Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution

The effect of dispersal on population size and stability is explored for a population that disperses passively between two discrete habitat patches. Two basic models are considered. In the first model, a single population experiences density-dependent growth in both patches. A graphical construction is presented which allows one to determine the spatial pattern of abundance at equilibrium for most reasonable growth models and rates of dispersal. It is shown under rather general conditions that this equilibrium is unique and globally stable. In the second model, the dispersing population is a food-limited predator that occurs in both a source habitat (which contains a prey population) and a sink habitat (which does not). Passive dispersal between source and sink habitats can stabilize an otherwise unstable predator-prey interaction. The conditions allowing this are explored in some detail. The theory of optimal habitat selection predicts the evolutionarily stable distribution of a population, given that individuals can freely move among habitats so as to maximize individual fitness. This theory is used to develop a heuristic argument for why passive dispersal should always be selectively disadvantageous (ignoring kin effects) in a spatially heterogeneous but temporally constant environment. For both the models considered here, passive dispersal may lead to a greater number of individuals in both habitats combined than if there were no dispersal. This implies that the evolution of an optimal habitat distribution may lead to a reduction in population size; in the case of the predator-prey model, it may have the additional effect of destabilizing the interaction. The paper concludes with a discussion of the disparate effects habitat selection might have on the geographical range occupied by a species.     

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.     

Dispersal delays, predator-prey stability, and the paradox of enrichment

It takes time for individuals to move from place to place. This travel time can be incorporated into metapopulation models via a delay in the interpatch migration term. Such a term has been shown to stabilize the positive equilibrium of the classical Lotka-Volterra predator-prey system with one species (either the predator or the prey) dispersing. We study a more realistic, Rosenzweig-MacArthur, model that includes a carrying capacity for the prey, and saturating functional response for the predator. We show that dispersal delays can stabilize the predator-prey equilibrium point despite the presence of a Type II functional response that is known to be destabilizing. We also show that dispersal delays reduce the amplitude of oscillations when the equilibrium is unstable, and therefore may help resolve the paradox of enrichment.     

Scales of dispersal and the biogeography of marine predator-prey interactions

Striking differences in the dispersal of coexisting species have fascinated marine ecologists for decades. Despite widespread attention to the impact of dispersal on individual species dynamics, its role in species interactions has received comparatively little attention. Here, we approach the issue by combining analyses of simple heuristic predator-prey models with different dispersal patterns and data from several predator-prey systems from the Pacific coasts of North and South America. In agreement with model predictions, differences in predator dispersal generated characteristic biogeographic patterns. Predators lacking pelagic larvae tracked geographic variation in prey recruitment but not prey abundance. Prey recruitment rate alone explained more than 80% of the biogeographic variation in predator abundance. In contrast, predators with broadcasting larvae were uncorrelated with prey recruitment or adult prey abundance. Our findings reconcile perplexing results from previous studies and suggest that simple models can capture some of the complexity of life-history diversity in marine communities.     

Dispersal as a regional process affecting the local dynamics of marine and stream benthic invertebrates

Recent work has shown that benthic invertebrate assemblages may be influenced in an ongoing fashion by dispersal. Water-column movements of meiofauna, juvenile insects and marine postlarvae are common and can act to alter greatly local dynamics such as predator-prey and competitive interactions in marine and stream ecosystems. These findings are important because past research on the role of dispersal in invertebrate dynamics has focused almost exclusively on how planktonic larval supply influences the establishment and maintenance of local assemblages, on the colonization of newly opened sites, or on the settlement success of new recruits. The emerging framework is that dispersal needs to be viewed as a regional process that may routinely influence local benthic dynamics, because fauna can move to and from water-column dispersal 'pools' and may do so at frequent intervals.     

Habitat destruction in a simple predator-prey patch model: how predators enhance prey persistence and abundance

We model a metapopulation of predator-prey patches using both spatially implicit or mean-field (MF) and spatially explicit (SE) approaches. We show that in the MF model there are parameter regimes for which prey cannot persist in the absence of predators, but can in their presence. In addition, there are parameter regimes for which prey may persist in isolation, but the presence of predators will increase prey patch density. Predators may thus enhance prey persistence and overall abundance. The key mechanism responsible for this effect is the occurrence of prey dispersal from patches that are occupied by both prey and predators. In addition, these patches should be either long-lived, such as that occurs when predators keep prey from overexploiting its local resource, or the presence of a predator on a patch should significantly enhance the prey dispersal out of that patch. In the SE approach these positive effects of predators on prey persistence and abundance occur for even larger parameter ranges than in the MF model. Prey dispersal from predator-prey patches may thus be important for persistence of both species as a community, independent of the modeling framework studied. Comparison of the MF and SE approaches shows that local dispersal constraints can have the edge over global dispersal for the persistence of the metapopulation in regimes where the two species have a beneficial effect on each other. In general, our model provides an example of feedback in multiple-species metapopulations that can make the implementation of conservation schemes based on single-species arguments very risky.     

Ant semiochemicals limit apterous aphid dispersal

Some organisms can manipulate the nervous systems of others or alter their physiology in order to obtain benefit. Ants are known to limit alate aphid dispersal by physically removing wings and also through chemical manipulation of the alate developmental pathway. This results in reduced dispersal and higher local densities of aphids, which benefit ants in terms of increased honeydew and prey availability. Here, we show that the walking movement of mutualistic apterous aphids is also reduced by ant semiochemicals. Aphids walk slower and their dispersal from an unsuitable patch is hampered by ants. If aphid walking dispersal has evolved as a means of natural enemy escape, then ant chemicals may act as a signal indicating protection; hence, reduced dispersal could be adaptive for aphids. If, however, dispersal is primarily a means to reduce competition or to maintain persistent metapopulations, then manipulation by ants could be detrimental. Such manipulation strategies, common in host-parasite and predator-prey interactions, may be more common in mutualism than expected.     

Competition and dispersal in predator-prey waves

Dispersing predators and prey can exhibit complex spatio-temporal wave-like patterns if the interactions between them cause oscillatory dynamics. We study the effect of these predator-prey density waves on the competition between prey populations and between predator populations with different dispersal strategies. We first describe 1- and 2-dimensional simulations of both discrete and continuous predator-prey models. The results suggest that any population that diffuses faster, disperses farther, or is more likely to disperse will exclude slower diffusing, shorter dispersing, or less likely dispersing populations, everything else being equal. It also appears that it does not matter whether time, space, or state are discrete or continuous, nor what the exact interactions between the predators and prey are. So long as waves exist the competition between populations occurs in a similar fashion. We derive a theory that qualitatively explains the observed behaviour and calculate approximate analytical solutions that describe, to a reasonable extent, these behaviours. Predictions about the cost of dispersal are tested. If strong enough, cost can reverse the populations' relative competitive strengths or lead to coexistence because of the effect of spiral wave cores. The theory is also able to explain previous results of simulations of coexistence in host-parasitoid models (Comins, H. N., and Massell, M. P., 1996, J. Theor. Biol. 183, 19-28).     

Discrete-time travelling waves: Ecological examples

Integrodifference equations are discrete-time models that possess many of the attributes of continuous-time reaction-diffusion equations. They arise naturally in population biology as models for organisms with discrete nonoverlapping generations and well-defined growth and dispersal stages. I examined the varied travelling waves that arise in some simple ecologically-interesting integrodifference equations. For a scalar equation with compensatory growth, I observed only simple travelling waves. For carefully chosen redistribution kernels, one may derive the speed and approximate the shape of the observed waveforms. A model with overcompensation exhibited flip bifurcations and travelling cycles in addition to simple travelling waves. Finally, a simple predator-prey system possessed periodic wave trains and a variety of travelling waves.     

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。     

The effect of spatial heterogeneity on the persistence of predator-prey interactions

The effect of spatially discontinuous environments on predator-prey systems is examined by using a computer simulation model. It is shown that increasing prey dispersal and decreasing predator dispersal do not necessarily have a stabilizing influence on the interaction, as had been concluded by previous workers. The stability of predator-prey interaction depends on the interaction of the dispersal process with normal reproduction and feeding of the predator and prey species.     

Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities

Although density-dependent dispersal and relative dispersal (the difference in dispersal rates between species) have been documented in natural systems, their effects on the stability of metacommunities are poorly understood. Here we investigate the effects of intra- and interspecific density-dependent dispersal on the regional stability in a predator-prey metacommunity model. We show that, when the dynamics of the populations reach equilibrium, the stability of the metacommunity is not affected by density-dependent dispersal. However, the regional stability, measured as the regional variability or the persistence, can be modified by density-dependent dispersal when local populations fluctuate over time. Moreover these effects depend on the relative dispersal of the predator and the prey. Regional stability is modified through changes in spatial synchrony. Interspecific density-dependent dispersal always desynchronizses local dynamics, whereas intraspecific density-dependent dispersal may either synchronize or desynchronize it depending on dispersal rates. Moreover, intra- and interspecific density-dependent dispersal strengthen the top-down control of the prey by the predator at intermediate dispersal rates. As a consequence the regional stability of the metacommunity is increased at intermediate dispersal rates. Our results show that density-dependent dispersal and relative dispersal of species are keys to understanding the response of ecosystems to fragmentation.     

The effect of predator avoidance and travel time delay on the stability of predator-prey metacommunities

The stability conditions for an isolated specialist predator-prey community are fairly well understood. The spatial coupling of several such systems through dispersal of individuals can generate new dynamic behavior that is not yet completely understood. Many factors are known to be stabilizing or neutral, e.g., random dispersal or time delays, while others may induce instabilities in some cases but not others, e.g., density-dependent movement. We study the combination of two stabilizing mechanisms in a two-patch Rosenzweig-MacArthur model with a novel density-dependent movement term. Specifically, we assume that prey move between patches according to their perceived predation risk, and we include travel time between patches as a time delay. We show that the combination of mechanisms may be destabilizing even though each mechanism by itself is stabilizing. Our results show that a detailed knowledge of mechanisms and their temporal scales is necessary to correctly predict the stability of a metacommunity.     

Stabilizing effects in spatial parasitoid-host and predator-prey models: a review

We review the literature on spatial host-parasitoid and predator-prey models. Dispersal on its own is not stabilizing and can destabilize a stable local equilibrium. We identify three mechanisms whereby limited dispersal of hosts and parasitoids combined with other features, such as spatial and temporal heterogeneity, can promote increased persistence and stability. The first mechanisms, "statistical stabilization", is simply the statistical effect that summing a number of out-of-phase population trajectories results in a relatively constant total population density. The second mechanism involves decoupling of immigration from local density, such that limited dispersal between asynchronous patches results in an effect that mimics density-dependence at the local patch level. The third mechanism involves altering spatially averaged parameter values resulting from spatial heterogeneity in density combined with non-linear responses to density. Persistence in spatially explicit models with local dispersal frequently associated with self-organized spatial patterning.     

Confronting the Paradox of Enrichment to the Metacommunity Perspective

Resource enrichment can potentially destabilize predator-prey dynamics. This phenomenon historically referred as the "paradox of enrichment" has mostly been explored in spatially homogenous environments. However, many predator-prey communities exchange organisms within spatially heterogeneous networks called metacommunities. This heterogeneity can result from uneven distribution of resources among communities and thus can lead to the spreading of local enrichment within metacommunities. Here, we adapted the original Rosenzweig-MacArthur predator-prey model, built to study the paradox of enrichment, to investigate the effect of regional enrichment and of its spatial distribution on predator-prey dynamics in metacommunities. We found that the potential for destabilization was depending on the connectivity among communities and the spatial distribution of enrichment. In one hand, we found that at low dispersal regional enrichment led to the destabilization of predator-prey dynamics. This destabilizing effect was more pronounced when the enrichment was uneven among communities. In the other hand, we found that high dispersal could stabilize the predator-prey dynamics when the enrichment was spatially heterogeneous. Our results illustrate that the destabilizing effect of enrichment can be dampened when the spatial scale of resource enrichment is lower than that of organismss movements (heterogeneous enrichment). From a conservation perspective, our results illustrate that spatial heterogeneity could decrease the regional extinction risk of species involved in specialized trophic interactions. From the perspective of biological control, our results show that the heterogeneous distribution of pest resource could favor or dampen outbreaks of pests and of their natural enemies, depending on the spatial scale of heterogeneity.     

Nonlinear Effect of Dispersal Rate on Spatial Synchrony of Predator-Prey Cycles

Spatially-separated populations often exhibit positively correlated fluctuations in abundance and other population variables, a phenomenon known as spatial synchrony. Generation and maintenance of synchrony requires forces that rapidly restore synchrony in the face of desynchronizing forces such as demographic and environmental stochasticity. One such force is dispersal, which couples local populations together, thereby synchronizing them. Theory predicts that average spatial synchrony can be a nonlinear function of dispersal rate, but the form of the dispersal rate-synchrony relationship has never been quantified for any system. Theory also predicts that in the presence of demographic and environmental stochasticity, realized levels of synchrony can exhibit high variability around the average, so that ecologically-identical metapopulations might exhibit very different levels of synchrony. We quantified the dispersal rate-synchrony relationship using a model system of protist predator-prey cycles in pairs of laboratory microcosms linked by different rates of dispersal. Paired predator-prey cycles initially were anti-synchronous, and were subject to demographic stochasticity and spatially-uncorrelated temperature fluctuations, challenging the ability of dispersal to rapidly synchronize them. Mean synchrony of prey cycles was a nonlinear, saturating function of dispersal rate. Even extremely low rates of dispersal (<0.4% per prey generation) were capable of rapidly bringing initially anti-synchronous cycles into synchrony. Consistent with theory, ecologically-identical replicates exhibited very different levels of prey synchrony, especially at low to intermediate dispersal rates. Our results suggest that even the very low rates of dispersal observed in many natural systems are sufficient to generate and maintain synchrony of cyclic population dynamics, at least when environments are not too spatially heterogeneous.     

Phytoseiid dispersal at plant to regional levels: a review with emphasis on management of Neoseiulus fallacis in diverse agroecosystems

Dispersal behaviors of phytoseiid and tetranychid mites are key factors in understanding predator-prey dynamics and biological control of pest mites at different spatial levels in agricultural and natural ecosystems. In this review, ambulatory and aerial dispersal of both mite groups are discussed at spatial levels of leaf, plant, crop and region. Emphasis is on dispersal of phytoseiids, and specifically, the specialist-predator, Neoseiulusfallacis (Garman), and two-spotted spider mite prey, Tetranychus urticae (Koch). Dispersal aspects that are discussed are ambulation on a leaf; plant or in a prey patch; aerial dispersal between plants; behavior and aerodynamics of aerial take-off; modeling vs. monitoring of dispersal distance; fates of dispersing mites that land on soil substrates; plants as take-off platforms and landing targets for dispersers; and regional dispersal patterns and integrated mite management.     

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.     

Phase locking, the Moran effect and distance decay of synchrony: experimental tests in a model system

Spatially separated populations of many species fluctuate synchronously. Synchrony typically decays with increasing interpopulation distance. Spatial synchrony, and its distance decay, might reflect distance decay of environmental synchrony (the Moran effect), and/or short-distance dispersal. However, short-distance dispersal can synchronize entire metapopulations if within-patch dynamics are cyclic, a phenomenon known as phase locking. We manipulated the presence/absence of short-distance dispersal and spatially decaying environmental synchrony and examined their separate and interactive effects on the synchrony of the protist prey species Tetrahymena pyriformis growing in spatial arrays of patches (laboratory microcosms). The protist predator Euplotes patella consumed Tetrahymena and generated predator-prey cycles. Dispersal increased prey synchrony uniformly over both short and long distances, and did so by entraining the phases of the predator-prey cycles. The Moran effect also increased prey synchrony, but only over short distances where environmental synchrony was strongest, and did so by increasing the synchrony of stochastic fluctuations superimposed on the predator-prey cycle. Our results provide the first experimental demonstration of distance decay of synchrony due to distance decay of the Moran effect. Distance decay of the Moran effect likely explains distance decay of synchrony in many natural systems. Our results also provide an experimental demonstration of long-distance phase locking, and explain why cyclic populations provide many of the most dramatic examples of long-distance spatial synchrony in nature.     

Relating coupled map lattices to integro-difference equations: dispersal-driven instabilities in coupled map lattices

Recently there has been a great deal of interest within the ecological community about the interactions of local populations that are coupled only by dispersal. Models have been developed to consider such scenarios but the theory needed to validate model outcomes has been somewhat lacking. In this paper, we present theory which can be used to understand these types of interaction when population exhibit discrete time dynamics. In particular, we consider a spatial extension to discrete-time models, known as coupled map lattices (CMLs) which are discrete in space. We introduce a general form of the CML and link this to integro-difference equations via a special redistribution kernel. General conditions are then derived for dispersal-driven instabilities. We then apply this theory to two discrete-time models; a predator-prey model and a host-pathogen model.