The dynamics of two diffusively coupled predator-prey populations

I analyze the dynamics of predator and prey populations living in two patches. Within a patch the prey grow logistically and the predators have a Holling type II functional response. The two patches are coupled through predator migration. The system can be interpreted as a simple predator-prey metapopulation or as a spatially explicit predator-prey system. Asynchronous local dynamics are presumed by metapopulation theory. The main question I address is when synchronous and when asynchronous dynamics arise. Contrary to biological intuition, for very small migration rates the oscillations always synchronize. For intermediate migration rates the synchronous oscillations are unstable and I found periodic, quasi-periodic, and intermittently chaotic attractors with asynchronous dynamics. For large predator migration rates, attractors in the form of equilibria or limit cycles exist in which one of the patches contains no prey. The dynamical behavior of the system is described using bifurcation diagrams. The model shows that spatial predator-prey populations can be regulated through the interplay of local dynamics and migration.     

Predation across spatial scales in heterogeneous environments

A hierarchy of scales is introduced to the spatially heterogeneous Lotka-Volterra predator-prey diffusion model, and its effects on the model's spatial and temporal behavior are studied. When predators move on a large scale relative to prey, local coupling of the predator-prey interaction is replaced by global coupling. Prey with low dispersal ability become narrowly confined to the most productive habitats, strongly amplifying the underlying spatial pattern of the environment. As prey diffusion rate increases, the prey distribution spreads out and predator abundance declines. The model retains neutrally stable Lotka-Volterra temporal dynamics: different scales of predator and prey dispersal do not stabilize the interaction. The model predicts that, for prey populations that are limited by widely ranging predators, species with low dispersal ability should be restricted to discrete high density patches, and those with greater mobility should be more uniformly distributed at lower density.     

Systems analysis of an acarine predator-prey system II: Interactions in discontinuous environment

An acarine predator-prey system in a circular stepping-stone environment was described with a simulation model to elucidate the factors responsible for persistence of the system. The main assumptions in this model are: (1) The prey are inevitably eliminated in patches in which predators exist. (2) The density of prey declines and becomes extinct by plant defoliation due to feeding by prey. In this regard this model is different from the models which mimickedHuffaker's (1958) experiments and assumed stable plant-prey relations. Analyses showed that the critical factor in persistence of the predator-prey system was the plant-prey relations, at any combination of other parameters involved in the model. The predator-prey system did not persist long under the unstable relationship of prey and plant. Otherwise the system persisted longer especially when I used a larger number of patches, a larger amount of plant in each patch, and long-distance-migrations of the prey. In particular, frequent emigration of the prey regardless of plant conditions was most effective.     

Does colonization asymmetry matter in metapopulations?

Despite the considerable evidence showing that dispersal between habitat patches is often asymmetric, most of the metapopulation models assume symmetric dispersal. In this paper, we develop a Monte Carlo simulation model to quantify the effect of asymmetric dispersal on metapopulation persistence. Our results suggest that metapopulation extinctions are more likely when dispersal is asymmetric. Metapopulation viability in systems with symmetric dispersal mirrors results from a mean field approximation, where the system persists if the expected per patch colonization probability exceeds the expected per patch local extinction rate. For asymmetric cases, the mean field approximation underestimates the number of patches necessary for maintaining population persistence. If we use a model assuming symmetric dispersal when dispersal is actually asymmetric, the estimation of metapopulation persistence is wrong in more than 50% of the cases. Metapopulation viability depends on patch connectivity in symmetric systems, whereas in the asymmetric case the number of patches is more important. These results have important implications for managing spatially structured populations, when asymmetric dispersal may occur. Future metapopulation models should account for asymmetric dispersal, while empirical work is needed to quantify the patterns and the consequences of asymmetric dispersal in natural metapopulations.     

Stabilization through spatial pattern formation in metapopulations with long-range dispersal

Many studies of metapopulation models assume that spatially extended populations occupy a network of identical habitat patches, each coupled to its nearest neighbouring patches by density-independent dispersal. Much previous work has focused on the temporal stability of spatially homogeneous equilibrium states of the metapopulation, and one of the main predictions of such models is that the stability of equilibrium states in the local patches in the absence of migration determines the stability of spatially homogeneous equilibrium states of the whole metapopulation when migration is added. Here, we present classes of examples in which deviations from the usual assumptions lead to different predictions. In particular, heterogeneity in local habitat quality in combination with long-range dispersal can induce a stable equilibrium for the metapopulation dynamics, even when within-patch processes would produce very complex behaviour in each patch in the absence of migration. Thus, when spatially homogeneous equilibria become unstable, the system can often shift to a different, spatially inhomogeneous steady state. This new global equilibrium is characterized by a standing spatial wave of population abundances. Such standing spatial waves can also be observed in metapopulations consisting of identical habitat patches, i.e. without heterogeneity in patch quality, provided that dispersal is density dependent. Spatial pattern formation after destabilization of spatially homogeneous equilibrium states is well known in reaction–diffusion systems and has been observed in various ecological models. However, these models typically require the presence of at least two species, e.g. a predator and a prey. Our results imply that stabilization through spatial pattern formation can also occur in single-species models. However, the opposite effect of destabilization can also occur: if dispersal is short range, and if there is heterogeneity in patch quality, then the metapopulation dynamics can be chaotic despite the patches having stable equilibrium dynamics when isolated. We conclude that more general metapopulation models than those commonly studied are necessary to fully understand how spatial structure can affect spatial and temporal variation in population abundance.     

Are spatially correlated or uncorrelated disturbance regimes better for the survival of species?

The survival of species in dynamic landscapes (characterised by patch destruction and subsequent regeneration) depends on both the species' attributes and the disturbance pattern. Using a spatially explicit model we explored how the mean time to extinction of a metapopulation depends on the spatial correlation of patch destruction in relation to the population growth and dispersal abilities of species. Two contrasting answers are possible. On the one hand, increasing spatial correlation of patch destruction increases the spatial correlation of population growth and this is known to decrease metapopulation persistence. On the other hand, spatially correlated patch destruction and regeneration can lead to clustered habitat patches and this is known to increase metapopulation persistence. Therefore, we hypothesised that some species are better off under spatially correlated and alternatively uncorrelated disturbance regimes. However, contrary to this hypothesis, in all kinds of cases spatial correlation reduced metapopulation persistence. We found this to be due to the fact that the spatial correlation of patch destruction causes increasing temporal fluctuations in the regional carrying capacity of the metapopulation and is hence generally disadvantageous for long-term persistence. The main consequence for conservation biology is that reducing spatial correlation in disturbances is likely to be a reliable strategy in a dynamic landscape that will benefit practically all species with a low risk of adverse side effects .     

Metapopulation persistence in dynamic landscapes: the role of dispersal distance

Species associated with transient habitats need efficient dispersal strategies to ensure their regional survival. Using a spatially explicit metapopulation model, we studied the effect of the dispersal range on the persistence of a metapopulation as a function of the local population and landscape dynamics (including habitat patch destruction and subsequent regeneration). Our results show that the impact of the dispersal range depends on both the local population and patch growth. This is due to interactions between dispersal and the dynamics of patches and populations via the number of potential dispersers. In general, long-range dispersal had a positive effect on persistence in a dynamic landscape compared to short-range dispersal. Long-range dispersal increases the number of couplings between the patches and thus the colonisation of regenerated patches. However, long-range dispersal lost its advantage for long-term persistence when the number of potential dispersers was low due to small population growth rates and/or small patch growth rates. Its advantage also disappeared with complex local population dynamics and in a landscape with clumped patch distribution.     

Host-parasitoid spatial ecology: a plea for a landscape-level synthesis

A growing body of literature points to a large-scale research approach as essential for understanding population and community ecology. Many of our advances regarding the spatial ecology of predators and prey can be attributed to research with insect parasitoids and their hosts. In this review, we focus on the progress that has been made in the study of the movement and population dynamics of hosts and their parasitoids in heterogeneous landscapes, and how this research approach may be beneficial to pest management programs. To date, few studies have quantified prey and predator rates and ranges of dispersal and population dynamics at the patch level--the minimum of information needed to characterize population structure. From host-parasitoid studies with sufficient data, it is clear that the spatial scale of dispersal can differ significantly between a prey and its predators, local prey extinctions can be attributed to predators and predator extinction risk at the patch level often exceeds that of the prey. It is also evident that populations can be organized as a single, highly connected (patchy) population or as semi-independent extinction-prone local populations that collectively form a persistent metapopulation. A prey and its predators can also differ in population structure. At the landscape level, agricultural studies indicate that predator effects on its prey often spill over between the crop and surrounding area (matrix) and can depend strongly on landscape structure (e.g. the proportion of suitable habitat) at scales extending well beyond the crop margins. In light of existing empirical data, predator-prey models are typically spatially unrealistic, lacking important details on boundary responses and movement behaviour within and among patches. The tools exist for conducting empirical and theoretical research at the landscape level and we hope that this review calls attention to fertile areas for future exploration.     

Predator-prey interactions in a nonequilibrium context: the metapopulation approach to modeling "hide-and-seek" dynamics in a spatially explicit tri-trophic system

Predators and prey are usually heterogeneously distributed in space so that the ability of the predators to respond to the distribution of their prey may have a profound influence on the stability and persistence of a predator‐prey system. A special type of dynamics is “hide‐and‐seek” characterized by a high turnover rate of local populations of prey and predators, because once the predators have found a patch of prey they quickly overexploit it, whereupon the starving predators either should move to better places or die. Continued persistence of prey and predators thus hinges on a long‐term balance between local extinctions and founding of new subpopulations. The colonization rate depends on the rate of emigration from occupied patches and the likelihood of successfully arriving at a suitable new patch, while extinction rate depends on the local population dynamics. Since extinctions and colonizations are both discrete probabilistic events, these phenomena are most adequately modeled by means of a stochastic model. In order to demonstrate the qualitative differences between a deterministic and stochastic approach to population dynamics, a spatially explicit tritrophic predator‐prey model is developed in a deterministic and a stochastic version. The model is parameterized using data for the two‐spotted spider mite (Tetranychus urticae) and the phytoseiid mite predator Phytoseiulus persimilis inhabiting greenhouse cucumbers.
Simulations show that the deterministic and stochastic approaches yield different results. The deterministic version predicts that the populations will exhibit violent fluctuations, implying that the system is fundamentally unstable. In contrast, the stochastic version predicts that the two species will be able to coexist in spite of frequent local extinctions of both species, provided the system consists of a sufficiently large number of local populations. This finding is in agreement with experimental results. It is therefore concluded that demographic stochasticity in combination with dispersal is capable of producing and maintaining sufficient asynchrony between local populations to ensure long‐term regional (metapopulation) persistence.     

Evolution of dispersal in a predator-prey metacommunity

Dispersal is crucial to allowing species inhabiting patchy or spatially subdivided habitats to persist globally despite the possibility of frequent local extinctions. Theoretical studies have repeatedly demonstrated that species that exhibit a regional metapopulation structure and are subject to increasing rates of local patch extinctions should experience strong selective pressures to disperse more rapidly despite the costs such increased dispersal would entail in terms of decreased local fitness. We extend these studies to consider how extinctions arising from predator-prey interactions affect the evolution of dispersal for species inhabiting a metacommunity. Specifically, we investigate how increasing a strong extinction-prone interaction between a predator and prey within local patches affects the evolution of each species' dispersal. We found that for the predator, as expected, evolutionarily stable strategy (ESS) dispersal rates increased monotonically in response to increasing local extinctions induced by strong predator top-down effects. Unexpectedly for the prey, however, ESS dispersal rates displayed a nonmonotonic response to increasing predator-induced extinction rates-actually decreasing for a significant range of values. These counterintuitive results arise from how extinctions resulting from trophic interactions play out at different spatial scales: interactions that increase extinction rates of both species locally can, at the same time, decrease the frequency of interaction between the prey and predator at the metacommunity scale.     

食饵庇护所对斑块环境下Leslie-Gower捕食系统的影响

建立了一个显式含有空间庇护所的两斑块Leslie-Gower捕食者-食饵系统。假设只有食饵种群在斑块间以常数迁移率迁移,且在每个斑块上食饵间的迁移比局部捕食者-食饵相互作用发生的时间尺度要快。利用两个时间尺度,可以构建用来描述所有斑块总的食饵和捕食者密度的综合系统。数学分析表明,在一定条件下,存在唯一的正平衡点,并且此平衡点全局稳定。进一步,捕食者的数量随着食饵庇护所数量增加而降低;在一定条件下,食饵的数量随着食饵庇护所数量增加先增加后降低,在足够强的庇护所强度下,两物种出现灭绝。对比以往研究,利用显式含有和隐含空间庇护所的数学模型所得结论不一致,这意味着在研究庇护所对捕食系统种群动态影响时,空间结构可能起着重要作用。     

Optimal patch use and metapopulation dynamics

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.     

Effects of competition, predation, and dispersal on species richness at local and regional scales

This study explores the consequences of predator-mediated coexistence among competitors for patterns of incidence and diversity at local and regional scales. We develop a model that draws on elements of metapopulation models of competitors and food chains by allowing competitors to coexist locally in the presence of predators but not in their absence. The model predicts that predators promote regional coexistence by greatly expanding the range of conditions under which two competitors persist at equilibrium. Predators could have positive or negative effects on mean local diversity within the region depending on their dispersal rates, those of the prey, and their effects on prey extinction rates. The presence of predators increased the abundance of inferior competitors, thereby expanding the conditions for positive relationships between local and regional diversity. The model also predicted positive correlations between local diversity of predators and prey. These predictions were supported by patterns of phytoplankton, zooplankton, and fish species richness among lakes. The model may help to resolve the apparent contrast between linear patterns of local and regional richness and experimental evidence for strong invasion resistance and rapid dispersal in zooplankton.     

Metapopulation persistence in fragmented landscapes: significant interactions between genetic and demographic processes

We formulated a mathematical model in order to study the joint influence of demographic and genetic processes on metapopulation viability. Moreover, we explored the influence of habitat structure, matrix quality and disturbance on the interplay of these processes. We showed that the conditions that allow metapopulation persistence under the synergistic action of genetic and demographic processes depart significantly from predictions based on a mere superposition of the effects of each process separately. Moreover, an optimal dispersal rate exists that maximizes the range of survival rates of dispersers under which metapopulation persists and at the same time allows the largest sustainable patch removal and patch‐size reduction. The relative impact of patch removal and patch‐size reduction depends both on matrix quality and the dispersal strategy of the species: metapopulation persistence is more affected by patch‐size reduction (patch removal) for low (high)‐dispersing species, in presence of a low (high) quality matrix. Avoidance of inbreeding, through increased dispersal when the rate of inbreeding in a population is large, has positive effects on low‐dispersing species, but impairs the persistence of high‐dispersing species. Finally, size heterogeneity between patches largely influences metapopulation dynamics; the presence of large patches, even at the expense of other patches being smaller, can have positive effects on persistence in particular for species of low dispersing ability.     

Spatial Heterogeneity in Habitat Quality and Cross-Scale Interactions in Metapopulations

Abstract Integration of habitat heterogeneity into spatially realistic metapopulation approaches reveals the potential for key cross-scale interactions. Broad-scale environmental gradients and land-use practices can create autocorrelation of habitat quality of suitable patches at intermediate spatial scales. Patch occupancy then depends not only on habitat quality at the patch scale but also on feedbacks from surrounding neighborhoods of autocorrelated patches. Metapopulation dynamics emerge from how demographic and dispersal processes interact with relevant habitat heterogeneity. We provide an empirical example from a metapopulation of round-tailed muskrats (Neofiber alleni) in which habitat quality of suitable patches was spatially autocorrelated most strongly within 1,000 m, which was within the expected dispersal range of the species. After controlling for factors typically considered in metapopulation studies—patch size, local patch quality, patch connectivity—we use a cross-variogram analysis to demonstrate that patch occupancy by muskrats was correlated with habitat quality across scales ≤1,171 m. We also discuss general consequences of spatial heterogeneity of habitat quality for metapopulations related to potential cross-scale interactions. We focus on spatially correlated extinctions and metapopulation persistence, hierarchical scaling of source–sink dynamics, and dispersal decisions by individuals in relation to information constraints.     

Metapopulation dynamics for spatially extended predator–prey interactions

Traditional metapopulation theory classifies a metapopulation as a spatially homogeneous population that persists on neighboring habitat patches. The fate of each population on a habitat patch is a function of a balance between births and deaths via establishment of new populations through migration to neighboring patches. In this study, we expand upon traditional metapopulation models by incorporating spatial heterogeneity into a previously studied two-patch nonlinear ordinary differential equation metapopulation model, in which the growth of a general prey species is logistic and growth of a general predator species displays a Holling type II functional response. The model described in this work assumes that migration by generalist predator and prey populations between habitat patches occurs via a migratory corridor. Thus, persistence of species is a function of local population dynamics and migration between spatially heterogeneous habitat patches. Numerical results generated by our model demonstrate that population densities exhibit periodic plane-wave phenomena, which appear to be functions of differences in migration rates between generalist predator and prey populations. We compare results generated from our model to results generated by similar, but less ecologically realistic work, and to observed population dynamics in natural metapopulations.     

Effects of habitat destruction and resource supplementation in a predator-prey metapopulation model

We developed a mean field, metapopulation model to study the consequences of habitat destruction on a predator-prey interaction. The model complements and extends earlier work published by Bascompte and Solé (1998, J. theor. Biol.195, 383-393) in that it also permits use of alternative prey (i.e., resource supplementation) by predators. The current model is stable whenever coexistence occurs, whereas the earlier model is not stable over the entire domain of coexistence. More importantly, the current model permits an assessment of the effect of a generalist predator on the trophic interaction. Habitat destruction negatively affects the equilibrium fraction of patches occupied by predators, but the effect is most pronounced for specialists. The effect of habitat destruction on prey coexisting with predators is dependent on the ratio of extinction risk due to predation and prey colonization rate. When this ratio is less than unity, equilibrial prey occupancy of patches declines as habitat destruction increases. When the ratio exceeds one, equilibrial prey occupancy increases even as habitat destruction increases; i.e., prey "escape" from predation is facilitated by habitat loss. Resource supplementation reduces the threshold colonization rate of predators necessary for their regional persistence, and the benefit derived from resource supplementation increases in a nonlinear fashion as habitat destruction increases. We also compared the analytical results to those from a stochastic, spatially explicit simulation model. The simulation model was a discrete time analog of our analytical model, with one exception. Colonization was restricted locally in the simulation, whereas colonization was a global process in the analytical model. After correcting for differences between nominal and effective colonization rates, most of the main conclusions of the two types of models were similar. Some important differences did emerge, however, and we discuss these in relation to the need to develop fully spatially explicit analytical models. Finally, we comment on the implications of our results for community structure and for the conservation of prey species interacting with generalist predators.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

Hops as a metapopulation landscape for tetranychid-phytoseiid interactions: perspectives of intra- and interplant dispersal

Population densities, distributions and dispersal of Neoseiulus fallacis (Garman) and Tetranychus urticae (Koch) on individual hop plants, Humulus lupulus L. were studied for attributes of metapopulations such as empty patches, asynchrony of subpopulations, extinction of subpopulations, and dispersal of predators and prey among patches. Occupancy of hop leaves by predators or prey was stable over a season with 69–75% of leaves having neither predators nor prey, 4–15% with prey mites only, 9–17% with both predators and prey mites and 6–10% with predaceous mites only. Stability of occupancy classes through time indicated that inherently unstable predator and prey subpopulations developed asynchronously. Flagged hop leaves showed the existence of many empty individual leaves, colonization of some by prey, then by predators, then extinction of both, and then recolonization by spider mites. This illustrated the existence of empty patches, extinction of subpopulations, and dispersal of predators and prey to empty patches. This differed from spider mites and phytoseiid predators on apple foliage where there was a progression of occupancy status, indicating synchronous development of subpopulations on individual plants. Studies of predator and prey dispersal between hop plants showed that removal of basal leaves to 1.5 m high, a common agronomic practice, greatly limited dispersal of the predaceous mites but not the spider mites. Retaining basal leaves facilitated interplant movement of predators and improved the extent and timing of biological control. Through management, N. fallacis dispersal may be adjusted so that the entire hop planting becomes a metapopulation landscape, leading to greater stability and persistence of predator–prey within a season.     

Effects of demographic parameters on metapopulation size and persistence: an analytical stochastic model

An analytical stochastic metapopulation model is developed. It describes how individuals will be distributed among patches as a function of density-dependent birth, death and emigration rates, and the probability of successful dispersal. The model includes demographic stochasticity, but not catastrophes, environmental stochasticity or variation in patch size and suitability. All patches are equally likely to be colonized by migrants. The model predicts: (a) mean and variance of the number of individuals per patch; (b) probability distribution of individuals per patch; (c) mean number of individuals in transit; and (d) turn-over rate and expected persistence time of a single patch. The model shows that (a) dispersal rates must be intermediate in order to ensure metapopulation persistence; (b) the mean number of individuals per patch is often well below the carrying capacity; (c) long transit times and/or high mortality during dispersal reduce the mean number of individuals per patch; (d) density-dependent emigration responses will usually increase metapopulation size and persistence compared with density-independent dispersal; (e) an increase in the per capita net growth rate can both increase and decrease metapopulation size and persistence depending on whether dispersal rates are high or low; (f) density-independent birth, death, and emigration rates lead to a spatial pattern described by the negative binomial distribution.