000
Snowshoe hare
30000
25000
20000
15000
10000
5000
0
Figure 3.1. a) The observed average winter densities of snowshoe hares (black), red
squirrels (grey), and lynx (dashed) in the Kluane study area, Yukon from 1987-2011. b)
Representative biomass of snowshoe hares (average biomass = 1.5kg) and red squirrels
(average biomass =0.25kg) in the Kluane lake study area from 1988-2001.
65
Figure 3.2. Comparison of the estimated densities (solid) and observed densites (dotted)
for a) Hares and b) Lynx using unweighted residuals in the Levenburg-Marquardt
algorithm.
66
Figure 3.3. Simulation of hare cycle at 3 different densities of squirrels, no squirrels (0
per 100 km2), average observed densities (23,000 per 100 km2), and maximum observed
densities (39,500 per 100 km2) showing a reduction in the amplitude from a peak of ~
20,000 hares per 100 km2 to ~17,000 hares per 100 km2 a reduction in period from 9.2
years to 7.9 as squirrel densities increase. In addition, we note an increase in the lynx
population lows from ~0.8 lynx per 100 km2 to ~ 3.5 lynx per 100 km2.
67
Figure 3.4. Elasticity analysis of system stability to proportional changes in parameter.
The real part of the eigenvalues of the jacobian matrix are plotted against values of the
parameter being shifted (i.e. 1 = best fit model estimate). The horizontal black line
indicates an eigenvalue of zero. A positive real eigenvalue represents an unstable
oscilating population (i.e. an unstable attractor) while a negative eigenvalue represents a
stable non-oscillating population (i.e. a fixed point attractor).
68
Figure 3.5. Elasticity analysis of the stability of the jacobian matrix in respect to
proportional changes in squirrel density (i.e. 1 = 23,000 squirrels per 100km2). As the
density of squirrels increases we see a shift in stability from an unstable attractor (positive
eigenvalues) to a stable fixed point (negative eigenvalues).
69
Figure 3.6. Elasticity analysis of system stability to proportional changes in parameters
and changes in squirrel density: no squirrels (light grey), average squirrel densities
(23,000 squirrels per 100 km2; medium grey) and 3 times average squirrel density (69,000
squirrels per 100 km2; black). The real part of the eigenvalues of the jacobian matrix are
plotted against values of the parameter being shifted (i.e. 1 = best fit model estimate).
Positive eigenvalues represent an unstable system (i.e. unstable attractor) while a negative
eigenvalues represent a stable system (i.e. fixed point attractor).
70
Chapter 4: General discussion
To restate the goal of this thesis, I wished to test the alternate prey hypothesis by
looking at the role of alternate prey and intraspecific competition (interference) on the
stability of an ecological community. In chapter 2, I examined the functional response of
2 similar-sized predators with differing dietary breadths on their primary and alternate
prey. I found that the lynx-hare functional response was best described by a preydependent hyperbolic or sigmoidal response, indicating possible active switching. In
contrast, the lynx-squirrel relationship was best described by either a linear ratiodependent response or an inverse (θ = 2) response providing further support for the
presence of active switching. Predation rates in coyotes were also found to be linear
response with strong evidence suggesting that these responses are influenced by
interference. I suspect that these linear interference relationships result indirectly from
the structure of predation pressure on the landscape (Poggiale et al. 1998), where predator
interference results from an aggregation predation pressure due to inducible prey
behaviors and refuges resulting in density-dependent reductions in the attack rate.
In chapter 3, I designed a predator-prey model using functional response data
from chapter 2 to incorporate alternate prey and predator interference. I found that the
addition of alternate prey can stabilize predator-prey dynamics via increased predator
competition rather than augmenting available resources. My findings suggest that
competition may have a larger role in the stability of complex ecosystems than diversity
and indicate to the potential importance of predator movement and seasonal variability in
prey choice on interference and stability.
Even though interference played a much larger role in determining the coyote
interactions and their predation rates on both primary and alternate prey, I excluded this
71
species from the third chapter due to the fact that: 1) coyotes are not considered
significant drivers of the predator prey cycle; further supported by the fact that lynx and
hare numbers showed cyclic fluctuations prior to the arrival of coyotes in the landscape
(Krebs et al. 2001a, Levy 2012); and 2) the primary focus of our analysis was to assess
the role of alternate prey, and to a lesser extent predator interference on population
dynamics. Although the role of alternate prey and predator interference are not explicitly
independent of each other, for my purpose, I considered that a simpler system involving a
single predator species was the appropriate approach for assessing population stability.
Our study comprised a series of data collected when hare population dynamics in
the Kluane study area were relatively stationary (Fig. 3.1). Recently this cycle, and other
well-known cycles have been seen to suffer from attenuation (Williams et al. 2004, Ims et
al. 2008, Cornulier et al. 2013) where the amplitude of the cycle has dampened over time.
At present, there is no universally accepted explanation for the disappearance of
population cycles and identifying the causes that have led to their disappearance will be
challenging as the mechanisms that lead to the development of population cycles are still
in debate (Ims et al. 2008). However, this study suggests that this may be due to the
decrease in suitable lynx and hare habitat (Peers et al. 2014) and a corresponding decrease
in the lynx‟s capture efficiency which may be particularly susceptible to shifts in climate
(Stenseth and Shabbar 2004, Row et al. 2014).
Collectively, my findings illustrate the complexity of multispecies interactions and
emphasize the importance of understanding the interactions the mechanisms driving these
relationships. Specifically, that interactions with prey may increase competition between
conspecific predators and that these factors should not be studied in isolation. As such, I
recommend that manipulative studies dealing with increasing numbers of alternate prey
72
should be cognizant of potential changes in competition between predators. While our
understanding predator-prey interactions has advanced substantially in recent years, the
ability to accurately predict the consequences of climate and habitat disruptions on
behavioral and population dynamics will remain incomplete in the absence of a more
robust understanding of the link between behavioral, landscape, and population ecology.
73
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Appendix I: Multispecies functional response derivation
To derive the multispecies functional response we follow the method outlined by
Murdoch (1973) using as a starting point the Holling equation for a hyperbolic functional
response. This equation states that the number of prey eaten, G, is the product of the prey
density (N), the searching efficiency a (also known as the attack rate or proportion of prey
encountered), and the time available for searching (T). Parameter h is defined as the
handling time (i.e. physical manipulation and digestion) and reduces searching efficiency
by decreasing the total amount of time available to the predator for active searching.
[
(
] (
(A.1)
This equation can be rearranged to form equation A.2a with the killing rate defined by
equation A.2b
(A.2a)
(
(A.2b)
In order to expand this to multiple prey species, we subtract the time taken to both
primary and alternate prey from the total time available for searching. A 2 prey system
would be described by equations A.3.
[
(
[
(
(
(
] (
(A.3a)
]
(A.3b)
(
From the equations A.3 we can derive the following equations for G1 and G2:
(
(A.4a)
(
(A.4b)
By substituting equation A.4 into equation A.3 and solving simultaneously, we can
derive:
87
(A.5a)
(A.5b)
With the generalized multi species functional response (A.6a) as well as the equation for
the killing rate (A.6b) shown below.
(A.6a)
∑
(
∑
(A.6b)
This formulation by Murdoch (1973) is limited by the assumption that all predator-prey
interactions are defined by a hyperbolic prey-dependent functional response. More recent
work has expanded this equation to include other functional response shapes such as the
sigmoidal response, by replacing N with Nθ, where θ = 1 represents a hyperbolic response
while θ = 2 is sigmoidal (Smout et al. 2010). Furthermore, Hassel and Varley (1969)
proposed that functional responses should be influenced by interference among predators
above full dependence on prey density.
To account for predator interference, we incorporate predator density into the functional
response using (
as a starting point rather than (
as employed by Holling. This
approach also was adopted by Arditi and Ginzburg (1989) in their formulation of the
ratio-dependent functional response, though they use the ratio of prey to predators, N/P,
rather than the more generalized form
proposed by Hassel and Varley. Here,
parameter m is a continuous variable defining the size of the predator interference (i.e. m
= 0, no interference; m < 0, greater interference) and mathematically, m = 1 is equivalent
to ratio-dependence. Combining both the shape parameter ( θ) and the interference
parameter (m) we can substitute N in eq. A.3 with
88
, and derive equation A.7a.
Following the previous steps used to derive the Murdoch multispecies response, we arrive
at a generalized predator-dependent multi-prey functional response (eq. A.7b). This
generalization allows for each prey item to have an individual functional response shape
(θ i) and interference component (mi). If interference is uniform between prey species then
equation A.7b can be reduced to A.7c.
∑
[
] (
(
(A.7a)
(
(
∑
(A.7b)
(
(
(
∑
(
(A.7c)
As we can see from this generalization, if the density of alternate prey approaches zero,
the functional response will reduce to the Hassel-Varley equation for a single species
predator-dependent response (eq. A.8). This can further be reduced to the Arditi-Ginzburg
ratio-dependent response (m = 1) or to the original Holling equation (Eq. A.2b), if there is
no predator interference (m = 0).
(
(A.8)
Inverse functional response derivation
Using the above equations, we can isolate the effect of primary prey density (Ni) on the
killing rate by removing the alternate prey (i.e., Nj = 0). However, to isolate the effect of
the alternate prey on the killing rate
(
), we cannot employ the same strategy but
instead must assume that density of the primary prey has no effect on killing rate (i.e.,
rate is constant), reducing equation A.7b to:
(
(
89
(A.9)
As it is impossible to estimate values for the parameters aj and hj due to the infinite
number of combinations, we can reduce equation A.10 to the following:
(
(
(A.10)
Where ci is the maximum killing rate of species i and dj is the product of the handling
time and attack rate of species j, collectively referred to as “non-focal prey” time. This
inverse relationship between alternate prey density and the killing rate of primary prey
has previously been described as a „mirror image‟ of the functional response (see Holling
1959a), although it has not been characterized mathematically.
90
Appendix II: Methods Testing
Working with field data, we are constrained by small sample sizes and an inability to
control the densities of study animals. As both the interference models and inverse
models are highly dependent on the inverse of predator and primary prey density
respectively, we were concerned that correlated predator prey numbers, and limited
sample sizes would adversely influence the ability of our AICc methodology to correctly
identify the drivers of predation in alternate prey. As a result, we wished to address 1)
whether co-varying predator-prey densities could result in an interference signature in
alternate prey predation and 2) determine the ability of our AICc methodology in
differentiating between the inverse response and interference responses, specifically, the
linear ratio-dependent response.
Methods
To test the validity of our methods, we simulated using a sine wave model with
the population equation below.
(
(
)
(
(A.11)
Here N is the density of species i at time t (in years), A is the amplitude or
deviation of the wave from 0 to peak, T is the period in years, φ is the phase or time lag in
radians, M is the mean of the population, and εi is a stochastic function based on a normal
distribution with mean μ and standard deviation ζ. We estimated μ and ζ for both prey
and predator by calculating the mean and standard deviation of the difference between the
mean observed hare densities and their maximum and minimum 95% confidence
intervals. As predator densities did not have associated confidence intervals, we
standardized ζ as a function of the amplitude (i.e. ζ = g*A). Alternate prey (red squirrel)
91
densities were simulated using only the stochastic function using the mean and standard
deviation of the observed squirrel densities. In the event that a simulated density was
found to be negative, this value was replaced with the lowest observed value for that
species. All data from which parameters are estimated are available online (see
http://www.zoology.ubc.ca/~krebs/kluane.html).
For each simulated population, either a linear ratio-dependent response (A.12a) or
an inverse (θ = 2) functional response (A.12b) was selected using a binomial distribution
to simulate a kill rate of alternate prey.
(
(
(A.12a)
(
)
(
(
(A.12b)
Noise was also added to simulated kill rates in the form on a stochastic function
εk. As kill rates estimated from snowtracking did not produce a confidence interval, we
estimated a high rate of noise as having μk of 0 with a standard deviation as 10% of the
maximum observed kill rate (ζk = 0.15 squirrels∙d-1). Simulated kill rates were replaced
with 0 in the event that the addition of noise resulted in a negative value. All data was
simulated for 10 years (n= 10) as a direct comparison to available data from the Kluane
project.
We selected 4 models for AICc to discriminate between: ratio-dependent linear,
predator dependent linear, inverse (θ = 1), inverse (θ = 2). From this we simulated 1000
time series and used our AICc methodology to identify the number of times a) the correct
model (e.g. ratio-dependent linear) was identified, and b) the correct class of model (i.e.
interference vs inverse) was identified using an ΔAICc cut-off of 2.
92
Results
We found in our 1000 simulations, AICc identified the correct model as the top model
55% of the time and assigned the correct class of model as the top model 94% of the time.
Within these results, we found that 69% of models only had 1 model under ΔAICc < 2
and 31% of simulations had multiple competing models with ΔAICc < 2. Of the
simulations that identified a single model as the best fit model, the correct model was
correctly assigned 60% of the time and the correct class was assigned 97% of the time. Of
simulations that had multiple models with ΔAICc < 2, the correct model was identified as
the best model 43% of the time and the correct class was identified as the best model 86%
of the time. Simulations with multiple models also had the correct model identified,
within ΔAICc < 2, 93% of the time and the correct class, within ΔAICc < 2, 96% of the
time.
93
Table A1. Parameters used to simulate population and kill rate data.
Parameter
Value
T
Period of Cycle in years
8
μi
Mean of normal distribution for stochastic
function
0.056*Ai
ζi
Standard deviation of normal distribution
for stochastic function
0.33*Ai
Hares (N1)
8000 hares/ 100km2
A1
Amplitude of observed hare population
φ1
Phase shift of hare population (lag)
0 (0 years)
M1
Mean of observed hare population
9500 hares/ 100km2
Lynx (P)
8.3 lynx/ 100km2
Ap
Amplitude of observed lynx population
φp
Phase shift of lynx population (lag)
2 (1 year)
Mp
Mean of observed lynx population
9.2 lynx/ 100km2
Red squirrels (N2)
μ2
Mean of red squirrel population
22816 squirrels/ 100km2
ζ2
Standard deviation of red squirrel
population
8087 squirrels/ 100km2
Kill rate
a2
Attack rate on red squirrels by lynx
0.0001 km2·P-1·d-1
c1
Maximum kill rate of red squirrels
1.47598998 N2·d-1
d2
“Hare” time in red squirrel kill rate
6.50 e-08
μk
Mean of normal distribution for stochastic
function on squirrel kill rate
0
ζk
Standard deviation of normal distribution
for stochastic function on squirrel kill rate
0.15 N2·d-1
94
Appendix III. Parameter estimates from functional response model fitting
Table A2. Parameter estimates for the lynx-hare functional response with 95% confidence intervals.
Model
Constant
Prey-Dependent
Linear
Hyperbolic (θ = 1)
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Ratio-Dependent
Linear
95
Hyperbolic (θ = 1)
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
constant
a
h
b
c
m1
m2
0.91 ± 0.08
-
-
-
-
-
-
-
8.9e-5 ± 8.3e-6
-
-
-
-
-
-
4.1e-4 ± 8.3e-6
0.66 ± 0.08
-
-
-
-
-
1.6e-7 ± 8.3e-6
0.81 ± 0.06
-
-
-
-
-
-
-
0.91 ± 0.65
0.0 ± 3.7e-5
-
-
-
-
-
0.91 ± 2.08
0.0 ± 4.1e-8
-
-
-
4.7e-4 ± 8.3e-6
-
-
-
-
-
-
2.1e-3 ± 8.3e-6
0.69 ± 0.13
-
-
-
-
-
4.0e-6 ± 8.3e-6
0.81 ± 0.94
-
-
-
-
-
-
-
1.26 ± 0.23
8.8 e-5 ± 6.0e-5
-
-
-
-
-
1.10 ± 0.13
7.4 e-9 ± 5.0e-9
-
-
-
1.5e-4 ± 8.3e-6
-
-
-
0.26 ± 0.15
-
-
4.3e-5 ± 8.3e-6
0.65 ± 0.13
-
-
0.05 ± 0.36
-
-
1.6e-7 ± 8.3e-6
0.81 ± 0.08
-
-
0.00 ± 0.57
-
-
-
-
1.08 ± 0.14
7.4e-4 ± 2.4e-3
-
3.66 ± 3.87
-
-
-
1.03 ± 0.11
2.3e-7 ± 1.1e-6
-
3.11 ± 2.86
Predator-Dependent
Linear
Hyperbolic (θ = 1)
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Table A3. Parameter estimates for the lynx-squirrel functional response with 95% confidence intervals.
Model
Constant
constant
a
h
b
c
m1
m2
0.47 ± 0.08
-
-
-
-
-
-
-
2.6 e-5 ± 7.1 e-6
-
-
-
-
-
-
2.6 e-4 ± 5.8 e-5
0.00 ± 4.29
-
-
-
-
-
1.4 e-9 ± 1.7 e-9
0.00 ± 2.05
-
-
-
-
-
-
-
2.40 ± 1.42
7.4 e-4 ± 7.8 e-4
-
-
-
-
-
1.48 ± 0.25
6.5 e-8 ± 3.9 e-8
-
-
-
1.1 e-4 ± 1.8 e-5
-
-
-
-
-
-
1.1 e-4 ± 6.0 e-5
0.00 ± 0.55
-
-
-
-
-
1.7 e-8 ± 8.8 e-9
0.23 ± 0.29
-
-
-
-
-
-
-
1.52 ± 1.32
1.3 e-3 ± 3.0 e-3
-
-
-
-
-
1.05 ± 0.43
8.4 e-7 ± 1.0 e-6
-
-
-
2.5 e-4 ± 1.6 e-4
-
-
-
1.87 ± 0.75
-
-
1.1 e-3 ± 3.9 e-3
0.33 ± 0.59
-
-
2.92 ± 2.88
-
-
8.3 e-8 ± 3.0 e-7
0.45 ± 0.41
-
-
1.61 ± 1.48
-
-
-
-
2.40 ± 1.64
7.4 e-3 ± 1.2 e-3
-
0.00 ± 0.61
-
-
-
1.47 ± 0.28
6.5 e-8 ± 9.4 e-8
-
0.00 ± 0.47
Prey-Dependent
Linear
Hyperbolic (θ = 1)
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Ratio-Dependent
Linear
Hyperbolic (θ = 1)
96
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Predator-Dependent
Linear
Hyperbolic (θ = 1)
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Table A4. Parameter estimates for the coyote-hare functional response with 95% confidence intervals.
Model
Constant
constant
a
h
b
c
m1
m2
0.89 ± 0.13
-
-
-
-
-
-
-
9.6 e-5 ± 9.5 e-6
-
-
-
-
-
-
1.7 e-4 ± 7.7 e-5
0.33 ± 0.21
-
-
-
-
-
9.1 e-8 ± 6.9 e-8
0.75 ± 0.12
-
-
-
-
-
-
-
1.96 ± 4.65
6.2 e-5 ± 2.7 e-4
-
-
-
-
-
1.30 ± 1.03
1.2 e-9 ± 3.2 e-9
-
-
-
3.5 e-4 ± 3.2 e-5
-
-
-
-
-
-
2.7 e-5 ± 2.7 e-5
0.00 ± 0.83
-
-
-
-
-
4.0 e-7 ± 1.8 e-7
0.57 ± 0.12
-
-
-
-
-
-
-
1.24 ± 0.40
5.4 e-5 ± 6.5 e-5
-
-
-
-
-
1.14 ± 0.21
4.0 e-9 ± 3.7 e-9
-
-
-
2.2 e-4 ± 4.9 e-5
-
-
-
0.57 ± 0.17
-
-
2.2 e-4 ± 7.6 e-5
0.00 ± 0.20
-
-
0.57 ± 0.18
-
-
5.1 e-8 ± 2.5 e-8
0.22 ± 0.12
-
-
0.56 ± 0.15
-
-
-
-
1.09 ± 0.18
7.2 e-4 ± 3.9 e-3
-
6.66 ± 14.9
-
-
-
1.09 ± 0.18
3.2 e-8 ± 2.1 e-7
-
3.33 ± 9.01
Prey-Dependent
Linear
Hyperbolic (θ = 1)
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Ratio-Dependent
Linear
Hyperbolic (θ = 1)
97
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Predator-Dependent
Linear
Hyperbolic (θ = 1)
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Table A5. Parameter estimates for the coyote-squirrel functional response with 95% confidence intervals.
Model
Constant
constant
a
h
b
c
m1
m2
0.18 ± 0.03
-
-
-
-
-
-
-
9.7 e-6 ± 1.6 e-6
-
-
-
-
-
-
9.7 e-6 ± 1.3 e-5
0.00 ± 6.91
-
-
-
-
-
4.9 e-10 ± 3.7 e-10
0.00 ± 3.44
-
-
-
-
-
-
-
0.50 ± 0.14
3.3 e-4 ± 2.0 e-4
-
-
-
-
-
0.37 ± 0.06
2.7 e-8 ± 1.4 e-8
-
-
-
2.3 e-5 ± 2.3 e-6
-
-
-
-
-
-
2.3 e-5 ± 8.4 e-6
0.00 ± 1.15
-
-
-
-
-
1.8 e-9 ± 8.7 e-10
0.19 ± 1.17
-
-
-
-
-
-
-
0.46 ± 0.21
7.9 e-4 ± 7.2 e-4
-
-
-
-
-
0.35 ± 0.09
2.3 e-7 ± 1.6 e-7
-
-
-
3.3 e-5 ± 7.5 e-6
-
-
-
1.78 ± 0.50
-
-
3.3 e-5 ± 6.7 e-5
0.00 ± 3.78
-
-
1.78 ± 1.64
-
-
1.4 e-9 ± 2.0 e-9
0.00 ± 2.67
-
-
0.80 ± 0.60
-
-
-
-
0.50 ± 0.18
3.3 e-4 ± 3.4 e-4
-
0.00 ± 0.56
-
-
-
0.37 ± 0.06
2.7 e-8 ± 3.02 e-8
-
0.00 ± 0.41
Prey-Dependent
Linear
Hyperbolic (θ = 1)
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Ratio-Dependent
Linear
Hyperbolic (θ = 1)
98
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Predator-Dependent
Linear
Hyperbolic (θ = 1)
Sigmoidal (θ = 2)
Inverse (θ = 1)
Inverse (θ = 2)
Appendix IV: Comparison of single species and multi-species functional
responses in a model selection framework
Methods
Lynx and coyotes were snowtracked during the winters of 1987-1988 to 1996-1997, and
hunting behavior, activity, habitat use and prey kills of each prey species were recorded.
These tracking sessions were conducted during late October and early March when snow
was appropriate for tracking. These data were used to estimate prey kill rates by
calculating distances traveled by lynx and comparing these to average daily activity of
lynx. Prey kill rates were adjusted for group size to obtain per capita kill rates of lynx on
snowshoe hares and red squirrels for parameterization of the lynx functional responses
(O‟Donoghue et al. 1998, Ch.2). These kill rates were fit with equation A.12 and its
nested single species models using the densities of lynx, snowshoe hare, and red squirrels
made available online (see http://www.zoology.ubc.ca/~krebs/kluane.html).
(
(
∑
(
(A.12)
We fit the models using the nls2 function in R (Grothendieck 2013) following the
methodology outlined by Vucetich et al. ( 2002). To reduce the number of models needed
to be tested (81 possible combinations of the multi-species response) and conserve only
biologically relevant functional response models, we chose to restrict testing of the single
species and combinations of the multispecies models a priori to models found within
ΔAICc < 4 in the single species analysis (Ch. 2). These models were the hyperbolic and
sigmoidal prey-dependent responses in hares and the linear and sigmoidal ratio-dependent
and inverse responses (θ = 1, θ = 2) in squirrels. Equation A.12 could not be fit in its
original form because aj hj only appears in the denominator as a product and the
99
individual parameters could not be estimated. Thus the product aj hj was combined as a
constant dj in order to be fit, where the 2 species equation is shown below.
(
(
(
(
(A.13)
The most parsimonious model was then identified from other candidate models using the
lowest ΔAICc, with models Δ<2 being considered as statistically indistinguishable
(Burnham and Anderson 2002).
100
Results
Table A6.The relative performance of multi-species functional response models fit to predation data from lynx (P) on hares (N1).
Hare (N1) response
Squirrel (N2) response
Form
Δ
Weight
Rank
Pseudo R2
Prey-dependent hyperbolic
-
a1N1/(1+a1h1N1)
0.0
0.51
1
0.67
Prey-dependent Sigmoidal
-
a1N12/(1+a1h1 N12)
0.3
0.44
2
0.69
Prey-dependent hyperbolic
Ratio-dependent sigmoidal
a1N1/(1+a1h1 N2++ d2N22 P-1)
6.0
0.03
3
0.67
Prey-dependent Sigmoidal
Ratio-dependent sigmoidal
a1N12/(1+a1h1 N12+d2N22P-1)
6.3
0.02
4
0.69
101
Table A7.The relative performance of multi-species functional response models fit to predation data from lynx (P) on squirrels (N2).
102
Hare (N1) response
Squirrel (N2) response
Form
Δ
Weight
Rank
Pseudo R2
-
Ratio-dependent linear
a2N2P-1
2.6
0.12
4
0.50
-
Ratio-dependent sigmoidal
a2N22P-1/(1+a2h2 N22)
5.7
0.03
7
0.51
Prey-dependent hyperbolic
Constant
c2/(1+d1N1)
5.4
0.03
6
0.56
Prey-dependent Sigmoidal
Constant
c2/(1+d1N12)
2.7
0.11
5
0.66
Prey-dependent hyperbolic
Ratio-dependent linear
a2N2P-1/(1+ d1N1)
2.2
0.15
2
0.66
Prey-dependent Sigmoidal Ratio-dependent linear
a2N2P-1/(1+a2h2 N22+ d1N12)
0.0
0.43
1
0.73
Prey-dependent hyperbolic
Ratio-dependent sigmoidal
a2N22P-1/(1+a2h2 N22)
7.5
0.01
8
0.73
Prey-dependent Sigmoidal
Ratio-dependent sigmoidal
a2N22P-1/(1+a2h2 N22+ d1N12)
2.6
0.12
3
0.81
Appendix V: Results of predator density-independent with predator densitydependent mortality model fitting and stability analysis.
Table A8. Parameter estimates of the density-independent with density-dependent
mortality numerical response model estimated using with the Levenburg-Marquardt
method.
Parameter
Value ± 95%CI
1.82 ± 6.03 y-1
r1
Intrinsic growth rate of hares
k1
Carrying capacity of hares
25,919 ± 89,374 N1
a
Attack rate (Hares)
0.15 ± 0.04 km2·y-1 *
h
Handling time (Hares)
a2
Attack rate (Rsq)
f
“Hare” time
0.0018 ± 0.0002 y·N1-1 *
0.062 ± 0.009 P-1·y-1 *
4.02 e-08 ± 2.57 e-08 km2·N1-1 *
χ1
Conversion rate of hares to lynx
0.0055 ± 0.0006 N1-1
χ2
Conversion rate of red squirrels to
0.0001 ± 0.001 N2-1
lynx
δ1
Hare mortality rate independent of
0.06 ± 6.21 y-1
lynx
H
Predator mortality rate
1.43 ± 0.49 y-1
δp
Predator mortality rate
0.009 ± 0.01 y-1
* These values were estimated from snowtracking data (see Appendix IV for full
methodology)
103
104
Figure A1. Elasticity analysis of system stability in the density-independent with density-dependent model to proportional changes in
parameter. The real part of the eigenvalues of the jacobian matrix are plotted against values of the parameter being shifted (i.e. 1 = best
fit model estimate). The horizontal black line indicates an eigenvalue of zero. A positive real eigenvalue represents an unstable
oscilating population (i.e. an unstable attractor) while a negative eigenvalue represents a stable non-oscillating population (i.e. a fixed
point attractor).
Appendix VI: Raw Data
Table A9. Raw data of Lynx kills for functional and numerical response fitting.
105
Year
1987-1988
1988-1989
1989-1990
1990-1991
1991-1992
1992-1993
1993-1994
1994-1995
1995-1996
1996-1997
1997-1998
1998-1999
1999-2000
2000-2001
Lynx/
100 km2
2.86
4.57
14.28
17.14
8.00
4.29
2.00
2.60
2.60
5.43
8.57
11.14
5.71
0.57
Hares/ 100
km2
5400
17300
16400
13500
7500
1100
1100
2700
6300
12100
19900
20200
6700
600
Squirrels/100
km2
16500
18500
19900
16700
16700
16100
23300
21400
22100
18300
26400
39400
35500
25600
Dist. Tracked
(km)
259.293
306.470
394.508
443.165
380.524
614.988
471.684
268.680
368.785
60.361
-
Lynx Time
(hrs)
661.46
791.59
1044.53
1213.18
984.67
1427.93
1119.40
633.91
889.59
166.71
-
Hare
Kills
19
32
47
62
40
25
16
11
50
11
-
Hare
kills/day
0.69
0.97
1.08
1.23
0.97
0.42
0.34
0.42
1.35
1.58
-
Squirrel
Kills
4
0
0
0
1
46
69
54
9
0
-
Squirrel
kills/day
0.15
0.00
0.00
0.00
0.02
0.77
1.48
2.04
0.24
0.00
-
Table A9. Raw data of coyote kills for functional response fitting.
106
Year
1987-1988
1988-1989
1989-1990
1990-1991
1991-1992
1992-1993
1993-1994
1994-1995
1995-1996
1996-1997
1997-1998
1998-1999
1999-2000
2000-2001
Coyotes per 100
km2
2.28
3.43
5.71
8.57
4.80
2.50
1.40
1.40
1.50
3.43
4.57
6.00
3.43
0.86
Hares/ 100
km2
5400
17300
16400
13500
7500
1100
1100
2700
6300
12100
19900
20200
6700
600
Squirrels/100
km2
16500
18500
19900
16700
16700
16100
23300
21400
22100
18300
26400
39400
35500
25600
Dist. Tracked
(km)
71.011
228.376
285.681
363.050
353.057
228.736
450.014
393.248
305.368
73.131
-
Coyote Time
(hrs)
83.76
279.56
351.03
466.55
399.83
229.16
469.32
404.80
342.16
77.46
-
Hare
Kills
5
27
15
15
8
3
5
11
6
4
-
Hare
kills/day
1.43
2.32
1.03
0.77
0.48
0.31
0.26
0.65
0.42
1.24
-
Squirrel
Kills
0
0
1
4
2
3
9
6
5
0
-
Squirrel
kills/day
0.00
0.00
0.07
0.21
0.12
0.31
0.46
0.36
0.35
0.00
-