Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation

Interplay between predator and prey is a complex process in ecosystems due to its nature. The population dynamics can be affected by many extrinsic and intrinsic factors. In this paper, we make an attempt to uncover the effects from environmental disturbances when populations are subject to habitat complexity and aggregation effect. We firstly propose a stochastic predator-prey model with habitat complexity and aggregation efficiency for prey. We then mathematically analyze the model, to demonstrate the existence, uniqueness and the stochastically ultimately boundedness of the global positive solution, and to establish sufficient conditions for the existence of ergodic stationary distribution of the solution. We also establish sufficient conditions under which either only predator population dies out or the entire predator-prey model becomes extinct. Our theoretical and numerical results indicate that: (1) the environmental noises are disadvantage for the survival of biological populations; (2) when the density of prey is greater than one, prey aggregation can heighten the capability of predator species to capture prey and reduce the effect of environmental fluctuations, while when the density of prey is less than one, the results are opposite; (3) habitat complexity is propitious to the survival of prey population and may seriously threaten the persistence of the predator population.     

On the structure and stability of mutualistic systems: Analysis of predator-prey and competition models as modified by the action of a slow-growing mutualist

Two basic models of mutualism are presented in which interactions among three species lead to mutualism between two of them. The models represent 2-species predator-prey or competition systems in which a third species acts as a mutualist with either the predator, the prey, or one of the competitors. The models include the assumptions that there is a cost of associating with the mutualist and that the mutualist population grows much more slowly than the other two populations. Special cases of these two models correspond to six qualitatively different types of mutualistic benefit, all of which are known to occur in nature: deterring predation, increasing prey availability, feeding on (or competing with) a predator, increasing competitive interactions, decreasing competitive interactions, and feeding on (or competing with) a competitor. These models and their special cases are subjected to a local stability analysis. The results show that mutualism based upon deterring predation, competing with a predator, or decreasing competitive interactions enhances local stability, while mutualism based upon increasing prey availability or increasing competitive interactions reduces local stability. These results clearly reject the idea that mutualism is an inherently unstable process, and reinforces the idea that each different kind of mutualism will have to be considered separately. Compared to 2-species models of mutualism, the 3-species models provide a more realistic representation of the structure of many mutualistic systems, the mechanisms by which one species benefits another, and the regulation of the interaction.     

Prey Selection,Vertical Migrations and the Impacts of Harvesting Upon the Population Dynamics of a Predator-Prey System

A model is developed to describe the interaction between a predator and two prey types located in different regions. Conditions for stability and persistence are analysed. The effects of harvesting the predators are investigated by making the predator mortality rate habitat dependent. Results demonstrate that for any given set of parameter values there is a value of the intrinsic preference of the predator for each prey type at which the system undergoes a Hopf bifurcation. Above this critical value the system evolves towards a stable equilibrium, whereas below it, stable limit cycles arise by Hopf bifurcations. Simulations demonstrate that the presence of demographic stochasticity may destabilise oscillatory populations, thereby causing population extinctions. An application of the model to the foraging behaviour of North Sea cod is described. It is shown that if the preferred prey is more productive, it is likely that the equilibrium will be stable, whereas if the less preferred prey is more productive, populations are likely to display cycles and in the stochastic case become extinct. As cod fishing mortality is increased, the point of bifurcation and region of parameter space for which the system is unstable decreases. An increased understanding of how cod behave may enable fish stocks to be managed more successfully, for example by indicating where marine reserves should be placed.     

The effects of habitat fragmentation on persistence of source-sink metapopulations in systems with predators and prey or apparent competitors.

We consider systems with one predator and one prey, or a common predator and two prey species (apparent competitors) in source and sink habitats. In both models, the predator species is vulnerable to extinction, if productivity in the source is insufficient to rescue demographically deficient sink populations. Conversely, in the model with two prey species, if the source is too rich, one of the prey species may be driven extinct by apparent competition, since the predator can maintain a large population because of the alternative prey. Increasing the rate of predator movement from the source population has opposite effects on prey and predator persistence. High emigration rate exposes the predator population to danger of extinction, reducing the number of individuals that breed and produce offspring in the source habitat. This may promote coexistence of prey by relaxing predation pressure and apparent competition between the two prey species. The number of sinks and spatial arrangement of patches, or connectivity between patches, also influence persistence of the species. More sinks favor the prey and fewer sinks are advantageous to the predator. A linear pattern with the source at one end is profitable for the predator, and a centrifugal pattern in which the source is surrounded by sinks is advantageous to the prey. When the dispersal rate is low, effects of the spatial structure may exceed those of the number of sinks. In brief, productivity in patches and patterns of connectivity between patches differentially influence persistence of populations in different trophic levels.     

Diseased Social Predators

Social predators benefit from cooperation in the form of increased hunting success, but may be at higher risk of disease infection due to living in groups. Here, we use mathematical modeling to investigate the impact of disease transmission on the population dynamics benefits provided by group hunting. We consider a predator–prey model with foraging facilitation that can induce strong Allee effects in the predators. We extend this model by an infectious disease spreading horizontally and vertically in the predator population. The model is a system of three nonlinear differential equations. We analyze the equilibrium points and their stability as well as one- and two-parameter bifurcations. Our results show that weakly cooperating predators go unconditionally extinct for highly transmissible diseases. By contrast, if cooperation is strong enough, the social behavior mediates conditional predator persistence. The system is bistable, such that small predator populations are driven extinct by the disease or a lack of prey, and large predator populations survive because of their cooperation even though they would be doomed to extinction in the absence of group hunting. We identify a critical cooperation level that is needed to avoid the possibility of unconditional predator extinction. We also investigate how transmissibility and cooperation affect the stability of predator–prey dynamics. The introduction of parasites may be fatal for small populations of social predators that decline for other reasons. For invasive predators that cooperate strongly, biocontrol by releasing parasites alone may not be sufficient.     

Restricting Prey Dispersal Can Overestimate the Importance of Predation in Trophic Cascades

Predators can affect prey populations and, via trophic cascades, predators can indirectly impact resource populations (2 trophic levels below the predator) through consumption of prey (density-mediated indirect effects; DMIEs) and by inducing predator-avoidance behavior in prey (trait-mediated indirect effects; TMIEs). Prey often employ multiple predator-avoidance behaviors, such as dispersal or reduced foraging activity, but estimates of TMIEs are usually on individual behaviors. We assessed direct and indirect predator effects in a mesocosm experiment using a marine food chain consisting of a predator (toadfish – Opsanus tau), prey (mud crab - Panopeus herbstii) and resource (ribbed mussel – Geukensia demissa). We measured dispersal and foraging activity of prey separately by manipulating both the presence and absence of the predator, and whether prey could or could not disperse into a predator-free area. Consumption of prey was 9 times greater when prey could not disperse, probably because mesocosm boundaries increased predator capture success. Although predator presence did not significantly affect the number of crabs that emigrated, the presence of a predator decreased resource consumption by prey, which resulted in fewer resources consumed for each prey that emigrated in the presence of a predator, and reduced the overall TMIE. When prey were unable to disperse, TMIEs on mussel survival were 3 times higher than the DMIEs. When prey were allowed to disperse, the TMIEs on resource survival increased to 11-times the DMIEs. We found that restricting the ability of prey to disperse, or focusing on only one predator-avoidance behavior, may be underestimating TMIEs. Our results indicate that the relative contribution of behavior and consumption in food chain dynamics will depend on which predator-avoidance behaviors are allowed to occur and measured.     

Adaptation in a hybrid world with multiple interaction types: a new mechanism for species coexistence

In ecological communities, numerous species coexist and affect each others’ population levels via various types of interspecific interactions. Previous ecological theory explaining multispecies coexistence tended to focus on a single interaction type, such as antagonism, competition, or mutualism, and its consequences on population dynamics. Hence, it remains unclear what, if any, contribution multiple coexisting interaction types have on the multispecies coexistence. Here, we show that the coexistence of multiple interaction types can be essential for multispecies coexistence. We present a simple model in which the exploiter and mutualist adaptively switch between two competing resource species. An adaptive mutualist, which favors the more abundant species, provides a mechanism of majority-advantage and, thus, potentially inhibits the coexistence of resource species. In the absence of an exploiter, an adaptive mutualist leads to competitive exclusion at the resource species level. However, the coexistence of an adaptive exploiter and a mutualist allows the coexistence of all species in the community, because the mutualist-mediated “winner” tends to be suppressed by the adaptive exploiter. The mutualist indirectly increases the abundance of the exploiter through mutualistic interactions, thereby indirectly supporting this coexistence mechanism. In fact, coexistence may occur even if the exploiter or mutualist alone cannot mediate the coexistence of two resources. We conclude that the coexistence of mutualism and antagonism may be the key to the persistence of the four-species module in the presence of adaptive switching.     

Stochastic predation exposes prey to predator pits and local extinction

Understanding how predators affect prey populations is a fundamental goal for ecologists and wildlife managers. A well-known example of regulation by predators is the predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to attain a high density equilibrium. While empirical evidence for predator pits exists, deterministic models of predator–prey dynamics with realistic parameters suggest they should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochasticity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likelihood of extinction became more prevalent. We found that incorporating stochasticity is essential to fully understand alternative stable states in ecological systems, and due to the interaction between top–down and bottom–up effects on prey populations, habitat management and predator control could help prey to be resilient to predation stochasticity.     

A synthetic Escherichia coli predator–prey ecosystem

We have constructed a synthetic ecosystem consisting of two Escherichia coli populations, which communicate bi‐directionally through quorum sensing and regulate each other's gene expression and survival via engineered gene circuits. Our synthetic ecosystem resembles canonical predator–prey systems in terms of logic and dynamics. The predator cells kill the prey by inducing expression of a killer protein in the prey, while the prey rescue the predators by eliciting expression of an antidote protein in the predator. Extinction, coexistence and oscillatory dynamics of the predator and prey populations are possible depending on the operating conditions as experimentally validated by long‐term culturing of the system in microchemostats. A simple mathematical model is developed to capture these system dynamics. Coherent interplay between experiments and mathematical analysis enables exploration of the dynamics of interacting populations in a predictable manner.     

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 anatomy of predator-prey dynamics in a changing climate

Humans are increasingly influencing global climate and regional predator assemblages, yet a mechanistic understanding of how climate and predation interact to affect fluctuations in prey populations is currently lacking. Here we develop a modelling framework to explore the effects of different predation strategies on the response of age-structured prey populations to a changing climate. We show that predation acts in opposition to temporal correlation in climatic conditions to suppress prey population fluctuations. Ambush predators such as lions are shown to be more effective at suppressing fluctuations in their prey than cursorial predators such as wolves, which chase down prey over long distances, because they are more effective predators on prime-aged adults. We model climate as a Markov process and explore the consequences of future changes in climatic autocorrelation for population dynamics. We show that the presence of healthy predator populations will be particularly important in dampening prey population fluctuations if temporal correlation in climatic conditions increases in the future.     

Spatiotemporal behavior of a prey–predator system with a group defense for prey

Group defense is a strategy widely employed by various species. We consider the effect of grouping on population persistence when animals join together in herds in order to provide a self-defense from predators. In literature, group defense is usually addressed in terms of individual behavioral responses. In this paper, we consider an alternative ‘mean-field’ approach which uses prey and predator densities as the dynamical variables. The model is essentially a predator–prey system but with an unconventional parametrization for the predation term. We discuss the outcomes of the ecosystem dynamics in terms of persistence and prey survival. In the spatially distributed model some specific spatio-temporal features are discovered.     

The influence of generalist predators in spatially extended predator–prey systems

The presence of generalist predators is known to have important ecological impacts in several fields. They have wide applicability in the field of biological control. However, their role in the spatial distribution of predator and prey populations is still not clear. In this paper, the spatial dynamics of a predator–prey system is investigated by considering two different types of generalist predators. In one case, it is considered that the predator population has an additional food source and can survive in the absence of the prey population. In the other case, the predator population is involved in intraguild predation, i.e., the source of the additional food of the predator coincides with the food source of the prey population and thus both prey and predator populations compete for the same resource. The conditions for linear stability and Turing instability are analyzed for both the cases. In the presence of generalist predators, the system shows different pattern formations and spatiotemporal chaos which has important implications for ecosystem functioning not only in terms of their predictability, but also in influencing species persistence and ecosystem stability in response to abrupt environmental changes. This study establishes the importance of the consideration of spatial dynamics while determining optimal strategies for biological control through generalist predators.     

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.     

Contemporary evolution and genetic change of prey as a response to predator removal

The ecological effects of predator removal and its consequence on prey behavior have been investigated widely; however, predator removal can also cause contemporary evolution of prey resulting in prey genetic change. Here we tested the role of predator removal on the contemporary evolution of prey traits such as movement, reproduction and foraging. We use EcoSim simulation which allows complex intra- and inter-specific interactions, based on individual evolving behavioral models, as well as complex predator–prey dynamics and coevolution in spatially homogenous and heterogeneous worlds. We model organisms' behavior using fuzzy cognitive maps (FCM) that are coded in their genomes which has a clear semantics making reasoning about causality of any evolved behavior possible. We show that the contemporary evolution of prey behavior owing to predator removal is also accompanied by prey genetic change. We employed machine learning methods, now recognized as holding great promise for the advancement of our understanding and prediction of ecological phenomena. A classification algorithm was used to demonstrate the difference between genomes belonging to prey coevolving with predators and prey evolving in the absence of predation pressure. We argue that predator introductions to naive prey might be destabilizing if prey have evolved and adapted to the absence of predators. Our results suggest that both predator introductions and predator removal from an ecosystem have widespread effects on the survival and evolution of prey by altering their genomes and behavior, even after relatively short time intervals. Our study highlights the need to consider both ecological and evolutionary time scales, as well as the complex interplay of behaviors between trophic levels, in determining the outcomes of predator–prey interactions.     

Female blue tits with brighter yellow chests transfer more carotenoids to their eggs after an immune challenge

Predicting the consequences of predator biodiversity loss on prey requires an understanding of multiple predator interactions. Predators are often assumed to have independent and additive effects on shared prey survival; however, multiple predator effects can be non-additive if predators foraging together reduce prey survival (risk enhancement) or increase prey survival through interference (risk reduction). In marine communities, juvenile reef fish experience very high mortality from two predator guilds with very different hunting modes and foraging domains—benthic and pelagic predator guilds. The few previous predator manipulation studies have found or assumed that mortality is independent and additive. We tested whether interacting predator guilds result in non-additive prey mortality and whether the detection of such effects change over time as prey are depleted. To do so, we examined the roles of benthic and pelagic predators on the survival of a juvenile shoaling zooplanktivorous temperate reef fish, Trachinops caudimaculatus, on artificial patch reefs over 2 months in Port Phillip Bay, Australia. We observed risk enhancement in the first 7 days, as shoaling behaviour placed prey between predator foraging domains with no effective refuge. At day 14 we observed additive mortality, and risk enhancement was no longer detectable. By days 28 and 62, pelagic predators were no longer significant sources of mortality and additivity was trivial. We hypothesize that declines in prey density led to reduced shoaling behaviour that brought prey more often into the domain of benthic predators, resulting in limited mortality from pelagic predators. Furthermore, pelagic predators may have spent less time patrolling reefs in response to declines in prey numbers. Our observation of the changing interaction between predators and prey has important implications for assessing the role of predation in regulating populations in complex communities.     

The evolution of traits affecting resource acquisition and predator vulnerability: character displacement under real and apparent competition

This article investigates some simple models of the evolutionary interaction between two prey species that share a common resource and a common predator. Each prey species is characterized by a trait that determines both the rate of resource capture and vulnerability to a predator. In a simple model of a three-species food chain, such traits usually increase in response to an imposed reduction in resource density. When the per capita growth rates of each of two prey species depend linearly on resource density, such traits will change in opposite directions when the two prey come into sympatry. In addition, the ratio of the effect of the predator on prey fitness to the effect of the resource on prey fitness will diverge from the corresponding ratio in a second prey species when those species coexist in sympatry. These simple predictions need not hold under several alternative assumptions, which may be more common in biological systems. Parallel changes in sympatry may occur if the relationship between resource consumption and prey growth is nonlinear, if the prey species have partial overlap in the set of resources used or in the set of predators that consume them, or if prey experience direct intraspecific competition. The responses to a second prey can also differ significantly from those predicted by the simplest model if separate traits affect vulnerability to predators and resource acquisition rate. It is important to determine whether examples of character displacement previously interpreted as responses to competition for resources might also reflect responses to altered predation risks in sympatry.     

Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions

Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.     

Size and scaling of predator-prey dynamics

We propose a scaled version of the Rosenzweig–MacArthur model using both Type I and Type II functional responses that incorporates the size dependence of interaction rates. Our aim is to link the energetic needs of organisms with the dynamics of interacting populations, for which survival is a result of a game-theoretic struggle for existence. We solve the scaled model of predator–prey dynamics and predict population level characteristics such as the scaling of coexistence size ranges and the optimal predator–prey size ratio. For a broad class of such models, the optimal predator–prey size ratio given available prey of a fixed size is constant. We also demonstrate how scaling predictions of prey density differ under resource limitation vs. predator drawdown. Finally, we show how evolution of predator size can destabilize population dynamics, compare scaling of predator–prey cycles to previous work, as well as discuss possible extensions of the model to multispecies communities.     

Dynamics of a intraguild predation model with generalist or specialist predator

Intraguild predation (IGP) is a combination of competition and predation which is the most basic system in food webs that contains three species where two species that are involved in a predator/prey relationship are also competing for a shared resource or prey. We formulate two intraguild predation (IGP: resource, IG prey and IG predator) models: one has generalist predator while the other one has specialist predator. Both models have Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type III functional response between IG prey and IG predator. We provide sufficient conditions of the persistence and extinction of all possible scenarios for these two models, which give us a complete picture on their global dynamics. In addition, we show that both IGP models can have multiple interior equilibria under certain parameters range. These analytical results indicate that IGP model with generalist predator has “top down” regulation by comparing to IGP model with specialist predator. Our analysis and numerical simulations suggest that: (1) Both IGP models can have multiple attractors with complicated dynamical patterns; (2) Only IGP model with specialist predator can have both boundary attractor and interior attractor, i.e., whether the system has the extinction of one species or the coexistence of three species depending on initial conditions; (3) IGP model with generalist predator is prone to have coexistence of three species.