We compared the metapopulation dynamics of predator-prey systems with (1) ... strong stabilizing influence on both within-patch and metapopulation dynamics.
Evolutionary Ecology, 1993, 7, 3 7 9 3 9 3
Optimal patch use and metapopulation dynamics J . M . F R Y X E L L 1. and P. L U N D B E R G 2 1Department of Zoology, University of Guelph, Guelph, Ontario, Canada NI G 2W1 2Department of Animal Ecology, The Swedish University of Agricultural Sciences, S-90I 83 Urne4, Sweden
Summary We compared the metapopulation dynamics of predator-prey systems with (1) adaptive global dispersal, (2) adaptive local dispersal, (3) fixed global dispersal and (4) fixed local dispersal by predators. Adaptive dispersal was modelled using the marginal value theorem, such that predators departed patches when the instantaneous rate of prey capture was less than the long-term rate of prey capture averaged over all patches, scaled to the movement time between patches. Adaptive dispersal tended to stabilize metapopulation dynamics in a similar manner to conventional fixed dispersal models, but the temporal dynamics of adaptive dispersal models were more unpredictable than the smooth oscillations of fixed dispersal models. Moreover, fixed and adaptive dispersal models responded differently to spatial variation in patch productivity and the degree of compartmentalization of the system. For both adaptive dispersal and fixed dispersal models, localized ('stepping-stone') dispersal was more strongly stabilizing than global ('island') dispersal. Variation among predators in the probability of dispersal in relation to local prey density had a strong stabilizing influence on both within-patch and metapopulation dynamics. These results suggest that adaptive space use strategies by predators could have important implications for the dynamics of spatially heterogeneous trophic systems. Keywords: dispersal; marginal value theorem; predator-prey dynamics; spatial structure; stability
Introduction Simple models of predator-prey dynamics tend to be inherently unstable over many, if not most, parameter combinations (May, 1972; Murdoch and Oaten, 1975; Tanner, 1975). A large body of theoretical work has been devoted to identifying factors that might tend to stabilize food web dynamics. Much of this work relates to the role of spatial complexity and/or heterogeneity on predator-prey interactions. Several types of models have been utilized to explore the role of spatial complexity on community dynamics (Taylor, 1988; Kareiva, 1990). One approach has focused on 'ideal-free' (Fretwell and Lucas, 1970) distribution of individuals among a variety of habitats that differ in quality (reviews in Pulliam and Danielson, 1991; Rosenzweig, 1991). In cases where some habitats are inadequate for subsistence, optimal dispersal patterns can be strongly stabilizing (Pulliam, 1988; Pulliam and Danielson, 1991). This approach has also suggested interesting mechanisms of competitive coexistence, due to fitness-maximizing habitat segregation (Rosenzweig, 1981, 1986; Abramsky et al., 1991). Similar arguments have been used to explain the stabilizing role of refugia from predators, when use of refuge habitats is density-dependent (Rosenzweig and MacArthur, 1963; McNair, 1986; Sih, 1987). * To whon-I correspondence should be addressed. 0269-7653
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A second approach to spatial complexity has been to model spatially-explicit subpopulations of predators and prey (termed 'metapopulations') that are linked either locally ('stepping stone models') or globally ('island models') in a larger network (e.g. Maynard-Smith, 1974; Hilborn, 1975; Hastings, 1977; Ziegler, 1977; Crowley, 1981; Nachman, 1987; Reeve, 1988). Alternatively, reaction-diffusion models have been used to model predator-prey dynamics in a homogeneous and continuous environment, while still restricting trophic interactions to a local spatial scale (e.g. Comins and Blatt, 1974; Mimura and Murray, 1978; Turchin, 1989). Finally, the role of covariation in local parasitoid and host distributions has been examined, without reference to explicit-spatial structure and mechanisms of dispersal (May, 1978; Chesson and Murdoch, 1986; Pacala et al., 1990; Hassell et al., 1991). A common theme emerges from these theoretical studies: spatial heterogeneity in predatorprey distributions induced by limitations on dispersal capabilities and non-random distributions of predators and/or prey often tends to have a stabilizing effect on metapopulation dynamics (Taylor, 1988; Hassell and Pacala, 1990; Kareiva, 1990). In this context, stabilizing refers to decreased amplitude of population fluctuations over time. However, most preceding studies of metapopulation dynamics neglect at least one aspect of the behaviour of adaptive predators - the potential for energy-maximizing strategies to influence dispersal behaviour. Optimality models suggest that patch residency should be strongly influenced by instantaneous rates of energy gain relative to expected rates of long-term gain elsewhere in the immediate environment (Charnov, 1976; Arditi and D'Acorogna, 1988) and many animals exhibit patch use behaviour consistent with these models (Stephens and Krebs, 1986). The only explicit theoretical consideration of adaptive patch use (on a non-renewable resource base) by predators in a spatially complex model showed that asymmetry in predator and prey distributions is a likely consequence of adaptive dispersal behaviour by predators with pronounced costs of interpatch travel (Bernstein et al., 1991). This suggests that predator dispersal, mediated by energy-maximizing behavioural strategies, could tend to stabilize trophic interactions in a spatially-complex environment. We explored this possibility using spatially-explicit models of subpopulations of mobile predators and stationary prey, in which the probability of predator dispersal was dictated by the marginal value theorem and local dynamics followed a standard Lotka-Volterra formulation. This approach extends the adaptive dispersal model of Bernstein et al. (1991) to a situation in which both predators and prey exhibit continuous renewal, the rate of which is contingent on local densities of each subpopulation. We compared attributes of our adaptive patch use model with a null metapopulation model with a constant fraction of dispersers from local predator populations. Furthermore, we compared the behaviour of metapopulation models with local vs global links among patches. This approach should yield information on both the role of adaptive dispersal as well as the impact of the degree of linkage among patches in the environment. Models
We modelled metapopulation dynamics according to four different scenarios: (1) adaptive dispersal with local links among subpopulations, (2) adaptive dispersal with global links, (3) fixed-fraction dispersal with local links and (4) fixed-fraction dispersal with global links. In all four cases, the internal dynamics within patches followed a common model structure, with population growth rates of predators (N) and prey (R) conforming to modified Lotka-Volterra equations. Because use of the marginal value theorem requires an explicit consideration of the time cost of dispersal between patches, we used an Euler approximation to the continuous
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Lotka Volterra equations. Internal dynamics within patch i were therefore calculated according to the following set of equations: gt
= rRi(1 - Ri/Ki)
gNi = [3iNi(cX i gt
-
-
[3iNiXi
d) + I i -
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where r is the maximum per capita recruitment rate of prey, K the prey carrying capacity, [3 the proportion of the initial predator population remaining in the patch, X the number of prey consumed per predator, c a proportionality constant converting per capita intake by predators to per capita birth rate by predators, d the predator mortality rate, E the emigrants from the patch, I the new predators immigrating into the patch and At is the reciprocal of the number of time intervals calculated for each generation. The number of emigrants was calculated as (1 - [3i) Ni. We further assumed that predators exhibit a type II functional response to changes in prey density: X i = aRi/(1 + ahRi)
where a is the encounter rate with prey and h is the mean time required to handle (consume and process) a single prey item. Emigrants from a given patch formed a 'cohort' that was specified to be in transit for m a t generations, where m was arbitrarily defined as an integer between one and five, depending on the length of time it took migrants to locate a new patch. For example, for At = 0.1 and m -- 5, migrants were specified to be in transit for 0.5 generations before finding a new patch. In the case of global dispersal, emigrants from all patches were pooled, whereas emigrants retained an explicit spatial identity in the case of local dispersal. While in transit, migrant predators did not consume prey, did not reproduce and suffered the same mortality rate (-d2it) as predators resident in patches. The number of immigrants to a given patch was specified differently according to each dispersal scenario. In the case of global dispersal, the number of immigrants arriving at each patch was calculated by dividing the total number of migrants exiting the pool by the total number of patches. Hence, each patch had an equal probability of receiving immigrants from the common pool. In the case of local dispersal, emigrants from a given patch were uniformly distributed among the adjacent patches. Patches were identified using Cartesian coordinates. For patches in the interior of the environment, this implied uniform movement in four cardinal directions, whereas movement was constrained to three directions for edge patches and two directions for corner patches. The proportion of individuals remaining resident within a given patch ([3i) was calculated according to two alternative scenarios. For adaptive dispersal, the proportion of resident predators in a given patch was predicted according to the marginal value theorem: J f3i = 1 if X i >~ 2~ Xj / (1 + mAt) J i=1 J [3i -- O if Xi < ~- Xj / (1 + mAt) J j=l
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where mAt is the time cost of movement between patches and J is the number of patches. Hence, all predators remained in a given patch when the instantaneous rate of consumption exceeded the expected long-term rate of gain over all J patches available, taking into account the time cost of travel to a new patch. For fixed-fraction dispersal, [3was always set at 0.95, regardless of predator or prey densities in a given patch. For all models, we set At = 0.1, implying ten time steps for each generation. We further assumed that there was variation in prey carrying capacity across patches, normally distributed with a specified coefficient of variation, using the algorithm outlined by Press et al. (1988: 216-17). In order to track dynamics in a 'typical' patch, however, we always specified that the first patch in the matrix had a mean carrying capacity. We used a similar Monte Carlo method to initially allocate predators among the available patches, with the mean predator density per patch a small fraction (/~/20) of the initial prey population density (/~/2). Simulations were conducted for 600 generations, with the first 200 generations discarded before analysis, to reduce the chance that transient behaviour might bias our conclusions. As there was a stochastic element in the initiation process, we used a constant seed for each simulation. Four parameters were varied in our simulations of the alternative metapopulation scenarios (Table 1): (1) the average carrying capacity of all patches, (2) the coefficient of variation of K across patches, (3) the number of patches in the environment and (4) the movement time by predators seeking new patches. In this paper, we largely confine our attention to the temporal dynamics of predators and prey averaged over all the patches in the environment. We will consider spatial dynamics (e.g. variation in N and R among patches) in detail in a companion paper.
Results
The mean densities of both predators and prey (averaged over time) were strongly affected by changes in the mean carrying capacity of patches (Fig. 1). All four models suggested positive curvilinear relationships between mean predator density, mean prey density, and the mean carrying capacity of prey. There was evidence that predator densities tended to level off and prey densities tended to increase as K approached high levels (Fig. 1). Variation in the three other general parameters tested (spatial variation in K, the time cost'of dispersal and the number of patches) had only minor effects on mean prey or predator densities (+ 2 for N or +50 for R). Changes in all four system parameters considerably affected the magnitude of temporal variation in predator and prey densities. For simplicity, we explicitly consider only variation in predator density, but in all cases there were analogous effects on prey density. For three of the models (the model with adaptive global dispersal and both of the models with fixed dispersal) there was a substantial increase in temporal variation in predator densities as mean carrying capacity increased (Fig. 2a). In contrast, the model with adaptive local dispersal showed little change in temporal variability of predator density as mean carrying capacity increased. In this sense, adaptive local dispersal showed the strongest stabilizing effect in relation to changes in environmental productivity. There was also little effect of spatial variation in carrying capacity on temporal variation in predator densities for the adaptive local dispersal model (Fig. 2b). In contrast, adaptive global dispersal resulted in a strong negative relationship between temporal variation in predator densities and the magnitude of spatial variation in carrying capacity. Both of the fixed dispersal models produced increased temporal variation in predator density as spatial variation in carrying capacity increased. Hence, pronounced spatial heterogeneity in prey carrying capacity tended to
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Patch use and metapopulation dynamics Table 1. Parameter values (mean carrying capacity of prey, coefficient of spatial variation in prey carrying capacity, time cost of migration and numbers of patches) used in numerical simulations of the trophic models with adaptive global, adaptive local, fixed global and fixed local dispersal by predators Parameter values Parameter test
/~
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M
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K1 K2 K3 K4 K5
600 800 1000 1200 1400
0.10 0.10 0.10 0.10 0.10
3 3 3 3 3
25 25 25 25 25
CV1 CV2 CV3 CV4 CV5
1000 1000 1000 1000 1000
0.00 0.05 0.10 0.15 0.20
3 3 3 3 3
25 25 25 25 25
M1 M2 M3 M4 M5
1000 1000 1000 1000 1000
0.10 0.10 0.10 0.10 0.10
1 2 3 4 5
25 25 25 25 25
J1 J2 J3 J4 J5
1000 1000 1000 1000 1000
0.10 0.10 0.10 0.10 0.10
3 3 3 3 3
1 9 25 49 81
Several additional parameters were held constant over all simulations; r = 0.40, a = 0.04, h = 0.1, c -- 0.15, d = 0.8 and At = 0.1. stabilize predator populations with adaptive global dispersal, destabilize predators with fixed dispersal strategies and have no effect on predators with adaptive local dispersal. Increasing movement time among patches resulted in decreased temporal variation in predator density for the systems with adaptive global, fixed global and fixed local dispersal (Fig. 2c). Predators with adaptive local dispersal were insensitive to the time costs of movement. Variation in the number of patches in the environment had contrasting effects on the adaptive vs fixed dispersal models (Fig. 2d). Both models with adaptive dispersal showed a sharp decline in temporal variation in predator density as the number of patches increased. The model with fixed global dispersal was unaffected by variation in patch number, whereas temporal variation in predator density increased with patch number for predators with fixed local dispersal. T h e r e were also differences in the temporal dynamics of predators among the four models. The models with fixed global and fixed local dispersal produced recurrent oscillations in predator densities over all parameter combinations tested (Fig. 3b and d). However, both of the models with adaptive dispersal showed more complex temporal dynamics. The model with adaptive global dispersal generally produced temporal dynamics with recurrent oscillations superimposed
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by small irregular deviations (Fig. 3a). Noisiness was most pronounced when spatial variation in carrying capacity was large or when there was a substantial time cost for dispersal among patches. In contrast, the cyclical component was more pronounced when the mean carrying capacity was large or when the number of patches was small. Predators with adaptive local dispersal showed little apparent periodicity, but rather fluctuated erratically over time (Fig. 3c). These differences in metapopulation dynamics were reflected by within-patch dynamics. For both of the fixed dispersal models, temporal fluctuations in adjacent patches were closely synchronized and smoothly cyclical (Fig. 4b and d). In contrast, there was less synchronization of within-patch temporal fluctuations in the adaptive dispersal models (Fig. 4a and c), particularly so for the model with adaptive local dispersal. Moreover, the adaptive dispersal models also showed more frequent oscillations, particularly for the model with adaptive local dispersal. Hence, the erratic temporal dynamics of the metapopulations of adaptively dispersing predators were associated with weak synchronization of dynamics among patches and higher frequency oscillations within patches. Discussion
Our numerical simulations of the metapopulation dynamics of a stationary prey and mobile predator point to a number of interesting observations. Perhaps most importantly, the function defining dispersal rates played a critical role in determining the dynamical characteristics of linked subpopulations of predators. Substantial differences in temporal dynamics were recorded between systems with adaptive (i.e. resource-dependent) dispersal and systems with a constant proportion of dispersers. In the former case, temporal fluctuations were rarely cyclical, particularly when stepping-stone movements occurred to neighbouring patches. In contrast, fixed dispersal invariably led to smooth oscillations in predator densities over time. These differences in temporal dynamics resulted from the all-or-nothing nature of dispersal dictated by the marginal value theorem. Predators remained resident in a given patch until the instantaneous rate of energy gain was less than the long-term expected gain elsewhere in the predator's universe, scaled to the time expenditure in locating a new patch. Hence, dispersal did not occur until the resource density in a given patch fell below a critical level (R), at which
aR 1 [ aR i _] + ah1~ - J(1 + 7) Z 1 + ahRiJ where ~"is the time cost of migration (mAt in our Euler approximation to a continuous case). One 1
way to visualize this effect is to use traditional isocline analysis of the predator-prey phase plane (Fig. 5a). The inclusion of a critical resource threshold leading to group desertion of a patch by all predators introduced a vertical tail in the otherwise hump-shaped prey isocline at/~ (Fig. 5b). Hence, there was a boundary condition that prevented completion of the normal limit cycle, which would ordinarily result in a protracted phase of low densities of both predators and prey. All other things being equal, this would produce asymmetrical, but persistent, oscillations of both predators and prey. However, asynchronous dynamics among the entire constellation of patches ensured that/~ was time-dependent and unpredictable, because prey density averaged over the entire environment was constantly changing. The end result was that the absorbing boundary in the prey isocline was constantly changing position, so that within-patch oscillations of predators and prey varied over time (Fig. 5b). Moreover, there were numerous immigration events arising unpredictably from other patches in the environment. It is therefore not surprising that population fluctuations were erratic, even though the internal dynamics within patches and the rules for dispersal were entirely deterministic. Hence, both immigration events and internal
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Figure 5. Phase-plane diagrams of the prey zero isocline (line) and the predator-prey trajectory (O) within a single patch using the adaptive global dispersal model. (a) One-patch system, (b) 25-patch system. (c) Predator dispersal is a continuous, rather than step, function of prey density (Z = 4). The prey zero isocline was calculated for the mean/~ recorded over the last 100 generations. Parameter values are listed under test K3 in Table 1.
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dynamics had an unpredictable, stochastic nature that led to erratic fluctuations over time. Previous models of spatially-distinct subpopulations with density- or resource-dependent dispersal rates have also produced unpredictable fluctuations (Nachman, 1987; Reeve, 1988), although this was partly due to stochastic variation in demographic parameters other than dispersal. The vertical tail in the prey zero isocline was a direct consequence of the all-or-nothing dispersal rule implicit in the marginal value theorem. It may be more reasonable to assume that the probability of dispersal varies among individual predators or that the behaviour of individual predators varies over time (Stephens and Krebs, 1986). To explore the impact of such variation, we reformulated the adaptive global model with the following dispersal function:
Rz ~-w+ ~R where tx =/~z. This produces a sigmoid function in which the proportion of predators dispersing from a given patch equals 0.5 when R i = / ~ . The parameter Z determines the goodness of fit to the optimal step function, with small Z implying gradual changes in [~ with varying resource density. As might be expected, within-patch dynamics were often strongly stabilized by a continuous dispersal function, as shown by a phase-plane plot of predator-prey dynamics within a typical patch (Fig. 5c). The continuous dispersal function converts the vertical tail in the prey zero isocline to one with negative slope. This density-dependence can result in local stability, provided that Z is small. Hence, wide variation among predators in the probability of dispersal in response to changes in local resource density should tend to have a strongly stabilizing influence on both within-patch and metapopulation dynamics. In both the adaptive and fixed dispersal models, temporal variation in predator densities was almost always less pronounced for local vs global dispersal models. This has been regularly reported in several previous simulation studies (Maynard-Smith, 1974; Roff, 1974; Nachman, 1987) with different model structures and hence appears a robust generalization. This is usually interpreted as a consequence of the increased synchrony among patches arising from global dispersal relative to local dispersal. In the context of our adaptive dispersal models, local dispersal enhanced the 'noisy' component of metapopulation trajectories, swamping the periodic component. This was apparently due to both asynchrony in dynamics among patches and accelerated frequency of oscillations within patches. The latter may have been due to the increased number of immigrants coming from neighbouring patches, rather than being evenly dispersed over the entire constellation of patches. This led to accelerated rates of prey depletion within patches, with shorter time before prey density within patches approached the critical value triggering wholesale emigration. For all of the models, variation in predator density was reduced by increasing the number of patches, which should promote predator persistence. This is a common feature of metapopulation models of predator-prey systems (Hilborn, 1975; Ziegler, 1977; Crowley, 1981; Nachman, 1987; Reeve, 1988) and appears to be a robust generalization. It is interesting, however, that patch dimension had a stronger stabilizing effect on the adaptive dispersal models, suggesting that habitat compartmentalization is more strongly stabilizing in systems with densitydependent dispersal than those with a constant probability of dispersal. For three of the models (adaptive global, fixed global and fixed local dispersal), variation in predator density was inversely related to the time cost of movement between patches. A pronounced movement cost implies that emigrants from dense patches are less quickly redistributed among neighbouring patches. This should tend to promote asynchrony among patches, which should tend to stabilize metapopulation dynamics. This process was unimportant
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in the model with adaptive local dispersal, perhaps because this system was already unsynchronized, even when time costs of movement were minor. Movement costs are rarely included in metapopulation models, but have already been shown to have effects on predator functional responses (Murdoch, 1977) and the degree of spatial covariation among predators and prey (Bernstein et al., 1991). Spatial variation in carrying capacity had a stabilizing effect in the model with adaptive global dispersal, no effect on the adaptive local dispersal model and a destabilizing effect on the fixed dispersal models. Previous metapopulation models have rarely incorporated true spatial heterogeneity in patch productivity (Taylor, 1988; Kareiva, 1990), but our results suggest that it probably plays a larger role in stabilizing systems with resource-dependent dispersal than those with constant probability of dispersal. Hence, spatial complexity plays a more important role than spatial heterogeneity per se in systems with fixed dispersal, whereas both features are important in systems with adaptive dispersal by predators. Many trophic models in which predator demographic parameters are solely a function of resource density predict that increased productivity of the system should translate into increased predator densities (albeit unstable), but little or no change in prey densities (Rosenzweig, 1971). This concept is central to dominant theories in community ecology, such as the 'exploitation hypothesis' (Fretwell, 1977; Oksanen et al., 1981; Oksanen, 1983) or the 'cascading effects hypothesis' (Carpenter et al., 1985), predicting changes in community structure in relation to productivity. Our simulations of spatially-distinct subpopulations of predators and prey produced a different pattern. Both predators and prey increased in density with increasing productivity, with decelerating curves for predators and accelerating curves for prey. Hence, simplistic predictions of biomass patterns in exploitation systems may be unjustified if spatial structure is present, even though predator dynamics are dictated entirely by resource density. It is also interesting to note that our patterns were similar to those predicted by mechanistic trophic models that have been proposed as an alternative to Lotka-Volterra trophic models (Schmitz, 1992). Hence, it may be difficult to distinguish between alternative underlying processes simply on the basis of measurements of trophic level abundance in relation to productivity. In sum, our results suggest that adaptive patterns of dispersal could have important demographic implications in predator-prey systems whose interactions are locally distinct. Coupled with our previous studies of adaptive diet choice by predators (J.M. Fryxell and Po Lundberg, unpublished manuscript) this indicates that an appreciation of fitness-maximizing behavioural strategies by predators may be instrumental in understanding trophic interactions.
Acknowledgements This work was supported through research operating grants from the Natural Sciences and Engineering Research Council of Canada (J.M. Fryxell), Renewable Resources Research Grant Program of Ontario (J.M. Fryxell), and the Swedish Natural Science Research Council (P. Lundberg). We thank Dick Green, Tom Nudds, Michael Rosenzweig and John Wilmshurst for constructive comments on the manuscript.
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