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Biol. Lett. (2012) 8, 1067–1069
doi:10.1098/rsbl.2012.0669
Published online 22 August 2012
Population ecology
Temporally varying
resources amplify the
importance of resource
input in ecological
populations
Alan Hastings*
Department of Environmental Science and Policy, University of
California, Davis, CA 95616, USA
*[email protected]
Temporally and spatially varying resource levels
are present in most ecological systems. Very
simple models incorporating the key features of
temporally varying resources and specific descriptions of survivorship for consumer species show the
overriding importance of the time dependence of
available resources and the role that allochthonous
inputs play as essentially insurance in allowing
species to persist. Persistence of species with lifetimes short relative to the timescale of resource
variability is determined by the geometric mean
of resource levels, while the persistence of species
where resources vary on a much shorter time
scale (or with exponential survivorship) are
determined by the arithmetic mean of resource
levels. Models that incorporate features of timevarying resources and explicit life histories
dramatically change our understanding of how
fluctuations in resource availability through time
and space will affect population persistence and
community dynamics.
Keywords: resource pulses; allochthonous inputs;
persistence; life histories
1. INTRODUCTION
Pulses [1–14] and allochthonous inputs [7] are
dramatic examples of temporally varying resource
availability in ecological systems, which can have large
impacts on the temporal and spatial abundances of
species. However, theoretical understanding of the
impact of these influences on population dynamics is
limited [2,10,15] since simple population dynamic
models typically ignore time-dependent aspects of
reproduction, survival and resource availability.
Although the study of age- and stage-structured populations through detailed models has progressed greatly
[16], these models are typically too complex to incorporate resource variability in time and simultaneously
to allow for general understanding. The role of resource
variability is a large and complex subject so here I focus
on the specific case of the interaction between allochthonous inputs and pulses. Note however that there are
many other ecological and evolutionary areas where
the life history of consumers plays a significant role,
and the approach developed here could be applied
to those.
Received 17 July 2012
Accepted 31 July 2012
The classical models in ecology include die-off of
consumers with exponentially distributed waiting
times in the absence of resources. This does not lead
to difficulties if resources do not vary on time scales
that would influence the survival of individuals, but it
will lead to difficulties if the time scale of variability
of resources is comparable to that of the lifetime of
individuals. Temporal variation of resources has only
been considered in specific cases [17,18]. Dramatically, pulsed input of resources [1– 14] is a key
component of the dynamics of many biological systems
with resources coming from within the system owing to
seasonal or annual or irregular fluctuations. Specific
examples include resources from outside the system
owing to allochthonous input [6,7], or in marine systems from upwelling. Other dramatic cases include
plant masting or the emergence of periodical cicadas
or other insect outbreaks [3]. Key biological questions
that arise are what controls persistence of species that
consume these irregular resources, and how do
pulses affect population dynamics.
A minimal model must incorporate the appropriate
time scales of the interaction between life histories of
consumers and temporal dynamics of resource availability. Including explicitly the effects of resources on
persistence of individuals is essential for scaling up
the role of resource pulses and allochthonous input on
the dynamics of populations as well as communities.
As the focus here is on the simplest models that can
be used to understand these issues, it is important to
identify the minimal biological issues that must be
included. In particular, the model must include the
possibility of death of all individual consumers in the
absence of resources. The second essential ingredient
that must be included is the time-dependent availability
of resources, including the possibility that there are
times where resources may be completely unavailable.
The key principles already emerge in simple models
where the availability of resources is periodic in time,
as in a seasonal environment, and the role of the
resources is implicit and appears through their impact
on fecundity and survival. More complex models
would treat resource pulses that arise stochastically
through time with large fluctuations and follow the
impact of the consumer on the resources.
2. METHODS AND RESULTS
The basic aspects of the modelling approach and the
consequences for understanding population dynamics
can easily be seen in the simplest setting of a single
resource and a single consumer species. Even for the
first model, I make a number of important simplifying
assumptions. I assume that although the resource may
last only a finite time (it disappears or becomes unavailable or unusable even if not consumed by the
species of interest), the quality of the resource is independent of the time since it was created (its ‘age’).
I ignore any issues of density dependence other than
those mediated through the resource.
The dynamics discussed here are those of a single
consumer and a single time-varying resource. A
simple model can be constructed for this system
using McKendrick – von Foerster equations. The
model is described by the following system of coupled
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This journal is q 2012 The Royal Society
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1068
A. Hastings Temporally varying resources
equations:
@n @n
þ
¼ mn ðRðtÞÞnðt; aÞ;
@t @a
ðA
nðt; aÞmðRðtÞ; aÞda;
nðt; 0Þ ¼
0
NðtÞ ¼
ðA
nðt; aÞda;
0
t1
@r @r
þ
¼ mc NðtÞrðt; aÞ mr ðt; aÞrðt; aÞ;
@t @a
rðt; 0Þ ¼ IðtÞ
and
RðtÞ ¼
consumer species has a fixed lifespan independent of
the resource level (resources affect only the birth
rate), i.e. l is one for ages less than 1 and 0 for ages
greater than 1. Then replacing m by R(s), setting l to
one for ages less than 1 and 0 for larger ages which
can be represented by changing the limits of
integration, we arrive at the model
ðt
NðsÞRðsÞds:
ð2:2Þ
NðtÞ ¼
ðB
rðt; aÞda;
0
where n(t,a) is a density function on age for the population at time t, N(t) is the total population size of the
consumer at time t, r(t,a) is a density function on age
for the resource at time t, R(t) is the total resource at
time t, mn is the age-dependent death rate of the consumer which depends on the total resource level, m is
the age-dependent birth rate and A is the maximum
age of the consumer. The resource is removed by the
consumer (linear functional response) at a rate mc,
and the resource disappears due to other factors at a
rate mr which can depend on the ‘age’ of the resource.
In the body of the paper, simpler versions of this model
are analysed which can be derived from the more
complete model under a variety of assumptions.
To understand issues of persistence, the first model
can be further simplified greatly, since, for persistence,
the consumer population would be small and the
effect on the resource could be ignored (essentially linearizing about the solution where the consumer
population level is zero). Recall that N(t) is the consumer population, R(t) is the resource level, and let
m(t,R(s)) be the per capita reproduction as a function
of the resource level, all at time t. This assumes that
reproduction depends only on the instantaneous level
of the resource. These assumptions mean that in the
McKendrick–von Foerster equations only the consumer
need be considered. Integrate the partial differential
equation for n(a,t) to find an explicit expression for
n(a,t), and substitute the answer and the boundary condition for n(0,t) into the equation for N(t). Then the
description of the consumer population dynamics
reduces to the single equation
ðt
NðsÞmðs; RðsÞÞlðt s; RðÞÞds;
ð2:1Þ
NðtÞ ¼
1
where the survival to age t 2 s, lðt s; RðÞÞ; depends on
the ongoing level of resources. The notation indicates
that the survivorship may depend explicitly on resources
available to the organism at all ages.
The easiest setting in which to analyse this simple
model is where the resource varies in time and the
dependence of the birth rate on the resource level is
linear. By scaling the resource level, we can assume
that the per capita birth rate is the resource level R(s).
I make the further simplifying assumption that the
Biol. Lett. (2012)
A complete analysis of model (2.2) is still not
straightforward, but two extreme cases are easy. If the
time scale over which the resource level varies is very
short, then R(s) in (2.2) can be replaced by its average
Rave, and the persistence condition simply is Rave . 1.
(This can be confirmed by replacing N(t) in (2.2) by
aelt and finding l). Conversely, if the resource level
changes on a long time scale only (i.e. R is fixed over
times much longer than 1), then the persistence
condition is essentially the geometric mean, i.e. that
½logðRÞave . 0:
ð2:3Þ
The result in (2.3) follows from observing that in this
case equation (2.2) reduces to essentially a discrete
time model with R chosen randomly; this is the classic
result [19] for discrete time models.
In contrast, the analogous model with an exponentially distributed lifetime can be found from equation
(2.1) by setting lðt sÞ ¼ ets ; m ¼ R, assuming the
mortality rate is one, and differentiating equation
(2.1) producing
dN
¼ RðtÞN N;
dt
which has the solution
Ðt
ðRðsÞ1Þds
:
NðtÞ ¼ Nð0Þe 0
ð2:4Þ
ð2:5Þ
From equations (2.3) and (2.5), we see the key result that
if the lifetime is exponentially distributed, persistence
depends only on the average resource level independent
of the time scale of the variability in the resource, while
if there is a fixed lifetime both the magnitude and the
time scale of variability in the resource matters.
A different model would be needed to examine the
importance of the resource for survival, rather than
fecundity. Assuming that the instantaneous survival
depends on the resource, if the resource level ever
drops completely to zero, the population goes extinct.
This reflects the observation that a period with no
births that is shorter in duration than the maximum
age span does not produce certain extinction, but any
period with no survival certainly leads to extinction.
3. DISCUSSION
The model framework developed here provides important insight into the general role of the dynamics of
resource pulses in population biology. This can be
used to understand the impact of alternative food
sources of varying kinds. As demonstrated here, the
role of allochthonous inputs, such as inputs into
islands [6,7], can be far more important than would
be suggested by considering the mean level of input
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Temporally varying resources A. Hastings
[15], as even small levels of input can greatly change
the geometric mean of resource levels available to consumers when the in situ resource levels fluctuate and
can reach very low levels. This can explain the observations [6,7] of the dramatic effect of allochtonous
input on the recipient community, and is in contrast
to the role of allochthonous input described in previous models [15] that do not include the temporal
aspects considered here.
Similarly, the survey [13] of the role played by pulses
and allochthonous inputs in general demonstrates the
dramatic influence on community dynamics. The
results from even the simplest models that incorporate
both life histories with finite age spans and temporally
varying resources are dramatically different from
models that incorporate only one or the other factor
(or neither). Incorporation of explicit temporal aspects
will be key to understanding the influence of temporal
and spatial variation, and changes in these, for population dynamics. For example, variability may affect
the dynamics of apparent competition [20]. The results
here are clearly consistent with the classic results [19]
that the geometric mean of the population growth rate
determines the long-term growth rate in a discrete
time model with non-overlapping generations, but the
extension to the continuous time case is only possible
with the explicit inclusion of consumer life histories
which have not typically been included in ecological
models. Although the results here are derived under
explicit extreme assumptions about the dynamics of
resources, the results clearly indicate behaviour under
less extreme assumptions as well. Understanding the
evolution of life-history characteristics, such as senescence [21], will certainly require including the role of
time scales and variable resources. This interaction
between temporal resource dynamics and consumer
life histories has been underappreciated.
I thank Carl Boettiger for helpful discussions and the referees
for very helpful comments. This research was supported by
US National Science Foundation grant EF-0742674.
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