Ecology, 89(1), 2008, pp. 280–288
Ó 2008 by the Ecological Society of America
BETWEEN DISCRETE AND CONTINUOUS: CONSUMER–RESOURCE
DYNAMICS WITH SYNCHRONIZED REPRODUCTION
E. PACHEPSKY,1 R. M. NISBET,
AND
W. W. MURDOCH
Department of Ecology, Evolution and Marine Biology, University of California, Santa Barbara, California 93106 USA
Abstract. In many consumer–resource systems the consumer population has synchronized reproduction at regular intervals (e.g., years) but consumes the resource and dies
continuously, while the resource population grows continuously or has overlapping
generations that are short relative to the time between consumer reproductive events. Such
systems require ‘‘semi-discrete’’ models that have both discrete and continuous components.
This paper defines and analyzes a canonical, semi-discrete model for a widespread class of
consumer–resource interactions in which the consumer is a discrete breeder and the resource
reproduction can be described continuously. The model is the analog of the Nicholson-Bailey
and Lotka-Volterra models for discrete and continuous systems, respectively. It thereby
develops the basis for understanding more realistic, and hence more complex, semi-discrete
models. The model can display stable equilibria, consumer–resource cycles, and single-specieslike overcompensation cycles. Cycles are induced by high maximum fecundity in the
consumer. If the resource grows rapidly and the consumer has high maximum fecundity, the
model reduces to a single-species discrete-time model of the consumer, which can exhibit
overcompensation cycles. By contrast, such cycles in discrete consumer–resource models
typically occur only in the resource once the consumer is extinct. Also unlike a common class
of discrete models that do not display consumer–resource cycles with periods below four years,
semi-discrete models can exhibit consumer–resource cycles with periods as short as two years.
Key words: consumer–resource cycles; consumer–resource dynamics; impulsive differential equations;
overcompensation; pulsed differential equations; semi-discrete models.
INTRODUCTION
Consumer–resource theory has two well-developed
approaches: discrete-time models and continuous-time
models. Discrete-time models describe the change of the
consumer and resource populations from one reproductive event to the next. They are designed to describe
systems where interaction between consumer and
resource occurs over a short period of time, and both
species breed at the same interval. The Nicholson-Bailey
model applied to host–parasitoid systems is the bestknown example. Continuous-time models, such as the
Lotka-Volterra model, are designed to describe systems
where consumer and resource interactions occur continuously, and reproduction in both consumer and resource
can be described as a continuous process (for example,
when both populations have overlapping generations).
Many consumer–resource systems do not fit either
framework. This paper focuses on one type of these
systems where the consumer is a discrete breeder
(reproducing typically once per year) while the resource
reproduces more or less continuously and has several,
Manuscript received 20 April 2007; accepted 21 May 2007.
Corresponding Editor: D. C. Speirs.
1 Present address: Microsoft Research, Computational
Ecology and Environmental Science Group, Roger Needham
Building, 7 JJ Thomson Avenue, Cambridge CB3 0FB UK.
E-mail: [email protected]
280
often overlapping, generations per year. Examples
include planktivorous fish continuously consuming
zooplankton in lakes, ungulates consuming grass (where
increase in biomass is continuous), ladybugs feeding on
aphids, and weasels attacking voles or other rodents.
Prey reproduction is usually not entirely continuous (it is
interrupted by winter, diapause, or some other seasonal
phenomenon), but a physiological time scale (e.g.,
degree days) can translate such phenology into resource
growth that is reasonably approximated by continuous
dynamics. Traditionally formulated, discrete-time models are not appropriate for such systems because it is not
possible to compact the interactions between consumers
and resource into a short time each year. Continuoustime models also cannot readily describe these systems
because consumer reproduction is not continuous, but
occurs in a burst every reproductive period, which for
convenience we hereafter refer to as one year.
To capture these life histories we need a hybrid model
that describes the continuous interactions of resource
and consumer throughout the year, punctuated by the
discrete reproduction event of the consumer. We refer to
these as ‘‘semi-discrete models.’’ We investigate how the
mix of continuous and discrete life histories affects their
dynamics; when they resemble continuous models and
when discrete. We ask if they can display the full range
of dynamics of continuous and discrete models, and
whether new dynamical features appear.
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SEMI-DISCRETE CONSUMER–RESOURCE DYNAMICS
We also examine the properties of population cycles,
which are of particular interest because their periods have
been used to infer the processes driving population
dynamics. For example, Murdoch et al. (2002) showed
that for a class of discrete models, the cycle period can be
used to sharply delineate ‘‘single-species’’ cycles from twospecies ‘‘consumer–resource’’ cycles. Single-species cycles
for the models they examine generally have a period of
one to four years if consumer reproduction occurs every
year, and can arise through direct or indirect density
dependence in consumer birth or death rates (Murdoch et
al. 2003). There are many biological mechanisms capable
of generating single-species dynamics, the best known
being delayed feedback that leads to overcompensation
and cycles with a period of at least two years. By contrast,
consumer–resource cycles have a period of at least six
years if both consumer and resource reproduce every year.
The equations for our semi-discrete consumer–resource model are examples of ‘‘impulsive equations’’ or
‘‘pulsed differential equations’’ (e.g., Lakshmikantham
et al. 1989). Several previous studies used this formalism
to examine the dynamics of predator–prey interactions
(e.g., Liu and Chen 2004, Liu et al. 2004, Zhang and
Chen 2005), bacteria in a chemostat (Funasaki and Kot
1993), and epidemiological systems (Hui and Chen
2004). In these works, the equations are sufficiently
simple that the differential equations for the within-year
dynamics can be solved explicitly; so the system is
expressed as a discrete-time model. Unfortunately, this
approach is not applicable to the nonlinear models that
arise with even a modicum of ecological realism.
Semi-discrete consumer–resource models are also
appropriate in systems where the consumer reproduces
approximately continuously and the resource discretely.
The most obvious members of this class are true
parasites and their hosts. Semi-discrete models have
been applied to insect host–pathogen systems (Briggs
and Godfray 1996, Bonsall et al. 1999) and a nematode–
host system (Dugaw et al. 2004). Briggs and Godfray
(1996) and Bonsall et al. (1999) found stability and
consumer–resource cycles, while Dugaw et al. (2004)
found overcompensation cycles. These studies had to
rely mainly on numerical analysis (sometimes with help
of analytical treatment of limiting situations) to
understand the behavior of the model, because the
analysis of full models was not possible.
This paper presents an analytical treatment of local
stability and oscillations in a semi-discrete model,
designed to incorporate the minimum degree of realism,
and that elucidates the range of possible dynamics. The
resource grows logistically and is eaten by the consumer at
a linear rate (type 1 functional response) throughout the
year. The consumers accumulate biomass used for
reproduction and also suffer mortality throughout the
year. At the end of each year, the surviving consumers
reproduce using the accumulated biomass. The analysis of
this model is more complex than for previous models
because the nonlinearity of the logistic function describing
281
the resource growth rate makes it impossible to translate
the model explicitly into a set of discrete-time equations.
Our overall aim is to propose a baseline, semi-discrete
model (in the nature of the Lotka-Volterra and
Nicholson-Bailey models) for interactions in which the
consumer is a discrete breeder and the resource breeds
continuously. We explore the possible dynamical behavior of this model (the stability of the equilibria and the
properties of population cycles), and constrain possible
dynamics by establishing reasonable ranges of parameter values. To elucidate the effects of the semi-discrete
life history on the stability and the likelihood and period
of population cycles, we compare the dynamics of our
semi-discrete model to continuous (damped LotkaVolterra) and discrete (Neubert and Kot 1992) models
with similar interactions, and finally to a limiting case of
the semi-discrete model under the assumption of fast
resource growth (Geritz and Kisdi 2004).
MODEL
The key distinguishing feature of our model is a time
interval, denoted by T, between pulses of consumer
reproduction. In many cases this interval is one year
and, though other intervals such as a lunar month are
possible, for simplicity we refer to annual reproduction
as ‘‘within-year’’ and ‘‘between-year’’ dynamics. Within
the year, the resource population grows, individual
consumers eat, store resource for reproduction, and may
die. Between years, consumers reproduce in a pulse. The
within-year dynamics are described by a system of
differential equations, and the between-year dynamics
are described by a system of difference equations. Years
are labeled by an integer, t, and we denote by s the time
within a year since consumer reproduction (0 s T ).
All state variables and parameters are defined in Table 1.
Within-year dynamics are described by three differential equations:
dF
F
dC
dB
¼r 1
F aFC
¼ mC
¼ aF:
ds
K
ds
ds
ð1Þ
Here, F(s) is the density of resource, and C(s) is the
density of consumers. The quantity B(s) is the total
amount of resource eaten over a year by an individual
consumer, some fixed proportion of which is assumed to
be stored for reproduction at the transition between
years. Parameter r is the intrinsic resource growth rate, K
is the resource carrying capacity, a is the consumer attack
rate, and m is the consumer death rate (Table 1). We
define quantities Vt and Wt to be resource and consumer
densities at the start of the year (i.e., immediately after
consumer reproduction). Thus each year, the three
equations are solved with the following initial conditions:
Fð0Þ ¼ Vt
Cð0Þ ¼ Wt
Bð0Þ ¼ 0:
ð2Þ
Consumers reproduce at the transition between years.
We assume this process to be instantaneous, so there is
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E. PACHEPSKY ET AL.
no change in resource density between years. At the
transition, therefore:
Vtþ1 ¼ FðT Þ
Wtþ1 ¼ ½hBðT Þ þ 1CðT Þ
ð3Þ
where F(T ), C(T ), and B(T ) are the resource
density, consumer density, and accumulated resource
consumed, respectively, at the end of the year. Parameter h is the consumer conversion efficiency (Table 1).
Scaled model
We non-dimensionalize the model to reduce the
number of parameters. Picking the scales for time within
year and consumer and resource densities,
ss ¼ T
Fs ¼ K
Cs ¼ Kh
ð4Þ
(where ss is the scale for time within year, Fs and Cs are
scales for consumer and resource densities respectively),
we define dimensionless quantities
s¼
s
ss
b¼
BCs
¼ hB
Fs
l ¼ mT
f ¼
F
Fs
c¼
vt ¼
C
Cs
Vt
Fs
q ¼ rT
wt ¼
Wt
Cs
ð5Þ
where s is the scaled time within a year, f is the scaled
resource density, c is the scaled consumer density, b is
the scaled amount of resource eaten by an individual
consumer during the year, vt is the scaled resource
density at the beginning of the year, wt is the scaled
consumer density at the beginning of the year, l is the
scaled consumer death rate, q is the scaled resource
intrinsic growth rate, and a is the scaled consumer
maximum fecundity (Table 1).
The within-year equations then take the following form:
df
¼ qð1 f Þf afc
ds
dc
¼ lc
ds
db
¼ af :
ds
ð6Þ
The update rule between years is
vtþ1 ¼ f ð1 Þ
wtþ1 ¼ ½bð1 Þ þ 1cð1 Þ:
ð7Þ
The dynamics in the scaled model are controlled by
three dimensionless parameter groups, q, l, and a. We
can interpret these as follows. The quantity eq is the
maximum (geometric) growth factor for the resource in
a consumer year. That is, if there were no consumers,
and no resource density dependence, this is the factor by
which the resource would grow in a year. The quantity
el is the year-to-year survival of the consumer. The
quantity a is the maximum number of offspring
(fecundity) produced by an individual consumer that
survives the year; to see this, consider a hypothetical
situation where throughout the year, F ¼ K, implying f ¼
1. It is related to the maximum geometric growth factor,
k, of the consumer population by
k ¼ ða þ 1Þel :
TABLE 1. Variables and parameters in the unscaled and scaled
models.
Parameter
ð8Þ
Definition
Unscaled model
t
years, integer
T
duration of a year
s
time within a year, s 2 [0, T ]
F
resource density during the year
C
consumer density during the year
B
amount of resource eaten by an individual
consumer during the year
resource density at the beginning of the year
Vt
consumer density at the beginning of the year
Wt
r
intrinsic growth rate of resource
K
resource carrying capacity
a
consumer attack rate
m
consumer death rate
h
consumer conversion efficiency
Scaled model
t
f
s
c
b
vt
wt
a ¼ ahKT
Ecology, Vol. 89, No. 1
q
a
l
years, integer
scaled resource density, F/K
scaled time within a year, s 2 [0, T ]
scaled consumer density, C/(Kh)
scaled amount of resource eaten by an
individual consumer during the year, hB
scaled resource density at the beginning of
the year, Vt/K
scaled consumer density at the beginning of
the year, Wt/(Kh)
scaled resource intrinsic growth rate, rT
scaled consumer maximum fecundity, ahKT
scaled consumer death rate, mT
MODEL DYNAMICS
The model has three possible equilibria: the trivial (0,
0) equilibrium, with no consumer and no resource; the
(1, 0) equilibrium, with no consumer, and resource at the
carrying capacity; and the equilibrium (v, w), with both
resource and consumer present. The derivation and the
linear stability analysis of all three equilibria are
presented in the Appendix.
The zero equilibrium (0, 0) is unstable if q is positive.
The no-consumer equilibrium (1, 0) is unstable if the
consumer can grow when the resource is at the carrying
capacity, which occurs when
ða þ 1Þel . 1:
ð9Þ
When both consumer and resource are present, the
model exhibits a range of behavior including a stable
nonzero equilibrium, consumer–resource cycles, and
consumer overcompensation cycles, including chaotic
dynamics (Fig. 1). We performed a linear stability
analysis and mapped out the boundaries of regions in
parameter space where consumer–resource cycles and
overcompensation cycles arise (Fig. 2). These stability
boundaries delineate the regions of local stability and of
potentially cyclic dynamics. We numerically checked
that the behavior of the model in those regions is
consistent with inferences from properties on the
boundaries.
Fig. 2 shows how the three parameter groups (i.e.,
scaled resource growth rate q, scaled consumer mortality
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SEMI-DISCRETE CONSUMER–RESOURCE DYNAMICS
283
FIG. 1. Possible behaviors of the model: (a, b) stable dynamics (a ¼ 10.0, l ¼ 1.5); (c, d) overcompensation cycles (a ¼ 11.0, l ¼
0.8); (e, f ) chaotic dynamics (a ¼ 13.5, l ¼ 0.7); and (g, h) consumer–resource dynamics (a ¼ 2.88, l ¼ 0.1). The density of the
consumers is shown at the end of the year, before reproduction. For all panels q ¼ 20. The scaled densities are dimensionless (see
Eq. 5). See Table 1 for parameter definitions.
rate l, and scaled maximum consumer fecundity a)
affect model dynamics. Fig. 2a is a plot of the stability
boundaries as a function of the scaled consumer death
rate l and the scaled maximum fecundity a for a fixed
value of q ¼ 20. The equilibrium is stable for higher
values of l and a (example in Fig. 1a and b). For high
values of l and low values of a, the consumer cannot
survive: the consumer extinction boundary is defined by
the relationship between l and a in Eq. 9.
For higher values of a, stable dynamics give way to
consumer–resource and overcompensation cycles. For
low values of l, the model displays consumer–resource
cycles (example in Fig. 1g and h). The amplitude of the
cycles is small close to the consumer–resource boundary,
and increases away from the boundary, suggesting that
the bifurcation is supercritical (e.g., Thieme 2003). The
cycle period of the consumer–resource cycles is shown in
Fig. 2b. On the boundary (i.e., as the cycles emerge), the
period is long for low values of l and a and decreases to
two at the point when the consumer–resource boundary
meets the overcompensation stability boundary (which
also has cycles with period 2; example in Fig. 1c and d),
at around l ¼ 0.8. The overcompensation boundary has
a U shape, with the right side facing the stability region
and the left side crossing the region of consumer–
resource cycles. In the region where the consumer–
resource and overcompensation regions overlap, the
model displays complex and possibly chaotic dynamics.
Fig. 2c shows how the stability boundaries shift with
changes in q. As q increases, the boundary for the onset
of consumer–resource cycles shrinks, and the overcompensation cycles occur for lower values of l. However,
when l is high enough, the overcompensation boundaries of the models for different values of q eventually
converge to the same line (as can be seen for the plots
with q ¼ 20 and 40).
Ecologically, the overcompensation cycles in our
model arise from delayed feedback involving indirect
density dependence in consumer reproduction. When
consumer density is above equilibrium at the start of a
year, the resource density is more strongly suppressed
throughout the year, each consumer accumulates less
resource biomass, and per head consumer reproduction
at the end of the year is lower, so consumer density at
the start of the following year is lower. The opposite
occurs the following year. Overcompensation cycles are
more likely when the resource has rapid growth, for
reasons explained in the next section.
CONNECTIONS
TO
OTHER MODELS
Semi-discrete models are a mix of continuous and
discrete processes. We now draw connections and
compare dynamics of the semi-discrete models to their
continuous and discrete counterparts. We compare the
dynamics of the model in this paper to three other
models: a purely continuous consumer–resource model,
a purely discrete analog of our model (Neubert and Kot
1992), and our model in a limiting situation of fast
resource response.
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E. PACHEPSKY ET AL.
Ecology, Vol. 89, No. 1
Connection to a continuous consumer–resource model
A continuous counterpart of the semi-discrete model
is a model with logistic resource growth and linear
consumer functional response:
dF
F
dC
¼ rF 1 aFC
¼ eaFC lC ð10Þ
dt
K
dt
where F is the resource density, C is the consumer
density, r is the intrinsic resource growth rate, a is the
consumer attack rate, l is the death rate of the
consumer, and e is the consumer conversion efficiency.
Note that the conversion efficiency, e, in the continuous
model is not the same as the conversion efficiency, h, in
the semi-discrete model because the former is instantaneous and the latter is invoked only once a year.
The expression for the resource equilibrium value of
the semi-discrete model (see Appendix: Eq. 13) involves
variables describing resource growth as well as consumer
growth. This is different from the continuous model
where the resource equilibrium value does not depend
on the parameters for resource growth, but is consistent
with results for discrete models (Murdoch et al. 2003:
Chapter 4).
In the continuous model the nonzero equilibrium of
consumer and resource is always globally stable (Nisbet
and Gurney 1982) if it exists (i.e., if K . l/(ea)). In the
semi-discrete model, the discrete consumer reproduction
adds a time lag to the dynamics of the consumer. This
can destabilize the equilibrium leading to the appearance
of consumer–resource cycles, as well as other possibilities discussed previously.
Connection to a fully discrete model with
density-dependent resource growth
A discrete model counterpart to our model is one with
density-dependent resource growth and linear functional
response in the consumer. A model of this type was
considered by Neubert and Kot (1992). It is described by
the following set of equations:
Ft
aFt Ct
Ftþ1 ¼ Ft þ rFt 1 K
Ctþ1 ¼ eaFt Ct þ ð1 dÞCt
FIG. 2. (a) The stability boundaries of the semi-discrete
model as a function of a and l; CR ¼ consumer–resource cycles
boundary (heavy solid line and dotted line), OC ¼ overcompensation cycles boundary (heavy dashed line), and consumer
extinction boundary (light solid line); scaled resource growth
rate q ¼ 20. (b) Cycle period along the consumer–resource
stability boundary. The period is shown as a function of l.
(Note that a changes for each value of l so that the model
remains on the consumer–resource boundary). Scaled resource
growth rate q ¼ 20. (c) Stability boundaries for different values
of q; consumer–resource cycles boundary (heavy solid line),
overcompensation cycle boundaries (heavy dashed line), and
consumer extinction boundary (light solid line). See Table 1 for
parameter definitions.
ð11Þ
where Ft is the resource density, Ct is the consumer
density, r and K are parameters for the Ricker resource
growth, a is consumer foraging efficiency, e is the
conversion efficiency of the consumer, and d is the
proportion of the consumer that dies over the year.
This model also displays consumer–resource cycles,
stable dynamics, and overcompensation cycles, but the
overcompensation cycles in this model are in the
resource, not the consumer population, and they
typically occur only when the consumer goes extinct.
That is, this model becomes a single-species model of the
resource with delayed density-dependent growth, which
produces overcompensation cycles. These cycles are not
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SEMI-DISCRETE CONSUMER–RESOURCE DYNAMICS
285
stable equilibrium, overcompensation cycles, or other
nonlinear ‘‘Ricker-like’’ dynamics (including chaos)
depending on the values of model parameters.
It is similarly possible to derive a one-species discrete
map that approximates the consumer dynamics in our
model when the resource responds quickly to consumer
densities (i.e., for very high values of q and a). Provided
the consumer density, wt, at the start of any given year is
not too high, pseudo-equilibrium implies that at all
times s, f(s) ’ 1 (awtels)/q and hence that
awt l
ð12Þ
ðe 1Þ :
bð1 Þ ¼ a 1 þ
ql
Substituting in Eq. 7 then yields
wtþ1 ¼ kwt ð1 kwt Þ
ð13Þ
where
k¼
a2 ð1 el Þ
qlð1 þ aÞ
and k is the consumer geometric growth factor
introduced in Eq. 8. Thus the consumer dynamics are
approximated by the logistic map, popularized in
ecology by May (1976) and analyzed in detail in many
textbooks (e.g., Kot 2001). For this map, the onset of
instability due to overcompensation occurs when k ¼ 3;
that is, when
a ¼ 3el 1:
FIG. 3. Stability boundary plots of maximum resource
fecundity vs. the resource growth rate for (a) the fully discrete
model from Neubert and Kot (1992: after their Fig. 1), and (b)
the semi-discrete model. Note that the stability plot is not along
the a and l axes as in previous plots, because when calculating
the stability boundaries in their paper, Neubert and Kot assume
that the consumer death rate d is 1, so they do not examine
changes of the stability boundaries as a function of the
consumer death rate. See Table 1 for parameter definitions.
ð14Þ
In Fig. 4 we compare the stability boundary
calculated from Eq. 14 with that of the full semi-discrete
model for a large, but still ecologically plausible, value
of q. Agreement is excellent, except for the smallest
values of a, when the semi-discrete model predicts
consumer–resource, not overcompensation, cycles.
possible in our model because in the absence of the
consumer the resource has continuous logistic growth
and hence has a stable equilibrium. In Fig. 3 we
illustrate the difference by comparing stability boundaries for the two models in terms of the resource growth
rate and maximum consumer fecundity.
Semi-discrete model with fast resource response
Geritz and Kisdi (2004) examine semi-discrete models
with a similar structure to the model presented in this
paper. In their analyses, they assume that the resource
responds to the consumer density very quickly. This
translates into an assumption that the resource is in
pseudo-equilibrium with the consumer throughout the
year. They show that the consumer dynamics then
resemble those of a single-species discrete model, with a
FIG. 4. Comparing stability boundaries for the semidiscrete model (heavy solid and dashed lines) with approximation of fast resource dynamics (dotted line); consumer
extinction ¼ light solid line; q ¼ 40. See Table 1 for parameter
definitions.
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E. PACHEPSKY ET AL.
FIG. 5. (a) Stability boundaries for a range of l, k, and q.
The area in white is the region of reasonable parameters. Solid
lines (consumer–resource boundary) and dashed line (overcompensation cycle boundary) delineate stability for different
values of q. Below the solid line is the stable region; above
the line the system is unstable, having crossed either a
consumer–resource boundary (heavy solid lines) or overcompensation boundary (heavy dashed lines). (b) Cycle periods on
the boundary for the different values of q. See Table 1 for
parameter definitions.
INTERPRETING
THE
RESULTS
To interpret our results, we define broadest possible
ranges for the model parameters. First we find a range
for the scaled intrinsic growth rate of the resource q.
Within systems that fit our model, one of the fastest
growing resources is zooplankton as prey for annual
fish. Using the well-studied zooplankton Daphnia as an
example, a reasonable maximum value of the intrinsic
growth rate r for Daphnia under optimal conditions in
nature is around 0.1–0.3/d. If these conditions were
sustained all year, the maximum value of q would be
about 100. Perhaps the slowest growing terrestrial prey
animals that breed continuously and still have overlapping generations are organisms like voles, with an r of
6/yr and hence q ¼ 6 (Turchin and Hanski 1997). The
range of q for animal resource species is thus between 6
Ecology, Vol. 89, No. 1
and 100. The growth rate of grasses may encompass that
range. Nielsen et al. (1996) present the range of daily r
values for various plants, and these translate into a
range of q for sedges, herbs, and shrubs between 0.3 and
100.
To estimate the scaled consumer death rate, l, it is
useful to think of 1/l as the average life span of the
consumer scaled by the time between reproductions. We
assume the minimum value of 1/l to be 1/2; this means
that by the end of the first year, 13% of the population
survives. This gives a maximum value of l ¼ 2.
Instead of estimating the range for the maximum
consumer fecundity, a, we constrain it using its
interpretation in Eq. 8. The relationship k ¼ (a þ 1)el
gives a relationship between a and l. There is already a
constraint that k must be .1 for consumer to survive
(Eq. 9). The largest biologically reasonable value for
yearly intrinsic growth rate, k, observed in natural
population is around 10 (Murdoch et al. 2002). This is
an upper constraint on a and l.
Fig. 5a shows the stability boundaries (as in Fig. 2c)
for parameter values in the specified ranges. Notice that
for q ¼ 5, the overcompensation cycles boundary does
not appear in the figure since overcompensation cycles
only arise for extremely large values of a and l.
A biological summary of our key results is as follows.
Consumers with short adult life spans (l large) and low
maximum fecundity enhance stability, provided they can
persist (Fig. 5a). When cycles exist, they are likely to be
consumer–resource cycles when the resource rate of
increase is low (e.g., q ¼ 5 in Fig. 5a). Shorter consumer
adult life span and higher consumer fecundity move the
cycles to the overcompensation type (e.g., q ¼ 20 in Fig.
5a). Note that our model is not appropriate for
univoltine predators because of the exponential survival
term (i.e., no matter how large the death rate of the
consumer, some proportion of consumers will survive to
the next year).
DISCUSSION
We analyzed a semi-discrete model where throughout
the year the resource grows and is consumed by the
consumer, and consumers may die, but consumers
reproduce once a year. Reproduction could be at
different intervals (e.g., every month), but we use yearly
reproduction to make the discussion simpler. The
resource grows logistically, and the consumer has a type
1 functional response. Our analysis showed that this
model can have a stable equilibrium with both consumer
and resource present, and can also display both
consumer–resource cycles (characteristic of two-species
dynamics) and consumer overcompensation cycles
(characteristic of one-species dynamics). The dynamics
of the model are driven by the resource growth rate, q,
death rate of the consumer, l, and a, the maximum
consumer fecundity. For low values of the resource
growth rate, q, consumer–resource cycles appear. For
higher resource growth rates, short consumer–resource
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SEMI-DISCRETE CONSUMER–RESOURCE DYNAMICS
cycles and overcompensation cycles (of period 2) occur.
Increased values of a destabilize the system. The semidiscrete model is less stable than the continuous model
(which is always stable). That is, not surprisingly, the lag
caused by discrete consumer reproduction has a
destabilizing effect.
The consumer overcompensation cycles in the semidiscrete model are caused by delayed indirect density
dependence in the consumer. They occur when the
consumer is effectively acting like a single species. This
situation is promoted by a rapidly growing resource,
which is then effectively in pseudoequilibrium throughout the year (see also Geritz and Kidsi [2004]). Our
treatment of this limiting situation (the argument
leading to Eq. 13) is similar to, but different in some
important mathematical details from, Geritz and Kisdi
(2004). We shall report on these in a subsequent
publication (R. M. Nisbet and E. Pachepsky, unpublished manuscript). Single-species-like cycles have also
been demonstrated in continuous-time consumer–resource models with developmental time delays in both
species; these models are written as delay differential
equations (Murdoch et al. 2003).
Interestingly, the cycle periods on the consumer–
resource boundary of our model range from very long
all the way down to period-two cycles. This contrasts
with results by Murdoch et al. (2002) who found that for
a class of discrete-time consumer–resource models, the
consumer–resource cycles will typically have a period
larger than six. Our model suggests that within-season,
continuous consumer–resource interaction eliminates
this hard boundary. However, the results in Fig. 5b still
suggest that consumer–resource cycles with a period less
than four times the consumer development time may not
be common in nature. Cycles this short require a
combination of an extremely rapidly growing resource
and a relatively short-lived, but nevertheless perennial,
consumer. We suggest zooplankton attacked by planktivorous fish are among the most likely to meet these
conditions. The examples we know about, however, are
from northern lakes. In these circumstances the zooplankton are not likely to have q greater than about 20.
Vendace is an example of a fish in such lakes that feeds
only on zooplankton, and it may have an average life
span of three to four years (individuals 13þ years have
been recorded [G. Mittelbach, personal communication];
i.e., l ¼ 0.25–0.3). This combination of values gives
consumer–resource periods equal to about four.
These analyses suggest caution when the cause of
cycles in real semi-discrete consumer–resource systems
(i.e., in which one species is a discrete breeder and the
other a continuous breeder) is inferred on the basis of
cycles arising in consumer–resource models that are
either fully discrete or fully continuous. The massive
literature on vole cycles, which are commonly interpreted as weasel–vole cycles, provides many examples. Since,
in season, voles breed more or less continuously,
whereas weasels are annual breeders, this interaction is
287
best described by a semi-discrete model, which is likely
to have different dynamical and cycle properties than
either a fully discrete or a fully continuous analog. This
further suggests that a match between the observed cycle
period and that seen in a purely continuous model, for
example, may be spurious. We expect that more realistic
models based on the framework developed in this paper
will provide insight into how the semi-discrete nature of
a system affects the possible range of cycles.
The model we explore has a very simple structure (a
logistically growing resource and a linear functional
response in the consumer); even so, its analysis is
complicated. The analysis provides an analytical framework for investigating more complex and realistic
versions, and we view it as the canonical case for mixed
life histories, analogous to the Lotka-Volterra and the
Nicholson-Bailey models for, respectively, continuoustime and discrete-time consumer–resource life histories.
An analogous study that uses semi-discrete models to
describe a host–parasitoid system where both species
have one generation per year, but where there are
continuous interactions within the year, has identified
situations where previous models make misleading
predictions regarding stability (Singh and Nisbet 2007).
A useful extension of the present model will explore
how the dynamics are altered if its structure is changed
or made more detailed; for example, if we replace the
type 1 with a type 2 functional response. Another
interesting avenue of research will be to examine a model
where the death rate of the consumer depends on the
resource density through the year, or when the consumer
growth is self-limiting. Such added realism will in
particular allow us to determine whether, and if so
when, we might see short-period consumer–resource
cycles in the real world.
ACKNOWLEDGMENTS
We thank Cherie Briggs and Mike Neubert for discussions,
and Abhyudai Singh and two anonymous reviewers for
commenting on an earlier draft of this paper. This research
was supported in part by a grant from NSF to WWM.
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APPENDIX
Stability of the logistic semi-discrete model (Ecological Archives E089-017-A1).
SUPPLEMENT
Mathematica code to calculate the overcompensation and consumer–resource stability boundaries of the semi-discrete model
(Ecological Archives E089-017-S1).