Complexity in a prey-predator model with prey refuge and diffusion

Prey-predator interaction is one of the most commonly observed relationships in ecosystem. In the study of prey-predator models, it is frequently assumed that the changes in population densities are only time-dependent and the dynamics is generally represented by coupled nonlinear ordinary differential equations. In natural system, however, either prey or predator or both move from one place to another for various reasons. In such a case, their dynamic interaction depends both on time and space and requires coupled nonlinear partial differential equations for its dynamic representation. It is also well documented that prey refuges affect the interaction between prey and predator significantly. In this paper, we studied the dynamics of a diffusive prey-predator interaction with prey refuge and type III response function. We have considered both one and two dimensional diffusivity in the model system and presented different stability results under the assumptions that one or both species may be mobile or sedentary. Our results showed that the system may exhibit different spatiotemporal (non-Turing) patterns, like spiral waves, patchy structures, spot pattern, or even spatiotemporal chaos depending on the refuge availability and diffusion rate of species. Another interesting finding was that the dynamic complexity in a prey-predator model increases in case of mobile predator and sedentary prey compare to mobile prey and sedentary predator while refuge availability is varied.     

Dynamical analysis of a prey-predator model incorporating a prey refuge with variable carrying capacity

A prey-predator model incorporating prey refuge with variable carrying capacity and Holling type-II functional response is proposed and analyzed. The model includes a case of increasing carrying capacity as well as a decreasing carrying capacity case. Sufficient conditions are derived to ensure the existence and local stability of the equilibrium points of the proposed model. Moreover, the occurrence of transcritical bifurcation as well as Hopf bifurcation are investigated. The effect of some model parameter related to the prey refuge and the variable carrying capacity on the prey-predator dynamics has been examined. Numerical simulations are presented to demonstrate the theoretical results and to illustrate the effect of these parameters on the model dynamics. Moreover, a comparison with the constant carrying case has been presented.     

具有捕食正效应的捕食-食饵系统

在经典的捕食食饵系统中考虑到由于捕食效应对食饵种群带来的正向调节作用后,提出了具有捕食正效应的捕食-食饵系统.通过对模型的动力学行为的分析,从理论上说明了正向调节作用对系统的影响,并就第一象限内平衡点存在时的相图解释了捕食正效应的作用.结果表明:(1)捕食系统中适当的正向调节作用会增加系统的稳定性;(2)当捕食正效应达到一定的程度后系统拥有一个不稳定的极限环;(3)当捕食正效应过大时会使系统的稳定性发生变化,使捕食者种群与食饵种群同时趋向无穷,出现了调节放纵现象.这些结果在保护生物学中具有重要的意义.     

Stochastic modelling of prey depletion processes

The modelling of prey-predator interactions is of major importance for the understanding of population dynamics. Classically, these interactions are modelled using ordinary differential equations, but this approach has the drawbacks of assuming continuous population variables and of being deterministic. We propose a general approach to stochastic modelling based on the concept of functional response for a prey depletion process with a constant number of predators. Our model could involve any kind of functional response, and permits a likelihood-based approach to statistical modelling and stable computation using matrix exponentials. To illustrate the method we use the Holling-Juliano functional response and compare the outcomes of our model with a deterministic counterpart considered by Schenk and Bacher [2002. Functional response of a generalist insect predator to one of its prey species in the field. Journal of Animal Ecology 71 (3), 524-531], who observed the depletion of Cassida rubiginosa due to its exclusive predator, Polistes dominulus. The predation was found to be Holling type III, reflecting the ability of the predator to regulate its prey. Our approach corroborates this result, but suggests that the prey depletion census should have been performed more often, and that predation features were significantly different between the two years for which data are available.     

Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator

In the present paper, we develop a simple two species prey-predator model in which the predator is partially coupled with alternative prey. The aim is to study the consequences of providing additional food to the predator as well as the effects of harvesting efforts applied to both the species. It is observed that the provision of alternative food to predator is not always beneficial to the system. A complete picture of the long run dynamics of the system is discussed based on the effort pair as control parameters. Optimal augmentations of prey and predator biomass at final time have been investigated by optimal control theory. Also the short and large time effects of the application of optimal control have been discussed. Finally, some numerical illustrations are given to verify our analytical results with the help of different sets of parameters.     

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r₀ as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r₀. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.     

Alternative prey can change model–mimic dynamics between parasitism and mutualism

Classical (conventional) Müllerian mimicry theory predicts that two (or more) defended prey sharing the same signal always benefit each other despite the fact that one species can be more toxic than the other. The quasi‐Batesian (unconventional) mimicry theory, instead, predicts that the less defended partner of the mimetic relationship may act as a parasite of the signal, causing a fitness loss to the model. Here we clarify the conditions for parasitic or mutualistic relationships between aposematic prey, and build a model to examine the hypothesis that the availability of alternative prey is crucial to Müllerian and quasi‐Batesian mimicry. Our model is based on optimal behaviour of the predator. We ask if and when it is in the interest of the predator to learn to avoid certain species as prey when there is alternative (cryptic) prey available. Our model clearly shows that the role of alternative prey must be taken into consideration when studying model–mimic dynamics. When food is scarce it pays for the predator to test the models and mimics, whereas if food is abundant predators should leave the mimics and models untouched even if the mimics are quite edible. Dynamics of the mimicry tend to be classically Müllerian if mimics are well defended, while quasi‐Batesian dynamics are more likely when they are relatively edible. However, there is significant overlap: in extreme cases mimics can be harmful to models (a quasi‐Batesian case) even if the species are equally toxic. A crucial parameter explaining this overlap is the search efficiency with which indiscriminating vs. discriminating predators find cryptic prey. Quasi‐Batesian mimicry becomes much more likely if discrimination increases the efficiency with which the specialized predator finds cryptic prey, while the opposite case tends to predict Müllerian mimicry. Our model shows that both mutualistic and parasitic relationship between model and mimic are possible and the availability of alternative prey can easily alter this relationship.     

Individual variation in prey choice in a predator-prey community

One predator-two prey community models are studied with an emphasis on individual variation in predator behavior. The predator behaves according to a well-known prey choice model. The behavioral model predicts that predators should always attack the primary prey (more profitable prey of the two), but only attack the alternative prey (less profitable prey of the two) when the density of the primary prey is below a threshold density. The predator that accepts the alternative prey does not discriminate between the primary and alternative prey (all-or-nothing preference for the alternative prey). However, empirical studies do not result in clear all-or-nothing responses. Previous models examined the relaxation of the all-or-nothing response by assuming partial preference (e.g., predators preferentially forage on the primary prey even when they also attack the alternative prey). In this study, I consider individual variation in two predator traits (prey density perception and handling time) as the sources of the variation in the threshold density, which can make empirical data appear deviated from the expectation. I examine how community models with partial preference and individual variation differ in their dynamics and show that the differences can be substantial. For example, the dynamics of a model based on individual variation can be more stable (e.g., stable in a wider parameter region) than that of a model based on partial preference. As the general statistical property (Jensen’s inequality) is a main factor that causes the differences, the results of the study have general implications to the interpretation of models based on average per-capita rates.     

Unidirectional prey-predator facilitation: apparent prey enhance predators' foraging success on cryptic prey

Food availability can strongly affect predator-prey dynamics. When change in habitat condition reduces the availability of one prey type, predators often search for other prey, perhaps in a different habitat. Interactions between behavioural and morphological traits of different prey may influence foraging success of visual predators through trait-mediated indirect interactions (TMIIs), such as prey activity and body coloration. We tested the hypothesis that foraging success of stream-dwelling cutthroat trout (Onchorhyncus clarki) on cryptically coloured, less-active benthic prey (larval mayfly; Paraleptophebia sp.) can be enhanced by the presence of distinctly coloured, active prey (larval stonefly shredder; Despaxia augusta). Cutthroat trout preyed on benthic insects when drifting invertebrates were unavailable. When stonefly larvae were present, the trout ate most of the stoneflies and also consumed a higher proportion of mayflies than under mayfly only treatment. The putative mechanism is that active stonefly larvae supplied visual cues to the predator that alerted trout to the mayfly larvae. Foraging success of visual predators on cryptic prey can be enhanced by distinctly coloured, active benthic taxa through unidirectional facilitation to the predators, which is a functional change of interspecific interaction caused by a third species. This study suggests that prey-predator facilitation through TMIIs can modify species interactions, affecting community dynamics.     

Bistability and limit cycles in generalist predator–prey dynamics

Since generalist predators feed on a variety of prey species they tend to persist in an ecosystem even if one particular prey species is absent. Predation by generalist predators is typically characterized by a sigmoidal functional response, so that predation pressure for a given prey species is small when the density of that prey is low. Many mathematical models have included a sigmoidal functional response into predator–prey equations and found the dynamics to be more stable than for a Holling type II functional response. However, almost none of these models considers alternative food sources for the generalist predator. In particular, in these models, the generalist predator goes extinct in the absence of the one focal prey. We model the dynamics of a generalist predator with a sigmoidal functional response on one dynamic prey and fixed alternative food source. We find that the system can exhibit up to six steady states, bistability, limit cycles and several global bifurcations.     

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.     

Effects of successive predator attacks on prey aggregations

We study the cumulative effect of successive predator attacks on the disturbance of a prey aggregation using a modelling approach. Our model intends to represent fish schools attacked by both aerial and underwater predators. This individual-based model uses long-distance attraction and short-distance repulsion between prey, which leads to prey aggregation and swarming in the absence of predators. When intermediate-distance alignment is added to the model, the prey aggregation displays a cohesive displacement, i.e., schooling, instead of swarming. Including predators, i.e. with repulsion behaviour for prey to predators in the model, leads to flash expansion of the prey aggregation after a predator attack. When several predators attack successively, the prey aggregation dynamics is a succession of expanding-grouping-swarming/schooling phases. We quantify this dynamics by recording the changes in the simulated prey aggregation radius over time. This radius is computed as the longest distance of individual prey to the aggregation centroid, and it is assumed to increase along with prey disturbance. The prey aggregation radius generally increases during flash expansion, then decreases during grouping until reaching a constant lowest level during swarming/schooling. This general dynamics is modulated by several parameters: the frequency, direction (vertical vs. horizontal) and target (centroid of the prey aggregation vs. random prey) of predator attacks; the distance at which prey detect predators; the number of prey and predators. Our results suggest that both aerial and underwater predators are more efficient at disturbing fish schools by increasing their attack frequency at such level that the fish cannot return to swarming/schooling. We find that a mix between aerial and underwater predators is more efficient at disturbing a fish school than a single type of attack, suggesting that aerial and underwater foragers may gain mutual benefits in forming foraging groups.     

捕食者具有厌食性反应且食饵具有Allee效应的捕食系统

在Dubis动力系统的基础上,建立了具有Allee效应的捕食系统模型。对系统的稳定性进行了分析,受Allee效应的影响,食饵种群可能因为种群大小处于临界点以下而趋于灭绝。通过对系统进行模拟,结果表明:不受Allee效应的影响,系统的演化属于一种理想化的情形系统到达P(平衡)点的时间较不受Allee效应影响时系统到达P点的时间短,不利于生物的进化,而在Allee效应的影响下,系统的演化将达到一个平衡状态。由此,说明Allee效应为濒临灭绝物种的管理提供了重要的理论依据,对管理部门的决策有参考指导作用。     

The advantage of alternative tactics of prey and predators depends on the spatial pattern of prey and social interactions among predators

Individual variation in behavioral strategies is ubiquitous in nature. Yet, explaining how this variation is being maintained remains a challenging task. We use a spatially-explicit individual-based simulation model to evaluate the extent to which the efficiency of an alternative spacing tactic of prey and an alternative search tactic of predators are influenced by the spatial pattern of prey, social interactions among predators (i.e., interference and information sharing) and predator density. In response to predation risk, prey individuals can either spread out or aggregate. We demonstrate that if prey is extremely clumped, spreading out may help when predators share information regarding prey locations and when predators shift to area-restricted search following an encounter with prey. However, dispersion is counter-selected when predators interact by interference, especially under high predator density. When predators search for more randomly distributed prey, interference and information sharing similarly affect the relative advantage of spreading out. Under a clumped prey spatial pattern, predators benefit from shifting their search tactic to an area-restricted search following an encounter with prey. This advantage is moderated as predator density increases and when predators interact either by interference or information sharing. Under a more random prey pattern, information sharing may deteriorate the inferior search tactic even more, compared to interference or no interaction among predators. Our simulation clarifies how interactions among searching predators may affect aggregation behavior of prey, the relative success of alternative search tactics and their potential to invade established populations using some other search or spacing tactics.     

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.     

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.     

Role of nutrient bound of prey on the dynamics of predator-mediated competitive-coexistence

The coexistence of competitive species with a shared predator is well established. The effect of 'food-value' on predator-prey dynamics has also received much attention. However, the study of a nutrient bound of prey, specifically on predator-mediated competitive-coexistence has not received much attention. Here we study the effects of the caloric content or a nutrient bound of prey on the dynamics of competitive-coexistence with the shared predator in a specific model. We propose and analyze a mathematical model for exploitative competition of two prey species with a shared predator. The change of dynamic stability due to the variation of a nutrient bound of each prey on predator-mediated dynamics is studied through extensive numerical experiments. Our analytical and numerical results demonstrate that variation in a nutrient bound promotes the switching of dynamics and may be treated as a driving force for the dynamics of competitive-coexistence with the shared predator.     

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.     

A powerful truncated tail strength method for testing multiple null hypotheses in one dataset

This article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a higher densities they were available to predators. Thus, their functional response was of the Holling type III. They analyzed this model and predicted existence of a limit cycle in predator-prey dynamics. In this article I show that their model is ill posed, because trajectories are not well defined. Using the Filippov method, I define and analyze solutions of the Gause model. I show that depending on parameter values, there are three possibilities: (1) trajectories converge to a limit cycle, as predicted by Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.     

Coevolution in a one predator-two prey system

Background

Our understanding of coevolution in a predator–prey system is based mostly on pair-wise interactions.

Methodology and Principal Findings

Here I analyze a one-predator–two-prey system in which the predator''s attack ability and the defense abilities of the prey all evolve. The coevolutionary consequences can differ dramatically depending on the initial trait value and the timing of the alternative prey''s invasion into the original system. If the invading prey species has relatively low defense ability when it invades, its defense is likely to evolve to a lower level, stabilizing the population dynamics. In contrast, if when it invades its defense ability is close to that of the resident prey, its defense can evolve to a higher level and that of the resident prey may suddenly cease to evolve, destabilizing the population dynamics. Destabilization due to invasion is likely when the invading prey is adaptively superior (evolution of its defense is less constrained and fast), and it can also occur in a broad condition even when the invading prey is adaptively inferior. In addition, invasion into a resident system far from equilibrium characterized by population oscillations is likely to cause further destabilization.

Conclusions and Significance

An invading prey species is thus likely to destabilize a resident community.