Plugging space into predator-prey models: an empirical approach

Extrapolating ecological processes from small-scale experimental systems to scales of natural populations usually entails a considerable increase in spatial heterogeneity, which may affect process rates and, ultimately, population dynamics. We demonstrate how information on the heterogeneity of natural populations can be taken into account when scaling up laboratory-derived process functions, using the technique of moment approximation. We apply moment approximation to a benthic crustacean predator-prey system, where a laboratory-derived functional response is made spatial by including correction terms for the variance in prey density and the covariance between prey and predator densities observed in the field. We also show how moment approximation may be used to incorporate spatial information into a dynamic model of the system. While the nonspatial model predicts stable dynamics, its spatial equivalent also produces bounded fluctuations, in agreement with observed dynamics. A detailed analysis shows that predator-prey covariance, but not prey variance, destabilizes the dynamics. We conclude that second-order moment approximation may provide a useful technique for including spatial information in population models. The main advantage of the method is its conceptual value: by providing explicit estimates of variance and covariance effects, it offers the possibility of understanding how heterogeneity affects ecological processes.     

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.     

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。     

Resolving discrepancies between deterministic population models and individual-based simulations

ABSTRACT This work ties together two distinct modeling frameworks for population dynamics: an individual-based simulation and a set of coupled integrodifferential equations involving population densities. The simulation model represents an idealized predator-prey system formulated at the scale of discrete individuals, explicitly incorporating their mutual interactions, whereas the population-level framework is a generalized version of reaction-diffusion models that incorporate population densities coupled to one another by interaction rates. Here I use various combinations of long-range dispersal for both the offspring and adult stages of both prey and predator species, providing a broad range of spatial and temporal dynamics, to compare and contrast the two model frameworks. Taking the individual-based modeling results as given, two examinations of the reaction-dispersal model are made: linear stability analysis of the deterministic equations and direct numerical solution of the model equations. I also modify the numerical solution in two ways to account for the stochastic nature of individual-based processes, which include independent, local perturbations in population density and a minimum population density within integration cells, below which the population is set to zero. These modifications introduce new parameters into the population-level model, which I adjust to reproduce the individual-based model results. The individual-based model is then modified to minimize the effects of stochasticity, producing a match of the predictions from the numerical integration of the population-level model without stochasticity.     

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.     

Explicit model for searching behavior of predator

The authors present an approach for explicit modeling of spatio-temporal dynamics of predator-prey community. This approach is based on a reaction-diffusion-adjection PD (prey dependent) system. Local kinetics of population is determined by logistic reproduction function of prey, constant natural mortality of predator and Holling type 2 trophic function. Searching behavior of predator is described by the advective term in predator balance equation assuming the predator acceleration to be proportional to the prey density gradient. The model was studied with zero-flux boundary conditions. The influence of predator searching activity on the community dynamics, in particular, on the emergence of spatial heterogeneity, has been investigated by linear analysis and numerical simulations. It has been shown how searching activity may effect the persistence of species, stabilizing predator-prey interactions at very low level of pest density. It has been demonstrated that obtaining of such dynamic regimes does not require the use of complex trophic functions.     

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.     

Intuition, functional responses and the formulation of predator-prey models when there is a large disparity in the spatial domains of the interacting species

1. The disparity of the spatial domains used by predators and prey is a common feature of many terrestrial avian and mammalian predatory interactions, as predators are typically more mobile and have larger home ranges than their prey. 2. Incorporating these realistic behavioural features requires formulating spatial predator-prey models having local prey mortality due to predation and its spatial aggregation, in order to generate a numerical response at timescales longer than the local prey consumption. Coupling the population dynamics occurring at different spatial scales is far from intuitive, and involves making important behavioural and demographic assumptions. Previous spatial predator-prey models resorted to intuition to derive local functional responses from non-spatial equivalents, and often involve unrealistic biological assumptions that restrict their validity. 3. We propose a hierarchical framework for deriving generic models of spatial predator-prey interactions that explicitly considers the behavioural and demographic processes occurring at different spatial and temporal scales. 4. The proposed framework highlights the circumstances wherein static spatial patterns emerge and can be a stabilizing mechanism of consumer-resource interactions.     

Consequences of size structure in the prey for predator-prey dynamics: the composite functional response

1. Current formulations of functional responses assume that the prey is homogeneous and independent of intraspecific processes. Most prey populations consist of different coexisting size classes that often engage in asymmetrical intraspecific interactions, including cannibalism, which can lead to nonlinear interaction effects. This may be important as the size structure with the prey could alter the overall density-dependent predation rates. 2. In a field experiment with damselfly and dragonfly larvae, 16 treatments manipulated the density of a small prey stage, the presence of large conspecific prey and the presence of heterospecific predators. 3. Size structure in the prey (i.e. when both prey stages were present) decreased the impact of the predator on overall prey mortality by 25-48% at mid and high prey densities, possibly due to density-dependent size-structured cannibalism in the prey. The predation rates on small prey stages were determined by the interaction of large prey and predators. Predation rates increased with prey density in the absence of large prey, but predation rates were constant across densities when large conspecifics were present. 4. The functional response for unstructured prey followed a Holling type III model, but the predation rate for size-structured prey was completely different and followed a complex pattern that could not be explained with any standard functional response. 5. Using additional laboratory experiments, a mortality model was developed and parameterized. It showed that the overall prey mortality of size-structured prey can be adequately predicted with a composite functional response model that modelled the individual functional responses of each prey stage separately and accounted for their cannibalistic interaction. 6. Thus, treating a prey population as a homogeneous entity will lead to erroneous predictions in most real-world food webs. However, if we account for the effects of size structure and the intraspecific interactions on functional responses by treating size classes as different functional groups, it is possible to reliably predict the dynamics of size-structured predator-prey systems.     

Impact of spatial heterogeneity on a predator-prey system dynamics

This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium.     

Intraguild predation with spatially structured interactions

Consumer–resource interactions with intraguild predation (IGP) were studied in a spatial setting (i.e., predators catch prey and individuals reproduce within local neighborhoods only). Pair approximation (a method for deriving ordinary differential equations that approximate the dynamics of a community that interacts in a lattice environment) was used to study the effect of spatially structured species interactions. An individual-based computer simulation was used to extend the study to a case with spatially variable resource densities. The qualitative results of the pair approximation model were similar to those of the corresponding non-spatial model. However, the spatial model predicted coex((istence over a wider range of parameters than the non-spatial model when intraguild prey are nutritionally valuable to intraguild predators. Spatially heterogeneous resource distributions and spatially structured interaction could overturn the qualitative predictions of non-spatial models.     

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.     

A shift from exploitation to interference competition with increasing density affects population and community dynamics

Intraspecific competition influences population and community dynamics and occurs via two mechanisms. Exploitative competition is an indirect effect that occurs through use of a shared resource and depends on resource availability. Interference competition occurs by obstructing access to a resource and may not depend on resource availability. Our study tested whether the strength of interference competition changes with protozoa population density. We grew experimental microcosms of protozoa and bacteria under different combinations of protozoan density and basal resource availability. We then solved a dynamic predator–prey model for parameters of the functional response using population growth rates measured in our experiment. As population density increased, competition shifted from exploitation to interference, and competition was less dependent on resource levels. Surprisingly, the effect of resources was weakest when competition was the most intense. We found that at low population densities, competition was largely exploitative and resource availability had a large effect on population growth rates, but the effect of resources was much weaker at high densities. This shift in competitive mechanism could have implications for interspecific competition, trophic interactions, community diversity, and natural selection. We also tested whether this shift in the mechanism of competition with protozoa density affected the structure of the bacterial prey community. We found that both resources and protozoa density affected the structure of the bacterial prey community, suggesting that competitive mechanism may also affect trophic interactions.     

Evolutionary dynamics of prey exploitation in a metapopulation of predators

In well-mixed populations of predators and prey, natural selection favors predators with high rates of prey consumption and population growth. When spatial structure prevents the populations from being well mixed, such predators may have a selective disadvantage because they do not make full use of the prey's growth capacity and hence produce fewer propagules. The best strategy then depends on the degree to which predators can monopolize the exploitation of local prey populations, which in turn depends on the spatial structure, the number of migrants, and, in particular, the stochastic nature of the colonization process. To analyze the evolutionary dynamics of predators in a spatially structured predator-prey system, we performed simulations with a metapopulation model that has explicit local dynamics of nonpersistent populations, keeps track of the number of emigrants entering the migration pool, assumes individuals within local populations as well as within the migration pool to be well mixed, and takes stochastic colonization into account. We investigated which of the predator's exploitation strategies are evolutionarily stable and whether these strategies minimize the overall density of prey, as is the case in Lotka-Volterra-type models of competitive exclusion. This was analyzed by pairwise invasibility plots based on short-term simulations and tested by long-term simulation experiments of competition between resident and mutant predator-types that differed in one of the following parameters: the prey-to-predator conversion efficiency, the per capita prey consumption rate, or the per capita emigration rate from local populations. In addition, we asked which of these three strategies are most likely to evolve. Our simulations showed that under selection for conversion efficiency the predator-prey system always goes globally extinct yet persists under selection for consumption or emigration rates and that the evolutionarily stable (ES) exploitation strategies do not maximize local population growth rates. The most successful exploitation strategy minimizes the overall density of prey but does not make it settle exactly at the minimum. The system did not settle at the point where the mean time to co-invasion (i.e., immigration of a second predator in a local prey population) equals the mean local interaction time (an idea borne out from studies on host exploitation strategies in host-pathogen systems) but rather where the mean time to co-invasion was larger. The ES exploitation strategies represent more prudent strategies than the ones that minimize prey density. Finally, we show that-compared to consumption-emigration is a more likely target for selection to achieve prudent exploitation and that prudent exploitation strategies can evolve only provided the prey-to-predator conversion efficiency is subject to constraints.     

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.     

A functional response model of a predator population foraging in a patchy habitat

1. Functional response models (e.g. Holling's disc equation) that do not take the spatial distributions of prey and predators into account are likely to produce biased estimates of predation rates. 2. To investigate the consequences of ignoring prey distribution and predator aggregation, a general analytical model of a predator population occupying a patchy environment with a single species of prey is developed. 3. The model includes the density and the spatial distribution of the prey population, the aggregative response of the predators and their mutual interference. 4. The model provides explicit solutions to a number of scenarios that can be independently combined: the prey has an even, random or clumped distribution, and the predators show a convex, sigmoid, linear or no aggregative response. 5. The model is parameterized with data from an acarine predator-prey system consisting of Phytoseiulus persimis and Tetranychus urticae inhabiting greenhouse cucumbers. 6. The model fits empirical data quite well and much better than if prey and predators were assumed to be evenly distributed among patches, or if the predators were distributed independently of the prey. 7. The analyses show that if the predators do not show an aggregative response it will always be an advantage to the prey to adopt a patchy distribution. On the other hand, if the predators are capable of responding to the distribution of prey, then it will be an advantage to the prey to be evenly distributed when its density is low and switch to a more patchy distribution when its density increases. The effect of mutual interference is negligible unless predator density is very high. 8. The model shows that prey patchiness and predator aggregation in combination can change the functional response at the population level from type II to type III, indicating that these factors may contribute to stabilization of predator-prey dynamics.     

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.     

Central assumptions of predator-prey models fail in a semi-natural experimental system

The relationship between the encounter rate of predators with prey and the density of this prey is fundamental to models of predator-prey interactions. The relationship determines, among other variables, the rate at which prey patches are depleted, and hence the impact of predator populations on their prey, and the optimal spatial distribution of foraging effort. Two central assumptions that are made in many models are that encounter rate is directly proportional to prey density and that it is independent of the proportion of prey already removed, other than via the decreased density. We show here, using captive great tits searching for winter moth caterpillars in their natural hiding positions, that neither of these assumptions hold. Encounter rate increased less than directly in proportion to prey density, and it depended not only on the current density of prey, but also on the proportion of prey already removed by previous foragers. Both of these effects are likely to have major consequences for the outcome of predator-prey interactions.     

Sensory Information and Encounter Rates of Interacting Species

Most motile organisms use sensory cues when searching for resources, mates, or prey. The searcher measures sensory data and adjusts its search behavior based on those data. Yet, classical models of species encounter rates assume that searchers move independently of their targets. This assumption leads to the familiar mass action-like encounter rate kinetics typically used in modeling species interactions. Here we show that this common approach can mischaracterize encounter rate kinetics if searchers use sensory information to search actively for targets. We use the example of predator-prey interactions to illustrate that predators capable of long-distance directional sensing can encounter prey at a rate proportional to prey density to the power (where is the dimension of the environment) when prey density is low. Similar anomalous encounter rate functions emerge even when predators pursue prey using only noisy, directionless signals. Thus, in both the high-information extreme of long-distance directional sensing, and the low-information extreme of noisy non-directional sensing, encounter rate kinetics differ qualitatively from those derived by classic theory of species interactions. Using a standard model of predator-prey population dynamics, we show that the new encounter rate kinetics derived here can change the outcome of species interactions. Our results demonstrate how the use of sensory information can alter the rates and outcomes of physical interactions in biological systems.     

Mechanisms of predator accumulation in a mixed crop system

Abstract. 1. In a previous study, Orius tristicolor (White), a generalist predator of soft-bodied insects and mites, invaded patches with a mixed plant assemblage at a faster rate than it did single-species stands. This study was designed to determine experimentally the underlying mechanisms of predator movement patterns that cause more intense colonization of crop mixtures.
2. Plots were established in a randomized complete block design. Treatments reflected different components of mixed crop patches: species richness, plant density, colour contrast, structural complexity, and volatile plant compounds.
3. O. tristicolor colonists were more abundant on squash intercropped with corn and cowpea than in squash monocultures even though early season prey densities were similar. Initial accumulation of the generalist predator was also higher in densely planted monocultures and in monocultures with artificially enhanced structural complexity than in normally-spaced squash monocultures.
4. Therefore, mechanisms underlying rates of predator colonization in a stand initially depended upon aspects of plant architecture and density and were independent of prey density and plant diversity. O. tristicolor densities at the end of the colonization period, however, were greater on squash ( Cucurbita pepo L.) in polyculture than in any other treatment.
5. These results suggest that attributes of the vegetation can influence the colonization rates of interacting organisms on different trophic levels and thus can alter predator-prey interactions and the development of community structure.