Uniqueness and stability of positive steady state solutions for a ratio-dependent predator–prey system with a crowding term in the prey equation

This paper deals with a ratio-dependent predator–prey system with a crowding term in the prey equation, where it is assumed that the coefficient of the functional response is less than the coefficient of the intrinsic growth rates of the prey species. We demonstrate some special behaviors of solutions to the system which the coexistence states of two species can be obtained when the crowding region in the prey equation only is designed suitably. Furthermore, we demonstrate that under some conditions, the positive steady state solution of the predator–prey system with a crowding term in the prey equation is unique and stable. Our result is different from those ones of the predator–prey systems without the crowding terms.     

HARVESTING A TWO-PATCH PREDATOR-PREY METAPOPULATION

ABSTRACT. A mathematical model for a two-patch predator-prey metapoplation is developed as a generalization of single-species metapopulation harvesting theory. We find optimal harvesting strategies using dynamic programming and La-grange multipliers. If predator economic efficiency is relatively high, then we should protect a relative source prey subpopulation in two different ways: directly, with a higher escapement of the relative source prey subpopulation, and indirectly, with a lower escapement of the predator living in the same patch as the relative source prey subpopulation. Numerical examples show that if the growth of the predator is relatively low and there is no difference between prey and predator prices, then it may be optimal to harvest the predator to extinction. While, if the predator is more valuable compared to the prey, then it may be optimal to leave the relative exporter prey subpopulation unharvested. We also discuss how a ‘negative’ harvest might be optimal. A negative harvest might be considered a seeding strategy.     

A foraging problem: Sit-and-wait versus active predation

The literature on foraging shows that some predators use a combination of ambush and active search to locate a prey. Let us suppose that a prey must go every day to some determined places to feed, and to another place, 0, to drink. A predator can stay at zone 0 waiting for the prey (sit-and-wait strategy) or it can move between the different places where the prey will go to eat (search strategy). If predator and prey meet each other in the same place, prey will be caught with a probability depending on the place. We study this problem in different situations, modelling them as two-person zero-sum games. We solve them in closed form, giving optimal strategies for prey and for predator and the value of the games.     

Conjugate coupling in ecosystems: Cross-predation stabilizes food webs

We study the dynamics of two predator–prey systems that are coupled via cross-predation, in which each predator consumes also the other prey. This setup constitutes a model system in which conjugate coupling emerges naturally and denotes the transition from two separate food chains to a food web. We show that cross-predation of a certain strength leads to amplitude death stabilizing the food web in a new equilibrium.     

A Leslie–Gower predator–prey model with disease in prey incorporating a prey refuge

In this paper, a predator–prey Leslie–Gower model with disease in prey has been developed. The total population has been divided into three classes, namely susceptible prey, infected prey and predator population. We have also incorporated an infected prey refuge in the model. We have studied the positivity and boundedness of the solutions of the system and analyzed the existence of various equilibrium points and stability of the system at those equilibrium points. We have also discussed the influence of the infected prey refuge on each population density. It is observed that a Hopf bifurcation may occur about the interior equilibrium taking refuge parameter as bifurcation parameter. Our analytical findings are illustrated through computer simulation using MATLAB, which show the reliability of our model from the eco-epidemiological point of view.     

Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure

We present a predator-prey model of Beddington-DeAngelis type functional response with stage structure on prey. The constant time delay is the time taken from birth to maturity about the prey. By the uniform persistence theories and monotone dynamic theories, sharp threshold conditions which are both necessary and sufficient for the permanence and extinction of the model as well as the sufficient conditions for the global stability of the coexistence equilibria are obtained. Biologically, it is proved that the variation of prey stage structure can affect the permanence of the system and drive the predator into extinction by changing the prey carrying capacity: Our results suggest that the predator coexists with prey permanently if and only if predator's recruitment rate at the peak of prey abundance is larger than its death rate; and that the predator goes extinct if and only if predator's possible highest recruitment rate is less than or equal to its death rate; furthermore, our results also show that a sufficiently large mutual interference by predators can stabilize the system.     

Stationary distribution and extinction of a stochastic predator–prey model with distributed delay

In this paper, we focus on a stochastic predator–prey model with distributed delay. We first obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Then we establish sufficient conditions for extinction of the predator population, that is, the prey population is survival and the predator population is extinct.     

Effect of emergent carrying capacity in an eco‐epidemiological system

The paper explores an eco‐epidemiological model of a predator–prey type, where the prey population is subject to infection. The model is basically a combination of S‐I type model and a Rosenzweig–MacArthur predator–prey model. The novelty of this contribution is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We explicitly separate the competition between non‐infected and infected individuals. This emergent carrying capacity is markedly different to the explicit carrying capacities that have been considered in many eco‐epidemiological models. We observed that different intra‐class and inter‐class competition can facilitate the coexistence of susceptible prey‐infected prey–predator, which is impossible for the case of the explicit carrying capacity model. We also show that these findings are closely associated with bi‐stability. The present system undergoes bi‐stability in two different scenarios: (a) bi‐stability between the planner equilibria where susceptible prey co‐exists with predator or infected prey and (b) bi‐stability between co‐existence equilibrium and the planner equilibrium where susceptible prey coexists with infected prey; have been discussed. The conditions for which the system is to be permanent and the global stability of the system around disease‐free equilibrium are worked out. Copyright © 2015 John Wiley & Sons, Ltd.     

Modelling a predator–prey system with infected prey in polluted environment

A predator–prey model with logistic growth in prey is modified by introducing an SIS parasite infection in the prey. We have studied the combined effect of environmental toxicant and disease on prey–predator system. It is assumed in this paper that the environmental toxicant affects both prey and predator population and the infected prey is assumed to be more vulnerable to the toxicant and predation compared to the sound prey individuals. Thresholds are identified which determine when system persists and disease remains endemic.     

Eco-epidemiological model with fatal disease in the prey

We investigate a model consisting of a predator population and both susceptible and infected prey populations. The predator can feed on either prey species but instead of choosing individuals at random the predator feeds preferentially on the most abundant prey species. More specifically we assume that the likelihood of a predator catching a susceptible prey or an infected prey is proportional to the numbers of these two different types of prey species. This phenomenon, involving changing preference from susceptible to infected prey, is called switching. Mukhopadhyay studied a switching model and proposed that the interaction of predators with infected prey is beneficial for the growth of the predator. In this model, we assume that the predator will eventually die as a result of eating infected prey. We find a threshold parameter $R_0$ and showed that the disease will be eradicated from the system if $R_0<1$.     

Classical predator–prey system with infection of prey population—a mathematical model

The present paper deals with the problem of a classical predator–prey system with infection of prey population. A classical predator–prey system is split into three groups, namely susceptible prey, infected prey and predator. The relative removal rate of the susceptible prey due to infection is worked out. We observe the dynamical behaviour of this system around each of the equilibria and point out the exchange of stability. It is shown that local asymptotic stability of the system around the positive interior equilibrium ensures its global asymptotic stability. We prove that there is always a Hopf bifurcation for increasing transmission rate. To substantiate the analytical findings, numerical experiments have been carried out for hypothetical set of parameter values. Our analysis shows that there is a threshold level of infection below which all the three species will persist and above which the disease will be epidemic. Copyright © 2003 John Wiley & Sons, Ltd.     

Influence of predator mutual interference and prey refuge on Lotka–Volterra predator–prey dynamics

A Lotka–Volterra predator–prey model incorporating a constant number of prey using refuges and mutual interference for predator species is presented. By applying the divergency criterion and theories on exceptional directions and normal sectors, we show that the interior equilibrium is always globally asymptotically stable and two boundary equilibria are both saddle points. Our results indicate that prey refuge has no influence on the coexistence of predator and prey species of the considered model under the effects of mutual interference for predator species, which differently from the conclusion without predator mutual interference, thus improving some known ones. Numerical simulations are performed to illustrate the validity of our results.     

Role of environmental disturbance in an eco‐epidemiological model with disease from external source

An eco‐epidemiological model with random environmental disturbance is proposed and analyzed. We assume that the susceptible prey population can acquire infection both from external sources and from internal transmission of the disease. It is also assumed that there is no recovery of the disease, and the consumption of diseased prey has a deleterious effect on the predator population. The conditions for the extinction of the predator and the prey populations are worked out. The most important observation of the present investigation is that oscillatory behavior of the populations observed in deterministic framework undergoes stable coexistence in the stochastic framework. Copyright © 2012 John Wiley & Sons, Ltd.     

Spreading speeds for time heterogeneous prey–predator systems with diffusion

We investigate the large time behavior for two components reaction–diffusion systems of prey–predator type in a time varying environment. Here we assume that these variations in time exhibit an averaging property, which will be called mean value in this work. This framework includes in particular time periodicity, almost periodicity and unique ergodicity. We describe the spreading behavior of the prey and the predator, wherein the two populations are able to co-invade the empty space. Our analysis is based the parabolic strong maximum principle for scalar equation and on the derivation of local pointwise estimates that are used to compare the solutions of the prey–predator problem with those of a KPP scalar equation on suitable spatio-temporal domains.     

具Ⅱ型Holling功能性反应强耦合椭圆系统的共存问题

本文研究带齐次Dirichlet边界条件的强耦合椭圆系统,首先证明了当食饵和捕食者的扩散率足够大,或者出生率足够小时,系统不存在共存现象,并给出半平凡解存在的充分条件．然后利用Schauder不动点定理,得到强耦合的椭圆问题至少有一个正解存在的充分条件．该条件说明只要捕食者的内部竞争强,物种的交叉扩散相对弱,或者捕获率足够小,物种的交叉扩散相对弱,强耦合系统就至少有一个正解存在．     

MANAGING ECOLOGICALLY INTERDEPENDENT SPECIES

ABSTRACT. We analyze open access harvesting of a predator species, while allowing for ecological interaction with herbivore species (the prey). In contrast to existing studies, we find that under some conditions open access harvesting may contribute to the abundance of predator and prey species. Particularly in fragmented habitats, moderate harvesting intensity may be a low‐cost substitute for management, and measures to reduce harvesting may result in collapse of predator and prey stock. These results highlight the importance of analyzing the ecological underpinnings of systems when manipulating economic parameters to promote conservation.     

The dynamics of an age structured predator–prey model with disturbing pulse and time delays

Many of the existing predator–prey models on stage structured populations are some ordinary differential equations (ODE) or models without a disturbing effect of human behavior. In reality, death of the juvenile during its immature stage and catching or poisoning for the prey or predator occur continuously. From this basic standpoint, we formulate a general and robust prey-dependent consumption predator–prey model with periodic harvesting (catching or poisoning) for the prey and stage structure for the predator with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and ecological study. We show that the conditions for global attractivity of the ‘predator-extinction’ (‘predator-eradication’) periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. We also show that constant maturation time delay and impulsive catching or poisoning for the prey can bring great effects on the dynamics of system by numerical analysis. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy–pest) model with age structure, exhibit a new modeling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

Effect of prey-taxis on predator’s invasion in a spatially heterogeneous environment

In this study, we consider a diffusive predator–prey system with prey-taxis and ratio-dependent functional responses in a spatially heterogeneous environment. Prey-taxis implies that the predator exhibits directed movement in the presence of prey. We claim that when the predator diffuses uniformly without prey-taxis, only the diffusion of the species plays a role in the predator’s invasion in a spatially homogeneous environment. However, when the predator disperses through uniform diffusion together with prey-taxis, we observe that both the diffusion of species and prey-taxis affect the invasion of the predator, which is not the case in a spatially homogeneous environment. The results are obtained by investigating the local stability of a semi-trivial solution, using an eigenvalue analysis.     

A discrete prey–predator model preserving the dynamics of a structurally unstable Lotka–Volterra model

Leslie's method to construct a discrete two dimensional dynamical system dynamically consistent with the Lotka–Volterra type of competing two species ordinary differential equations is applied in a newly extended manner for the Lotka–Volterra prey–predator system which is structurally unstable. We show that, independently of the time step size, the derived discrete prey–predator system is dynamically consistent with the continuous counterpart, keeping the nature of neutrally stable periodic orbit. Further, we show that the extended method to construct the discrete prey–predator system can provide a dynamically consistent model also for the logistic Lotka–Volterra one.     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.