Download How does adaptive consumer movement affect population dynamics

Journal of Theoretical Biology 277 (2011) 99–110
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Journal of Theoretical Biology
journal homepage: www.elsevier.com/locate/yjtbi
How does adaptive consumer movement affect population dynamics
in consumer–resource metacommunities with homogeneous patches?
Peter A. Abrams n, Lasse Ruokolainen 1
Department of Ecology and Evolutionary Biology, University of Toronto, 25 Harbord St. Toronto, ON, Canada M5S 3G5
a r t i c l e i n f o
abstract
Article history:
Received 29 October 2010
Received in revised form
20 February 2011
Accepted 22 February 2011
Available online 1 March 2011
This article uses simple models to explore the impact of adaptive movement by consumers on the
population dynamics of a consumer–resource metacommunity consisting of two identical patches.
Consumer–resource interactions within a patch are described by the Rosenzweig–MacArthur predator–
prey model, and these dynamics are assumed to be cyclic in the absence of movement. The per capita
movement rate from one patch to the other is an increasing function of the difference between the per
capita birth minus death rate in the destination patch and that in the currently occupied patch. Several
variations on this model are considered. Results show that adaptive movement frequently creates antiphase cycles in the two patches; these suppress the predator–prey cycle and lead to low temporal
variation of the total population sizes of both species. Paradoxically, even when movement is very
sensitive to the fitness difference between patches, perfect synchrony of patches is often much less
likely than in comparable systems with random movement. Under these circumstances adaptive
movement of consumers often generates differences in the average properties of the two patches. In
addition, mean global densities and responses to global perturbations often differ greatly from similar
systems with no movement or random movement.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
Dispersal
Metacommunity
Predator–prey system
Synchrony
Stability
1. Introduction
The question posed by the title is important because most natural
consumer–resource interactions occur in spatially subdivided ecological systems (metacommunities) connected by the dispersal of
individuals. While movement behavior is likely to be adaptive
(Bowler and Benton, 2005), relatively little is known about the effects
of adaptive movement on the dynamics of metacommunities. Habitat
selection models have usually assumed that individuals moving
towards better habitat patches will achieve a stable ‘ideal free’
distribution (Fretwell and Lucas, 1970), in which their fitness is equal
across all occupied habitats (e.g., Křivan, 1997; Morris, 2003). If
consumer–resource interactions within a patch are structurally
identical across patches, equality of consumer fitness implies that
resource densities are also equal across patches. This means that if the
populations in the patches fluctuate, they must fluctuate in synchrony. However, some of the few studies that have explicitly
modeled adaptive movement in sets of interacting species have found
that ideal free distributions are often not necessarily attained, even
when patches are equivalent in all parameter values (Schwinning and
Rosenzweig, 1990; Bernstein et al., 1999; Abrams et al., 2007;
n
Corresponding author.
E-mail address: [email protected] (P.A. Abrams).
1
Current address: University of Helsinki, Viikinkaari 1, P.O. Box 68, 00014
Helsinki, Finland.
0022-5193/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2011.02.019
Abrams, 2007). Other types of non-diffusive movement have been
shown to affect stability differently than random movement in simple
systems with interacting species (Huang and Diekmann, 2001;
Amarasekare, 2008). The present article is a step towards a more
systematic understanding of how adaptive movement affects
dynamics and distributions of individuals in metacommunities.
Most work on metacommunities (e.g., all but one chapter
in Holyoak et al. (2005)) has assumed that movement consists of
random dispersal. Although random movement is often thought to
synchronize fluctuations across patches, this is not always true, even
in very simple metacommunities consisting of two patches and two
species (Jansen, 2001) or in single-species models with cycles driven
by discrete density-dependent dynamics (Gyllenberg et al., 1993;
Hastings, 1993). In metacommunity systems with three or more
interacting species, small numbers of patches, and endogenously
driven population cycles within a patch, it is even more likely that
asynchronous dynamics will arise from random movement between
patches (Koelle and Vandermeer, 2005; Wilson and Abrams, 2005).
Nevertheless, for the simple case of two similar patches, each
containing a consumer–resource (predator–prey) pair, most population growth parameters and movement rates imply that random
movement synchronizes the fluctuations in different patches (Jansen
and de Roos, 2000; Jansen, 2001; Goldwyn and Hastings, 2008;
Vasseur and Fox, 2009). However, Jansen (2001) noted that antisynchronized cycles are possible for some parameter values in this
case. The fact that consumer movement towards a patch with more
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P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
resource should reduce the difference in resource densities might
suggest that synchronization would be more likely with adaptive
than with random movement. This was found by Ruxton and
Rohani (1998) in a study of fitness-dependent movement within a
discrete time single species metapopulation model; they concluded that such movement was more likely to stabilize and
synchronize dynamics than was random movement. They also
found that fitness-dependent movement generally simplified
fluctuations when it did not produce point stability; chaotic
dynamics and complex cycles were usually converted to simple
cycles. The work we describe here does not support either the
intuition or the earlier results from discrete models.
Defining ‘fitness’ and ‘adaptive’ can be problematic, particularly in
the context of movement. However, for cost free movement of
unstructured populations, whose dynamics are represented by ordinary differential equations, an individual that is always located in the
patch that confers the largest per capita birth minus death rate will
outcompete all others. ‘Adaptive’ movement in the models considered
here means that the per capita movement rate towards the patch
currently characterized by the larger per capita birth minus death rate
is higher than the movement in the reverse direction. Although a few
previous models of metacommunities have assumed this type of
movement, most have considered scenarios more complicated than
just consumer movement in a two-patch consumer resource system.
In some, both consumers and resources move adaptively (Abrams,
2007); others involve systems with very many patches (Armsworth
and Roughgarden, 2008), or systems in which the biological community has more than two species (Abrams, 2007; Amarasekare, 2007,
2008, 2009; Křivan et al., 2008). These studies have shown that in
specific cases, adaptive movement in a set of interacting species can
produce dynamics that differ greatly from the dynamics arising via
random movement and may not always equalize fitness of the
moving species across patches. Understanding the generality of these
findings requires further studies to determine how adaptive movement alters the dynamics of the simplest ecological communities.
The present study will analyze several differential equation
models of population dynamics in two identical patches, each
containing the same consumer and the resource species. The resource
is assumed to be immobile, while the consumer moves adaptively
between patches. We concentrate on cases in which the consumer–
resource system within an isolated patch would undergo limit cycle
dynamics. This assumption of cyclic dynamics is appropriate because
movement in general is an adaptation to changing conditions. The
other population dynamical assumptions mirror those of much of the
previous work on random movement in coupled 2-patch consumer–
resource systems (Jansen, 1995, 2001; Goldwyn and Hastings, 2008,
2009). The purpose of the analyses is to determine how consumer
movement affects the variation of the populations of both species,
their synchrony across patches, and how closely the consumers
approach an ideal free distribution. Our results show that moving
towards the patch with more resource often produces anti-synchrony, in which patch dynamics are 1801 out of phase, and the
variability in total density of each species is greatly reduced. A variety
of other outcomes are possible, and many parameter sets can have
two or more alternative attractors. We compare these results to those
for similar systems with random movement, and examine how
adaptive movement changes the response of the entire system to
global environmental variables that affect both patches in the same
way. Similar questions for systems with non-equivalent patches are
considered elsewhere (Ruokolainen et al., submitted for publication).
2. The primary model
The following consumer–resource (predator–prey) model is
the familiar Rosenzweig-MacArthur (1963) model with the
addition of fitness-dependent
Abrams (2000, 2007):
consumer
dRi
CRi Ni
¼ Ri ðrkRi Þ
dt
1 þ ChRi
dNi
bCRi
¼ Ni
d mNi exp lðWi Wj Þ
dt
1 þ ChRi
þmNj exp lðWj Wi Þ
bCRi
d
i, j ¼ 1,2, where Wi ¼
1 þChRi
movement
as
in
ð1a; bÞ
These equations describe resource (1a) and consumer (1b)
population growth rate in patch i, where Ri and Ni are resource
and consumer densities, respectively, in patch i, and Wi is
consumer fitness (instantaneous per capita birth minus death)
in patch i. In the absence of the consumer, resources have logistic
growth. The parameterization of the logistic used here separates
the per capita resource growth rate into a maximum per capita
growth rate r and a per capita reduction in that rate of k (Rueffler
et al., 2006). Under this parameterization, the equilibrium
resource density in the absence of consumption is r/k. Consumption of resources is given by a Holling type II functional response
with an attack rate C and handling time h. Consumed resources
are converted to additional consumers (numbers or biomass) with
an efficiency of b. The parameter d defines the (constant) per
capita mortality of the consumer. The form of the movement
function assumes that a consumer individual has the knowledge
of resource densities in both patches. Information about conditions in the other patch may be obtained by cues that may be
detected at a distance (e.g., diffusing chemicals) or by shortduration non-foraging visits. Eq. (1b) implies that a baseline per
capita movement rate m applies when consumer fitness (i.e., per
capita growth rate W) is equal in the two patches. This is modified
by an exponential function of the difference in fitness between
the patches. A larger value of the fitness sensitivity (l) makes the
movement rate increase more rapidly with the fitness difference
between patches. A larger l means that the consumer moves into
a more-rewarding patch at a greater rate and moves into a lessrewarding patch at a lower rate; l ¼0 implies random movement.
Alternative forms for both the movement function and population
dynamics are considered in Appendix B, which is summarized
in Section 3.6.
Here we consider movements that occur on a behavioral time
scale, which is assumed to be more rapid than the population
dynamical time scale. As a result, in choosing parameter values,
we require that the maximum per capita movement rate (which
occurs when Ri ¼r/k and Rj ¼0; i.e., m exp[bC(r/k)/(1þCh(r/k))])
exceeds the maximum per capita population growth rate within a
patch (i.e., bC(r/k)/(1þCh(r/k)) d). The ratio of these two quantities is referred to as the ‘movement ratio’ (MR). We assume a
low value of the baseline rate m, reflecting the idea that ‘adaptive’
implies low rates of movement to poorer quality patches. The
analysis examines a wide range of the fitness sensitivity parameter l. The minimum l considered in each case is one that
implies MR ¼1, although most behavioral movement is likely to
have MR b1. The maximum l considered in all cases represents a
MR of at least 106. Many of these maximum movement rates may
be too large to be realistic in most biological systems. However,
the large differences between resource densities required to
produce near-maximal movement do not occur except during
transient dynamics for some initial conditions. The realized
movement rates and their effects are discussed for the example
in Fig. 1C and D. Appendix B discusses several alternative movement functions, including one with an upper limit to the
movement rate.
The population dynamic parameters are constrained by the
assumption that consumer and resource populations in a single
P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
101
Fig. 1. Population dynamics for three different fitness sensitivity parameters for Eq. (1) with parameters values, b¼ 0.25; C¼r ¼k ¼1; h ¼ 3; m ¼0.0005; d¼ 0.02. For panels
A through C the left side gives resource densities and the right side gives consumer densities (solid line is patch 1 and short dashed line is patch 2; the long dashed line on
the right is the average of the two consumer populations). Panel D is based on a short time span from panel C; the left side displays the fitness difference, W1 W2, and the
right side gives the corresponding fraction of consumers located in patch 1.
isolated patch exhibit limit cycle dynamics. This requires that
d
Chrk
o
b
hðChr þ kÞ
ð2Þ
For cycles to exist for some values of b and d, it is necessary
that Chr4k. We are interested in examining the impact of
movement for different types of population cycles. It is possible
to reduce the number of parameters in a single-patch model to
three by rescaling variables (e.g., Gurney and Nisbet, 1998). This
scaling and numerical analysis (e.g., Abrams and Roth, 1994;
Abrams et al., 1998; Abrams and Holt, 2002) has shown that the
primary determinants of cycle shape and amplitude are: (i) the
value of Chr/k, which gives the ratio of handling to search time for
a consumer when the resource is at its equilibrium density;
(ii) the relative speeds of consumer and resource dynamics, which
can be varied by proportional changes in both b and d; and
(iii) the distance of a stability-determining parameter (mortality,
d, is the parameter used here) from its stability threshold value
(inequality (2)), which determines the period and amplitude of
cycles. We study the impact of adaptive movement on dynamics
for a range of values of each of these composite parameters.
Numerical methods are outlined in Appendix A.
3. Results
We begin with a heuristic treatment of how consumer movement might affect resource dynamics. This paragraph also
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P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
summarizes some of the basic features of the simulation results,
which are detailed subsequently. All of the simulations assume
parameters that produce limit cycle dynamics in a single isolated
patch. If a single patch is stable, neither random nor adaptive
movement produces cycles in a two-patch system. The movement
function implies that little directional movement occurs when the
fitness difference is small. In these circumstances, detecting a
difference is difficult and, even if the difference is assessed
correctly, the reward from moving is small. Slow movement with
a small fitness difference implies a time lag between when the
identity of the best patch changes and when any significant
fraction of the consumer population has relocated. If the lag in
moving is sufficiently long, it may drive cycles, and may interact
in complex ways with consumer–resource cycles that are generated within one or both patches. Decreasing the baseline movement rate m increases the effective lag because m is the
approximate per capita movement rate when the patches are
similar in resource density. Lags can produce cycles that are 1801
out-of-phase in the two patches; this is one of the natural modes
of oscillation of coupled oscillators. These anti-phase cycles
suppress large amplitude consumer–resource cycles because
consumer individuals do not stay in a patch for long enough to
depress the resource to very low density. Thus, only small
amplitude fluctuations occur in total consumer population size.
Larger amplitude consumer–resource cycles at the global scale
usually re-emerge when the fitness sensitivity parameter l
becomes large enough; however, the lag inherent in the adaptive
movement model can still prevent perfect synchrony, and usually
results in the system frequently being far from an ideal free
distribution at many points in time. For some parameter sets, a
sufficiently large sensitivity l does result in perfect synchrony;
such synchrony occurs for a wider range of other parameter
values when the baseline movement rate m is relatively large,
since this reduces the time required for many individuals to shift
to a patch that has recently become better. Many parameter sets
imply the existence of two or more alternative attractors for a
wide range of fitness sensitivities.
species; these fluctuations have high but imperfect correlation
between patches, as shown in Fig. 1C (l ¼350). Fig. 1C shows that
some asynchrony persists in spite of relatively rapid movement
towards the higher fitness patch. This is due to the significant lag in
consumer population shift when both patches have very low (but
different) resource densities or very high, but different resource
densities. In both cases, the absolute difference in fitness can be
quite small. The lag is illustrated in Fig. 1D, which shows the fitness
difference between patches (scaled to the maximum possible
fitness difference) and the consumer distribution for 30 time units
of Fig. 1C, beginning at a time with nearly equal populations of
both species across the two patches. Note that a significant
redistribution of consumers is not visible until time 7681, although
the fitness difference between patches exceeds 5% of the maximum
possible difference 4.76 time units earlier. An almost even distribution of consumers across patches shifts to one having fewer
than 1% of individuals in patch 1 between t¼7685 and 7691.
Although MR4107 for the parameters in Fig. 1C, the maximum per
capita movement rate (approximately 2.5) that occurs on the
attractor is only about 40 times larger than the maximum per
capita growth rate within a patch (and is many orders of magnitude less than the theoretical maximum movement rate). If l is
increased to 1000 in this example (MR¼1.64 1025), perfect
synchrony is still not observed, and the maximum movement rate
on the attractor is still less than 100-fold larger than the maximum
per capita growth rate within a patch.
Fig. 2 shows the coefficient of variation in global densities of
each species and the correlation coefficient between patches for
both species for the range of l values shown in Fig. 1. As expected,
high correlations between patches are accompanied by an
3.1. A reference example
Here we consider a system having intermediate values of the
population dynamic parameters. The parameter set is: {r ¼1;
k¼1; C ¼1; h ¼3; b¼0.25; d ¼0.02; m ¼0.0005}. The value of d/b
is slightly less than 1/2 the stability threshold value given by
Eq. (2). The initial analysis examines how different values of the
fitness sensitivity parameter (l) affect the dynamics. The lowest
sensitivity considered is l ¼70, which produces a movement ratio
(MR) of approximately one. Dynamics were investigated for a
regular array of l-values up to l ¼350 (Fig. 1C; MR ¼3.7 107),
and also for a few larger values up to l ¼ 1000 (MR¼1.64 1025).
Alternative attractors were not observed for any value of l within
this range. There are simple anti-synchronized cycles in the two
patches, like those illustrated in Fig. 1A for relatively low movement rates (roughly 70 o l o125; MR roughly between 1 and 29).
Over this range there is very little variation in total consumer
density, and variation decreases as l increases. Fig. 1B illustrates
resource and consumer dynamics for l just above this zone of
asynchrony (l ¼130), where a longer period, larger amplitude
global consumer–resource cycle that is more similar to the singlepatch cycle is beginning to emerge. (The period of the cycle in
total consumer population size in this case is very close to that in
a single isolated patch.) The variation in total consumer density
begins to increase with l at this point.
High fitness sensitivities (roughly l 4150) are characterized by
irregular longer-period cycles in the total populations of both
Fig. 2. (A) Coefficients of variation (panel A) and between-patch correlation
coefficients (panel B) as a function of fitness sensitivity, for the populations of
both species in the system illustrated in Fig. 1.
P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
103
3.2. Proximity to the stability threshold
Fig. 3 illustrates the change in the coefficient of variation (CV)
of consumer density as a function of the fitness sensitivity
parameter for mortality rates that are one-half (Fig. 3A) and twice
(Fig. 3B) of those in Figs. 1 and 2. The mortality rate in Fig. 3B is
only 4% less than the value at which the equilibrium becomes
stable. In both of the cases considered in Fig. 3, alternative
attractors exist for a wide range of l-values, so the observed CV
depends on initial densities. The lower line in both figures reflects
anti-synchronized dynamics similar to those shown in the top
panels of Fig. 1. For this attractor, the CV of consumer densities
decreases as l increases because both the period and amplitude of
the back-and-forth cycles of consumers decrease. The attractor
corresponding to upper line in Fig. 3A has complex dynamics with
a positive temporal correlation between the population sizes of a
given species in the two patches. Fitness sensitivities greater than
the largest shown in Fig. 3A continue to have positively correlated, but asynchronous fluctuations in the two patches, with no
strong trend in the CV, even for l double the maximum value
shown in the figure. The upper line in Fig. 3B represents two
alternative attractors in which one of the two patches has more
variable populations of the two species than does the other.
Because the patches are identical, symmetry implies that there
must be an alternative ‘mirror image’ (MI) attractor for which the
identities of the higher and lower variability patch are reversed.
One of these two for l ¼200 is illustrated in Fig. 4B. The
qualitative form of dynamics in Fig. 3B changes when l exceeds
Fig. 3. Coefficients of variation of consumer density in systems that are similar to
those illustrated in Fig. 1 and 2 except that d ¼ 0.01 in panel A, and d ¼0.04 for
panel B. The two lines in each case reflect alternative attractors, with the bottom
line representing a case of anti-synchrony, similar to that illustrated in the top
panels of Fig. 1.
increased coefficient of variation (CV) in total abundance of each
species (lower stability). The anti-synchrony at low l is reflected
in large negative correlation coefficients of populations in different patches and in the greatly reduced global variation in total
density, particularly at the consumer level. The coefficients of
variation of consumer and resource densities change very little for
values of l larger than those shown; all are slightly less than the
CVs for a perfectly synchronized system (0.349 and 1.359 for
consumer and resource, respectively). Thus, very high l values
produce stability and synchrony measures close to those of a
system with random movement, although the population trajectories within a patch usually differ considerably.
The qualitative change in dynamics with increase in fitness
sensitivity illustrated by Figs. 1 and 2 is the one most commonly
observed in simulations of Eq. (1) using low baseline movement
rates (m). However, different population dynamics parameters in
some cases changed these patterns. We next explore the impact
of changing three population growth parameters that alter the
three important determinants of cycle form listed in the previous
section. We alter d in the system as shown in Figs. 1 and 2 so that
it is either closer to or further from the stability threshold,
resulting in smaller or larger population fluctuations. We change
b and d proportionally to examine the impact of the relative speed
of consumer and resource dynamics. We explore smaller and
larger handling times to examine the impact of the maximum
saturation of the consumer’s functional response.
Fig. 4. (A) Resource densities and (B) consumer densities; both for the system
used in Fig. 3B (d ¼0.04) when l ¼ 200. The solid line denotes a population in patch
1, the short dashed line is for patch 2, and the long-dashed line in panel B is the
mean consumer population
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P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
290 (MR¼1.65 106). At this point the lower (anti-synchronized)
attractor looses stability, and the two MI attractors represented
by the upper line in Fig. 3B coalesce into a single attractor with
both populations undergoing identical synchronized population
cycles in each patch; this persists for larger l.
Simulations over a range of l were carried out for numerous
other values of the death rate d with the remaining parameters
identical to those in Fig. 1. Patterns observed for this set of
simulations were: (1) anti-synchronized attractors with very low
variation in total densities occurred for a significant range of low
to intermediate l for all death rates except those that were
extremely close to the stability threshold; (2) alternative attractors occurred for the widest range of l for high and low mortality
rates within the unstable range; (3) the maximum l for which a
low-variability anti-synchronized attractor existed was also the
greatest at relatively high and low death rates (up to l ¼290
for d ¼0.04 and l ¼330 for d ¼0.005); (4) high enough fitness
sensitivities l always led to dynamics with high positive correlation in the resource densities between patches; and (5) perfect
synchrony was only observed at very large movement ratios, or
when the consumer death rate was very close to the stability
threshold.
3.3. Relative speed of consumer and resource population dynamics
The conversion efficiency parameter used in the preceding
sections (b ¼0.25) implies that the maximum per capita growth
rate of the consumer is roughly one-twentieth that of the
resource. We investigated values of b ranging from 0.01 to 10,
with d values corresponding to the same percentage of the
stability threshold as in Figs. 1, and 3A and B. All other parameters
were identical to the values shared by these figures. These
changes do not alter the equilibrium points, but do change the
nature of population fluctuations. Systems with b40.25 in most
cases have the same sequence of changes in qualitative dynamics
as a function of l. However, more rapid consumer dynamics
means: (1) a wider range of movement ratios (but often not a
wider range of l) produces anti-synchrony; (2) a smaller range of
MRs are characterized by alternative attractors; and (3) fewer
parameter sets with reasonable values of MR produce perfect
synchrony. In general, slower population dynamics reduces the
range of MRs characterized by anti-synchrony. In fact with
b¼0.01 and d¼0.0016 (4% below the stability threshold), perfectly synchronized dynamics occur for all l for which MR 41.
When consumer dynamics are slower, the death rate, d, must be
further from its stability threshold value (implying larger amplitude cycles in an isolated patch) for anti-phase cycles to occur
over some range of l values.
Slow population dynamics can also change the simple
anti-synchrony seen in the previous sections to anti-phase complex cycles; this is shown in Fig. 5A for b ¼0.1, l ¼116, and d 4%
below the stability threshold. Larger values of l in this case
produce smaller negative correlations with higher variance in
consumer density, although the patches are still antisynchronized. Fig. 5B shows the dynamics for the same system
when l ¼400 (MR¼1224); this has high positive correlations
between patches in the densities of each species although the
dynamics in the two patches are still phase shifted by 1801.
Perfect synchrony is not achieved until approximately l ¼870
(MR¼1.55 108). Systems having b ¼0.1 with death rates farther
below the stability threshold exhibit simple anti-synchronized
cycles like those shown in the top panel of Fig. 1 for low-tomoderate movement ratios, and complex cycles with positively
correlated densities for larger MRs. There is often a range of l
values producing alternative attractors, and perfect synchrony is
often not observed for realistic MRs.
3.4. Handling time
We present results for two of the handling times we examined: a relatively large value (h¼10) and a relatively small value
(h ¼1.5). In these and other cases, we ran simulations across a
Fig. 5. Left column: Resource densities, Right column: Consumer densities (A) l ¼ 116 and (B) l ¼ 400. The solid line represents patch 1 and the short dashed line
represents patch 2; the long dashed line for consumers is the average consumer density.
P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
105
Fig. 6. (A) Consumer (dashed) and resource (solid) CVs (B) Between patch correlations of consumers (dashed); resources (solid), (C) N1 (dashed); N2 (solid) for l ¼ 240 and
(D) N1 (dashed); N2 (solid) for l ¼ 500.
wide range of movement sensitivities for systems that were at
different distances from the stability threshold, beginning with d
approximately 4% below the stability threshold to match
the Fig. 3B example. The fitness sensitivity and other parameters
corresponded to those in Fig. 1.
With high handling time, h¼10, and d¼0.0196 (4% below the
stability threshold d), there were anti-synchronized cycles for low
fitness sensitivities and positively correlated cycles for higher
values. Fig. 6A and B shows the resulting coefficients of variation
of total population size and between-patch correlations as a
function of l. We did not find any parameters with alternative
synchronized and anti-synchronized cycles. However, there were
many cases in which there were alternative mirror-image pairs of
asymmetrical cycles. Two examples of the latter are illustrated in
panels C and D of the figure. Perfect synchrony was not an
attractor until l exceeded a threshold between 1100 and 1200
(MR41010). The transition to positive between patch correlations
of the consumer occurs at approximately l ¼300 (MR¼146). The
range of l values producing anti-synchronized cycles expands as
d is lowered. Alternative attractors (other than mirror image
pairs) were rare when h was relatively large.
Smaller handling times generally favor synchrony. A handling
time 41 is needed for cyclic dynamics to exist for any d, given
the other parameters assumed here. If h¼1.5 (given C¼r ¼k¼ 1),
d ¼0.032 represents a value 4% below the stability threshold; in
this case, perfectly synchronized cycles were the only dynamics
observed for all l with MR41, given m¼0.0005. The more
common pattern of anti-synchrony followed by positively correlated cycles as l increases emerged for lower death rates; there is
also often a range where both types of attractors exist as
alternatives. This occurs, for example when d¼0.016; anti-synchrony is the only attractor for l up to 125, and exists as one of
the two alternative attractors for 125o l o145 (1597 oMRo
11,802). Perfect synchrony, when it is possible, usually requires
unrealistic movement ratios.
3.5. Summary of results for low baseline movement in Eq. (1)
The above analysis has basically explored one-dimensional
lines through the space of potential population dynamic parameters, and many additional parameter combinations are possible. We have analyzed systems with various combinations of the
features explored above, such as different speeds of consumer
dynamics in systems with larger handling time. Different values
of C, r, and k were simulated, but under similar circumstances;
these produced dynamics qualitatively similar to those described
above. The important features are: (1) anti-synchronized
(approximately 1801 out of phase) simple cycles in the two
patches with a relatively short period and low variation in total
density were the most common dynamic pattern at low-tomoderate fitness sensitivities; there were spatially correlated
but imperfectly synchronized predator–prey cycles at high fitness
sensitivity; (2) alternative attractors were common at intermediate fitness sensitivities; (3) perfect synchrony was not a common
outcome, but was observed in several different circumstances,
either at very high fitness sensitivities, very slow consumer
dynamics, or in systems that were either close to the stability
threshold or had a value of Chr/k close to the limiting value of one.
3.6. Summary of alternative movement parameters and functions
Appendix B considers the impact on dynamics of different
baseline movement rates, m, and of several alternative movement
or population dynamics functions. It shows that sufficiently large
baseline movement rates make perfect synchrony the dominant
form of dynamics by effectively homogenizing the system. The
main alternative movement functions considered are: (1) ones in
which the per capita movement rate approaches an asymptote at
relatively large fitness differences (‘‘constrained rate’’ movement); (2) ones in which movement rates decrease to zero when
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P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
the other patch confers equal or lower fitness (‘‘no-error’’ movement); and (3) ones in which only fitness in the current patch
influences movement (‘‘local fitness’’ movement). These alternatives produce some qualitative differences in the patterns of
dynamics with different movements. Constrained rate movement
is more likely to produce anti-synchrony, and no-error movement
is less likely to produce either anti-synchrony or perfect synchrony than is the movement function in Eq. (1). Alternative
attractors under the no-error model are usually mirror image
attractors with positive between-patch density correlations. Local
movement rules produce alternative anti-synchronized and perfectly synchronized attractors in many cases, but are more likely
to produce only perfectly synchronized dynamics when there was
a relatively high rate of movement from a patch having a per
capita growth rate of zero. Appendix B should be consulted for
more details on these and other models.
3.7. Effects of random and adaptive movement on system-level
properties
Random movement produces perfectly synchronized dynamics
for all of the parameter combinations reported above for Eq. (1),
when the per capita random movement rate is given by the
baseline m¼0.0005 value of the adaptive function; synchrony is
also produced by all other m values up to m¼0.5 (higher values
were not examined because m¼0.5 made the consumer distribution nearly uniform across patches, regardless of the resource
levels). Thus, for consumer–resource systems like those considered here, asynchrony of patches under adaptive movement is
much more likely than under random movement.
The different dynamics under random and adaptive movement
lead to differences in system-level properties, such as the response
to enrichment or harvesting of either the resource or the consumer.
Both harvesting and enrichment of the resource can be modeled by
changes in the maximum resource per capita growth rate, r.
Increasing r (enrichment) increases the rate of resource dynamics
relative to those of the consumer, increases instability within a
patch, and increases the maximum saturation level of the consumer’s functional response. Thus, the pattern of change in dynamics
with change in r combines the three types of parameter variation
considered in the main analysis of Eq. (1). A full treatment of the
impacts of adaptive movement on response to fertilization is beyond
the scope of this article. However, we will present one example of
the impact of increase in r using the system illustrated in Figs. 3B
and 4. Under random movement, the two patches become synchronized, and consumer variability increases with r due to increase in
amplitude of the underlying predator–prey cycles (Fig. 7A). Fig. 7B
assumes adaptive movement based on Eq. (1) with l ¼ 100; here, the
two patches have negative density correlations over the same range
of r illustrated in Fig. 7A. The negative correlation produces a much
lower coefficient of variation in N. (Note that the y-axis of Fig. 7B
spans only 1/4 the range of Fig. 7A.) The CV of consumer density is
proportionally more sensitive to r under adaptive than under
random movement, and its direction of change with r is no longer
a monotonic increase. In Fig. 7B, the low part of the curve between
r¼1 and 1.9 consists of simple anti-synchronized cycles where the
predator–prey oscillation is suppressed. Anti-synchronized cycles
return at r¼3, but have a much longer period, and the CV then
increases rapidly with further increases in r.
The lower variation in the system with adaptive movement
also implies that the temporal mean consumer density is greater
with adaptive than with random consumer movement. At r ¼1,
there is only a 1% difference in density between randomly and
adaptively moving consumers, but for 3 or o5, an adaptively
moving consumer achieves a mean population density
Fig. 7. Coefficient of variation of total consumer population size vs. resource
intrinsic growth rate, r, for the population dynamics model used in Figs. 3B and 4.
(A) Random movement with m¼0.0005 (any rate producing synchronized
patches yields the same result). (B) Adaptive movement according to Eq. (1) with
m¼0.0005 and l ¼100. Note that the y-axis covers a much smaller range in
panel B
approximately 50% greater than that of a randomly moving
consumer. This also means that harvesting the resource in a
2-patch metapopulation with a large initial r-value will produce a
much greater absolute decrease in the consumer population in a
system with adaptive than with random consumer movement.
The difference between random and adaptive populations is even
larger for mean resource densities, with random movement
leading to an approximate 2.4-fold greater mean resource density
when r ¼5, but less than a 1% difference when r ¼1. In this same
example, if the initial r is 5, harvesting consumers (increasing the
death rate from 0.04 to slightly less than the stability threshold of
d¼0.073) produces a paradoxical increase in consumer density
under both types of movement (the ‘‘hydra effect’’; Abrams,
2009). There is a much greater increase (and a much greater
proportional increase) in consumer density over most of this
range of mortalities when movement is adaptive than when it is
random. There are likely to be many other system-level responses
that differ under random and adaptive movement.
4. Discussion
It has long been known that adaptive movement can alter
metapopulation dynamics, even when only a single species is
involved. However, these differences have only been noted in
single-species models when patches differ from one another
(Hastings, 1982; Holt, 1985). The common occurrence of various
P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
types of asynchronous dynamics in 2-species systems with
homogeneous patches demonstrated here shows that the impacts
of adaptive movement on population dynamics are likely to be
greater and more varied in systems with interacting species than
in single-species systems. Asynchronous dynamics have been
found in several lab studies of simple metacommunities having
similar patches (e.g., Holyoak and Lawler, 1996; Janssen et al.,
1997), but the cause has not been clearly identified. Modeling
studies (e.g., Abrams et al., 2007; Křivan et al., 2008) have shown
that instability occurs in some metacommunity models of two
competing species, and Abrams (2007) has shown that it is also
possible in 3-level predator–prey–resource systems that incorporate adaptive movement of both predator and prey. However, the
present work suggests that spatial heterogeneity in fitness also
arises from the simplest homogeneous-patch consumer–resource
(predator–prey) systems having adaptive consumer movement. In
fact, inequality between patches is far more likely to arise in such
a homogenous patch system when consumers actively move
towards the patch having the higher resource density than it is
for systems in which movement is random. Adaptive habitat
choice may create rather than remove spatial differences in
consumer fitness. Huang and Diekmann (2001) have shown that,
if predators do not move when handling prey, fluctuations are
never significantly decreased relative to the synchronous system.
This type of movement could be considered adaptive movement,
as high prey densities will lead to more time spent handling. It
suggests that much more needs to be done to understand the
range of effects of different movement rules.
In the models we have discussed, the asynchrony due to
adaptive movement in most cases reduces the global instability
(coefficient of variation of global population size) for both species
in the consumer–resource system, relative to a similar system in
which there is a low rate of random movement. When there are
anti-synchronized patch dynamics, the CV of total consumer
population size may be two or more orders of magnitude less
than the CV in a single isolated patch or in a two-patch
system having random movement between patches. This stabilization occurs primarily because the dynamics suppress the
normal consumer–resource cycle. Analogous types of anti-synchronized dynamics occur in three-patch systems, and also occur
in two-patch models with heterogeneous patches (Ruokolainen
et al., submitted for publication). Even when dynamics are not
perfectly out of phase, most cases of asynchronous dynamics
produce a lower CV in total consumer density than what is
observed in a single patch or more patches coupled by random
movement.
Asynchronous dynamics do not occur in cases with a high
baseline movement rate, m, which decreases the effective time lag
between the point when the identity of the more-rewarding
patch changes, and when a significant number of consumers have
moved. Models of switching behavior of consumers between two
different resources within a single patch (Abrams, 1999; Abrams
and Matsuda 2003, 2004) have shown the same key role of time
lags in preventing synchronization.
It is difficult to generalize about the impact of habitat selection
on system-wide dynamics. Very rapid and accurate movement
(high fitness sensitivity l) usually leads to positive between-patch
density correlations and high variation in total population sizes of
both species. However this is generally the result of a transition
from anti-synchrony to positively correlated cycles over a narrow
range of l. Higher l actually leads to lower CVs in the antisynchronous range, and leads to irregular changes within the
positively correlated range. Depending on the initial l, a higher
value can move the system into or out of a region with alternative
attractors. In a stochastic environment, switching between attractors could make total densities most variable within the
107
intermediate range of l having alternative attractors, but this
requires additional analysis.
Adaptive movement often produces systems having very
different global properties than similar systems in which movement is random. We have provided an example in which the
relative stability of systems with adaptive movement translated
into much greater densities of the consumer and a much greater
sensitivity of the consumer to either harvesting or enrichment of
the resource. Abrams et al. (submitted for publication) show that
patch-specific harvesting of consumers in 2-patch consumer–
resource systems has different impacts depending on whether
movement is adaptive or random. It is likely that responses of
community dynamics to environmental changes or species additions will differ significantly in most multispecies metacommunities having adaptive rather than random movement by any
major subset of species.
Much evidence suggests that movement is broadly adaptive in
many species and systems, but there is generally insufficient
evidence to determine functional forms. The results presented
here show that the dynamics of a consumer–resource system can
differ significantly with changes in the form and/or parameter
values of the function describing adaptive movement. We hope
that our demonstration that the details of movement decisions
can greatly alter system dynamics will spur more empirical work
on the dynamics of adaptive movement in systems with two or
more species in two or more patches.
The work presented above needs to be extended in several
directions. Systems with more patches and more species are
obviously of interest. We have not examined transient dynamics
here. Doing so would be of interest because Goldwyn and
Hastings (2008) show that long-term transients involving asynchronous dynamics often exist in 2-patch consumer–resource
systems with random movement. We have concentrated on
systems in which only the top trophic level moves. Koelle and
Vandermeer (2005) and Abrams (2007), respectively, found
strong effects of movement by bottom- and mid-level species
on stability in 3-trophic-level models. Investigating larger food
webs and models with explicit and/or continuous space is likely
to uncover additional differences between the community-level
effects of random and adaptive movement.
Acknowledgments
This work was supported by a Strategic Project Grant and a
Discovery Grant from the Natural Sciences and Engineering
Research Council of Canada. We thank the reviewers, B.J. Shuter,
A.J. Golubski, and P. Amarasekare for comments on earlier
versions of this work.
Appendix A. Numerical methods
Most of the numerical integration was carried out using the
NDSolve command in Mathematica 7.0.1 (Wolfram Research,
2009), with an Accuracy Goal setting of Infinity. In some cases
the ability to integrate successfully depended on initial values of
the variables. A selection of cases were checked using a FORTRAN
routine, RADAU5 (Hairer and Wanner, 1996). The standard run
was for 8000 time units and average densities were obtained for
the last 4000 time units. These durations were lengthened as
needed when there were long-lasting transient dynamics or when
the dynamics underwent long-period cycles or chaos. For very
long period cycles, the period was determined and average
densities were obtained for exactly one period. Simulations
investigating dynamics as a function of the fitness sensitivity
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P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
parameter, l, generally used an interval of 5 for successive values
of the parameter; a wide range of initial densities was used for l
values corresponding to MR ¼1 and for a very large MR, and
attractors were followed as l was either increased or decreased
by small intervals using a slightly perturbed set of final values for
the previous l as initial densities for the next l.
Appendix B. Different movement parameters and functions
Different baseline movement, m, in Eq. (1)
The standard baseline per capita movement rate, m¼0.0005,
means that movement between two patches with equal resource
density is almost two orders of magnitude smaller than the per
capita death rates. Many systems are likely to have more frequent
movement under these conditions. If baseline movement is
increased 100-fold, m ¼0.05, the majority of parameter sets
described in text only have a single attractor with perfect
synchrony. However, this movement rate is greater than the
maximum per capita growth rate within a patch for the parameters used in Figs. 1–3, and it produces similar numbers of
consumers in both patches, even when one patch is a sink. If m
has the intermediate value of 0.005, the occurrence of perfect
synchrony is increased (relative to 0.0005), but anti-synchrony
and various forms of asynchrony are still reasonably common.
When m ¼0.005 and other parameters are as in Figs. 1 and 2, the
transition from anti-synchrony to positively correlated cycling
occurs at a much lower movement ratio (15.4 rather than 138.7).
Perfect synchrony is achieved at a lower l and MR than
in Figs. 1 and 2, but it still requires a l close to 400 (MRE109).
The synchronizing influence of a larger m has a larger impact
when the cyclic tendency within a patch is weaker. Thus, if d is
increased to 0.04 (with m¼0.005 and other parameters as
in Fig. 3B), a perfect synchrony is observed for all l values
representing MR41. Lowering m ten-fold to 0.00005 increases
the occurrence of anti-synchrony for parameters near the stability
threshold, but in general has relatively small effects on the types
of dynamics and the MR values at which they are observed.
Increase in m means more movement under all conditions.
Random movement can also be increased by adding a random
movement term without changing the value of m in the adaptive
movement function of Eq. (1). This increases random movement
equally for all magnitudes of fitness difference. It has less of a
synchronizing impact than does increasing m for parameters
producing the same movement rate at fitness equality (i.e.,
m0 ¼uþ m, where m0 is the increased value of the baseline movement parameter). Such independent random movement reduces
the occurrence of anti-synchrony considerably, but does not
greatly increase the occurrence of perfect synchrony. For the
system considered in Figs. 3B and 4 (d ¼0.04), adding random
movement at a rate u ¼0.0045 (so that m0 ¼0.005) produces
complex positively correlated cycles with quasi-periodicity for
most of the parameter range producing anti-synchronized cycles
in the absence of the random movement. Perfect synchrony
occurs at a somewhat lower l than in the system lacking random
movement (l ¼ 210 rather than 290). This contrasts with perfect
synchrony at all l when m is simply raised to 0.005 in the
adaptive movement function. Nevertheless, some synchronizing
effect of added random movement was seen across all of the
parameter sets we explored.
Constrained rate movement
The movement formula in Eq. (1b) can produce an extremely
high rate of movement when both l and the fitness difference
between patches are large. There is likely to be a maximum rate of
transition between patches. This can be accommodated by changing the rate of movement from patch i to j from the one
discussed above, to
mExp½lðWj Wi Þ
,
1þ amExp½lðWj Wi Þ
ðB:1Þ
where a is a small (5 1)positive constant. The movement rate
approaches a maximum of 1/a as l(Wj–Wi) becomes large, and is
referred to as a ‘‘constrained rate’’ movement function. When the
fitness difference is small, this has little impact on the movement
described in the previous model, as the exponential term in the
denominator will be small relative to 1. The previous model
represents a special case of Eq. (B.1) in which a ¼0. Even quite
small values of a greatly increase the range of other parameters
that produce anti-synchronized dynamics. If the example in Fig. 1
is changed to incorporate movement function (B.1) with a ¼0.01,
the maximum movement rate is 100 (which is still much greater
than the maximum per capita birth minus death rate of 0.0425).
However, this changes the dynamics by greatly expanding the
range of movement sensitivities that permit anti-synchronized
dynamics (up to l 41350), and creates a wide range of parameters (very roughly 400 o l o1400) where alternative attractors
exist, the anti-synchronized attractor, and a single or two mirror
image attractors, each with highly positively correlated cycles in
the two patches. Larger values of m again tend to produce
synchronous cycles, while larger a favors anti-synchronized
cycles. This is in part due to the lower absolute movement rates
implied by a nonzero a.
No-error movement
Here, we assume that movement to a poorer quality patch is
impossible. Cressman et al. (2004) investigated some of the
properties of such ‘no-error’ rules. A continuous function having
this property also implies no movement between patches conferring identical consumer fitness. A function with these properties is
(
)
mNi ðexpðlðWj Wi ÞÞ1Þ if Wj 4 Wi
Mij ¼
ðB:2Þ
0
if Wi Z Wj
The dynamics produced for the reference example in Fig. 2
(d ¼0.02) are very similar in terms of CVs and between-patch
correlations, although the transition to higher variance and
correlation occurs at slightly lower values of l and the MR. When
d¼0.04, movement function (B.2) also produces a set of dynamics
that is very similar to the comparable system illustrated
in Figs. 3B and 4 in the text as l is increased. The main difference
is that the transition to perfect synchrony requires a somewhat
higher MR in this no-error model.
Many other potential movement functions share (with
Eq. (B.2)) the property that mistakes (movements that reduce
fitness) never occur. The exponential function in Eq. (B.2) can be
replaced by any increasing, accelerating function of (Wj–Wi) to
produce adaptive movement having the desired property that
movement rates increase faster than linearly with the fitness
benefit of moving. An alternative in which the fitness difference is
raised to a power Z2 was discussed briefly in the appendix
of Abrams (2007). The movement from patch j to patch i is then
mNi(Wi–Wj)z (where zZ2) if Wi oWi, and is zero if the inequality
is reversed. This produces the same range of dynamics as Eq. (B.2),
although the MR values producing anti-synchronized dynamics
are somewhat smaller, particularly when z¼2. Perfect synchrony
can occur, but it often requires higher movement ratios than in
the model discussed in the text. Slightly asymmetrical mirror
P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
image attractors with positive correlations commonly occur at
moderately high MR.
Local fitness movement
Another alternative movement function is based on the assumption that only local conditions can be detected by the consumer. This
requires eliminating the Wj from the formula for the movement of
consumers in patch i in Eq. (1b), producing a function similar to the
one used by Ives (1992). In this case, it is reasonable to assume a
relatively low movement rate when conditions are good enough
that births balance deaths, but movement should go up rapidly
when the resource becomes scarce. The maximum movement rate
out of patch i occurs when fitness (instantaneous per capita growth
rate) is equal to di. In calculating a movement ratio, the movement
rate corresponding to this case is compared to the maximum per
capita growth rate (that occurs when R¼r/k) within a patch. This
alternative model produces some changes in the dynamics of the
system. It is clear that when both patches have little resource, there
will be frequent movement between patches, which is not the case
when m is small in Eq. (1b). Nevertheless, anti-synchronized
dynamics also occur in this system. In some systems with relatively
low per capita movement rates, dynamics are still anti-synchronized
(and the normal predator–prey cycle is suppressed), even at the
highest movement rates that are realistic. For example, if the
example in Fig. 3B is changed to eliminate the effect of the other
patch on movement, an anti-synchronized attractor exists for the
entire range of biologically reasonable l values (1oMRo1010). A
perfectly synchronized attractor also exists for l Z190 (MRZ44).
Similar alternative attractors are observed when the Fig. 3B example
is changed so the consumer has slower dynamics (b¼0.1, d¼0.016);
109
again the anti-synchronized attractor exists for all l values, while
perfect synchrony is an alternative for MRZ75. Parameters further
from the stability threshold are characterized by the loss of the antisynchronous attractor at smaller l. Systems with a larger handling
time often have an asynchronous attractor with high positive
correlations between patches as an alternative to the anti-synchronized attractor. In all systems, anti-synchrony becomes rare or
absent when the movement constant, m, is large enough. However,
the general features of the attractors and how they change with
parameters are roughly similar to the model with movement based
on between-patch comparisons. Fig. B1 presents some examples of
dynamics for a system similar to that in Fig. 3B, but with a slightly
lower death rate. Here the period of the anti-synchronize cycle is
much shorter than for the comparable examples based on Eq. (1b),
and the corresponding variation in total density is much smaller.
Alternative synchronized and anti-synchronized attractors exist for
quite a wide range of l, from approximately 150 to 490.
Other variations on Eq. (1)
Two-patch systems may have unusual properties because of
the fact that emigrants have only one place to go. We have run a
limited number of simulations of 3- and 4-patch systems. These
assumed that the patch leaving decision was based on local
conditions only, and examined cases in which the emigrants from
a patch were divided evenly between the other patches. Parameter sets that displayed anti-synchronized dynamics in the
2-patch models often led to the equivalent (each patch 1201 out
of phase with each other) in the 3-patch case. 3-patch dynamics
also included cases in which two patches were perfectly
Fig. B1. The dynamics of a system in which movement is based on local patch conditions only, using Eq. (1b) after removing the fitness of the unoccupied patch from the
movement function. Parameters are as in Fig. 3B, except that mortality, d, is 0.035 rather than 0.04. Panel A shows the coefficient of variation for both the synchronous (top
line) and the anti-synchronous (lower line) attractors across a range of l; the anti-synchronous attractor loses stability above l ¼490, while the synchronous one persists
for l 4500. Panel B is an example of consumer (left hand side) and resource (right hand side) dynamics for the anti-synchronous attractor (l ¼ 125). The solid line is the
population in patch 1, the short dashed line is the patch 2 population, and the long-dashed line in the consumer graph is the mean population across both patches.
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P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99–110
synchronized and the third was out of phase with these two.
Clearly, there is more scope for alternative attractors with three
patches.
A final variant of the main model adds random movement by the
resources. A high enough resource movement rate leads to perfectly
synchronized consumer (and resource) dynamics for most parameter sets, whether or not the consumer moves adaptively or
randomly. However, resource movement at low-to-moderate rates
does not always eliminate the asynchronous dynamics caused by
adaptive consumer movement, although it generally reduces the
range of l values yielding anti-synchronous dynamics. As shown
in Abrams (2007), adaptive (fitness-dependent) movement by
resources usually promotes anti-synchrony in two-patch models
when the predator has a saturating functional response.
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