Effects of Density Dependent Migrations on the Dynamics of a Predator Prey Model

We study the effects of density dependent migrations on the stability of a predator-prey model in a patchy environment which is composed with two sites connected by migration. The two patches are different. On the first patch, preys can find resource but can be captured by predators. The second patch is a refuge for the prey and thus predators do not have access to this patch. We assume a repulsive effect of predator on prey on the resource patch. Therefore, when the predator density is large on that patch, preys are more likely to leave it to return to the refuge. We consider two models. In the first model, preys leave the refuge to go to the resource patch at constant migration rates. In the second model, preys are assumed to be in competition for the resource and leave the refuge to the resource patch according to the prey density. We assume two different time scales, a fast time scale for migration and a slow time scale for population growth, mortality and predation. We take advantage of the two time scales to apply aggregation of variables methods and to obtain a reduced model governing the total prey and predator densities. In the case of the first model, we show that the repulsive effect of predator on prey has a stabilizing effect on the predator-prey community. In the case of the second model, we show that there exists a window for the prey proportion on the resource patch to ensure stability.     

The Effects of Predator Evolution and Genetic Variation on Predator–Prey Population-Level Dynamics

This paper explores how predator evolution and the magnitude of predator genetic variation alter the population-level dynamics of predator–prey systems. We do this by analyzing a general eco-evolutionary predator–prey model using four methods: Method 1 identifies how eco-evolutionary feedbacks alter system stability in the fast and slow evolution limits; Method 2 identifies how the amount of standing predator genetic variation alters system stability; Method 3 identifies how the phase lags in predator–prey cycles depend on the amount of genetic variation; and Method 4 determines conditions for different cycle shapes in the fast and slow evolution limits using geometric singular perturbation theory. With these four methods, we identify the conditions under which predator evolution alters system stability and shapes of predator–prey cycles, and how those effect depend on the amount of genetic variation in the predator population. We discuss the advantages and disadvantages of each method and the relations between the four methods. This work shows how the four methods can be used in tandem to make general predictions about eco-evolutionary dynamics and feedbacks.     

Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds

The population-dispersal dynamics for predator–prey interactions and two competing species in a two patch environment are studied. It is assumed that both species (i.e., either predators and their prey, or the two competing species) are mobile and their dispersal between patches is directed to the higher fitness patch. It is proved that such dispersal, irrespectively of its speed, cannot destabilize a locally stable predator–prey population equilibrium that corresponds to no movement at all. In the case of two competing species, dispersal can destabilize population equilibrium. Conditions are given when this cannot happen, including the case of identical patches.     

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.     

Remarks on the number of persistent states to a fragmented predator–prey model system

We consider a predator–prey model system for spatially distributed species over patches. Each predator species has a unique preferred patch (shelter and reproduction site) and travel for chasing prey. Its individuals are split into resident from the preferred patch and travelers. Further there is at most one resident predator species per patch. Depending on the availability of local anthropized resources not related to local prey on the preferred patch, one distinguishes between well-fed and starving predators. We assume prey species do not disperse at the predator scale.In this study we are interested in the number of persistent stationary states for the resulting ordinary differential equations model system. There exists at most one persistent predator–prey stationary state when there is exactly one starving resident predators per patch provided all functional responses to predation are Lotka–Volterra like or when a single starving resident predators is available. Else multiple persistent predator–prey stationary state are likely to exist. A specific emphasis is put on toy-model systems with 2 or 3 patches. Slow–fast dynamical methodology is also used for locally asymptotically stable purposes.Numerical experiments suggest that several scalings may govern the dynamics at stabilization.     

When can dispersal synchronize populations?

While spatial synchrony of oscillating populations has been observed in many ecological systems, the causes of this phenomenon are still not well understood. The most common explanations have been the Moran effect (synchronous external stochastic influences) and the effect of dispersal among populations. Since ecological systems are typically subject to large spatially varying perturbations which destroy synchrony, a plausible mechanism explaining synchrony must produce rapid convergence to synchrony. We analyze the dynamics through time of the synchronizing effects of dispersal and, consequently, determine whether dispersal can be the mechanism which produces synchrony. Specifically, using methods new to ecology, we analyze a two patch predator-prey model, with identical weak dispersal between the patches. We find that a difference in time scales (i.e. one population has dynamics occurring much faster than the other) between the predator and prey species is the most important requirement for fast convergence to synchrony.     

Handling time promotes the coevolution of aggregation in predator-prey systems

Predators often have type II functional responses and live in environments where their life history traits as well as those of their prey vary from patch to patch. To understand how spatial heterogeneity and predator handling times influence the coevolution of patch preferences and ecological stability, we perform an ecological and evolutionary analysis of a Nicholson-Bailey type model. We prove that coevolutionarily stable prey and searching predators prefer patches that in isolation support higher prey and searching predator densities, respectively. Using this fact, we determine how environmental variation and predator handling times influence the spatial patterns of patch preferences, population abundances and per-capita predation rates. In particular, long predator handling times are shown to result in the coevolution of predator and prey aggregation. An analytic expression characterizing ecological stability of the coevolved populations is derived. This expression implies that contrary to traditional theoretical expectations, predator handling time can stabilize predator-prey interactions through its coevolutionary influence on patch preferences. These results are shown to have important implications for classical biological control.     

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.     

Foraging behavior of an estuarine predator, the blue crab Callinectes sapidus in a patchy environment

To define general principles of predator‐prey dynamics in an estuarine subtidal environment, we manipulated predator density (the blue crab, Callinectes sapidus) and prey (the clam, Macoma balthica) patch distribution in large field enclosures in the Rhode River subestuary of the central Chesapeake Bay. The primary objectives were to determine whether predators forage in a way that maximizes prey consumption and to assess how their foraging success is affected by density of conspecifics. We developed a novel ultrasonic telemetry system to observe behavior of individual predators with unprecedented detail. Behavior of predators was more indicative of optimal than of opportunistic foraging. Predators appeared responsive to the overall quality of prey in their habitat. Rather than remaining on a prey patch until depletion, predators appeared to vary their patch use with quality of the surrounding environment. When multiple (two) prey patches were available, residence time of predators on a prey patch was shorter than when only a single prey patch was available. Predators seemed to move among the prey patches fairly regularly, dividing their foraging time between the patches and consuming prey from each of them at a similar rate. That predators more than doubled their consumption of prey when we doubled the number of prey (by adding the second patch) is consistent with optimizing behaviors ‐ rather than with an opportunistic increase in prey consumption brought about simply by the addition of more prey. Predators at high density, however, appeared to interfere with each other's foraging success, reflected by their lower rates of prey consumption. Blue crabs appear to forage more successfully (and their prey to experience higher mortality) in prey patches located within 15–20 meters of neighboring patch, than in isolated patches. Our results are likely to apply, at least qualitatively, to other crustacean‐bivalve interactions, including those of commercial interest; their quantitative applicability will depend on the mobility of other predators and the scale of patchiness they perceive.     

PREY ADAPTATION AS A CAUSE OF PREDATOR-PREY CYCLES

We analyze simple models of predator-prey systems in which there is adaptive change in a trait of the prey that determines the rate at which it is captured by searching predators. Two models of adaptive change are explored: (1) change within a single reproducing prey population that has genetic variation for vulnerability to capture by the predator; and (2) direct competition between two independently reproducing prey populations that differ in their vulnerability. When an individual predator's consumption increases at a decreasing rate with prey availability, prey adaptation via either of these mechanisms may produce sustained cycles in both species' population densities and in the prey's mean trait value. Sufficiently rapid adaptive change (e.g., behavioral adaptation or evolution of traits with a large additive genetic variance), or sufficiently low predator birth and death rates will produce sustained cycles or chaos, even when the predator-prey dynamics with fixed prey capture rates would have been stable. Adaptive dynamics can also stabilize a system that would exhibit limit cycles if traits were fixed at their equilibrium values. When evolution fails to stabilize inherently unstable population interactions, selection decreases the prey's escape ability, which further destabilizes population dynamics. When the predator has a linear functional response, evolution of prey vulnerability always promotes stability. The relevance of these results to observed predator-prey cycles is discussed.     

Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics

Most classical prey-predator models do not take into account the behavioural structure of the population. Usually, the predator and the prey populations are assumed to be homogeneous, i.e. all individuals behave in the same way. In this work, we shall take into account different tactics that predators can use for exploiting a common self-reproducing resource, the prey population. Predators fight together in order to keep or to have access to captured prey individuals. Individual predators can use two behavioural tactics when they encounter to dispute a prey, the classical hawk and dove tactics. We assume two different time scales. The fast time scale corresponds to the inter-specific searching and handling for the prey by the predators and the intra-specific fighting between the predators. The slow time scale corresponds to the (logistic) growth of the prey population and mortality of the predator. We take advantage of the two time scales to reduce the dimension of the model and to obtain an aggregated model that describes the dynamics of the total predator and prey densities at the slow time scale. We present the bifurcation analysis of the model and the effects of the different predator tactics on persistence and stability of the prey-predator community are discussed.     

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.     

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.     

The ecology of fear and inverted biomass pyramids

In inverted biomass pyramids (IBPs) prey are outnumbered by their predators when measured by biomass. We investigate how prey should behave in the face of danger from higher predator biomass, and how anti-predator behavior (in the form of vigilance) can, in turn, affect the predator–prey system. In this study, we incorporate anti-predator behaviors into a Lotka–Volterra predator–prey model in the form of fixed and facultative vigilance. Facultative vigilance models behavior as a dynamic foraging game, allowing us to assess optimal behavioral responses in the context of IBPs using a dynamical fitness optimization approach. We model vigilance as a tradeoff between safety and either the prey's maximum growth rate or its carrying capacity. We assess the population dynamics of predators and prey with fear responses, and investigate the role fear plays on trophic structure. We found that the ecology of fear plays an important role in predator–prey systems, impacting trophic structure and the occurrence of IBPs. Fixed vigilance works against IBP structure by always reducing the predator–prey biomass ratio at equilibrium with increasing levels of vigilance. Facultative vigilance can actually promote IBPs, as prey can now adjust their vigilance levels to cope with increased predation and the costs associated with vigilance. This is especially true when the effectiveness of vigilance is low and predators are very lethal. In general, these trends are true whether the costs of vigilance are felt on the prey's maximum growth rate or its carrying capacity. Just as the ecology of fear, when first introduced, was used to explain why top carnivores are rare in terrestrial systems, it can also be used to understand how big fierce predators can be common in IBPs.     

Emergence of donor control in patchy predator—prey systems

We study a general predator—prey system in a spatially heterogeneous environment. The predation process, which occurs on a behavioural time-scale, is much faster than the other processes (reproduction, natural mortality and migrations) occurring on the population dynamics time-scale. We show that, taking account of this difference in time-scales, and assuming that the prey have a refuge, the dynamics of the system on a slow time-scale become donor-controlled. Even though predators may control the prey density locally and on a behavioural fast time-scale, nevertheless, both globally and on a slow time-scale, the prey dynamics are independent of predator density: the presence of predators generates a constant prey mortality. In other words, in heterogeneous environments, the prey population dynamics depend in a switch-like manner on the presence or absence of predators, not on their actual density.     

Minimax strategies for a predator—Prey game

Suppose prey are distributed in patches. The predator knows the fraction of patches containing 0, 1, 2,… prey, but not how many prey a particular patch contains. It searches each patch randomly, at constant speed. It leaves a patch when the intercapture times satisfy a formula designed to maximize the number of prey eaten per unit time. We show that, if the prey distribution is Poisson, the predator should stay in each patch for the same time, regardless of what happens there. Accordingly, the prey can minimize the predator's maximum intake by choosing the Poisson distribution, and the predator can maximize its minimum intake (against a “smart” prey) by choosing the constant-time strategy.     

Predatory responses of selected lines of developmental variants of ladybird,Propylea dissecta (Coleoptera: Coccinellidae) in relation to increasing prey and predator densities

Individuals of the same species, population and generation frequently exhibit sub-maximal and significant genetic and phenotypic variation in their rate of development, showing slow and fast developers. Fast developers commonly have higher foraging and predation rates than slow developers. The consequence of such differences and foraging for the efficacy of biocontrol species remains under-explored. Slow and fast developers from a population of the ladybird, Propylea dissecta were separated and selected experimentally for F15 generations, and the predatory response of fourth instar larvae of control and experimentally selected slow and fast developers was then assessed at differing levels of prey (pea aphid, Acyrthosiphon pisum) and conspecific predator abundance. All individuals, whether slow or fast developers, showed a Type-II functional response, decrease in proportion of prey consumed with increasing prey biomass and an increase in proportion of prey consumed with increasing predator density. The proportion of prey consumed was highest in experimental fast developers and lowest in experimental slow developers. Attack rate was highest and handling time longest in slow developers of control/experimental groups. Mutual interference was least while area of discovery was highest in experimental fast developers. Thus, selection of fast developers for F15 generations led to higher functional responses, slower attack rates and faster prey consumption. This lower mutual interference and high searching efficiency indicates that they can be experimentally selected and used for better control of the pea aphids. This study is the first attempt to evaluate predatory responses of selected lines of an aphidophagous ladybird.     

Habitat choice in predator-prey systems: spatial instability due to interacting adaptive movements

The role of habitat choice behavior in the dynamics of predator-prey systems is explored using simple mathematical models. The models assume a three-species food chain in which each population is distributed across two or more habitats. The predator and prey adjust their locations dynamically to maximize individual per capita growth, while the prey's resource has a low rate of random movement. The two consumer species have Type II functional responses. For many parameter sets, the populations cycle, with predator and prey "chasing" each other back and forth between habitats. The cycles are driven by the aggregation of prey, which is advantageous because the predator's saturating functional response induces a short-term positive density dependence in prey fitness. The advantage of aggregation in a patch is only temporary because resources are depleted and predators move to or reproduce faster in the habitat with the largest number of prey, perpetuating the cycle. Such spatial cycling can stabilize population densities and qualitatively change the responses of population densities to environmental perturbations. These models show that the coupled processes of moving to habitats with higher fitness in predator and prey may often fail to produce ideal free distributions across habitats.     

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

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.