A delayed prey–predator model with Crowley–Martin‐type functional response including prey refuge

In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin‐type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

Wave features of a hyperbolic prey–predator model

A hyperbolic predator–prey model is proposed within the context of extended thermodynamics. The nature of the steady state solutions for the uniform and non‐uniform perturbations are analyzed. The existence of smooth traveling wave‐like solutions, related to the invasion of the predator population into a prey‐only state is discussed. Validation of the model in point is also accomplished by searching for numerical solutions of the system, which also points out limit cycles in the populations. Copyright © 2010 John Wiley & Sons, Ltd.     

On Nonautonomous Prey predator Patchy System

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｅｒｅｓｔｈａｓｂｅｅｎｇｒｏｗｉｎｇｉｎｔｈｅｓｔｕｄｙｏｆｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｓｏｆｐｏｐｕｌａｔｉｏｎｓｄｉｓｐｅｒｓ－ｉｎｇａｍｏｎｇｐａｔｃｈｅｓｉｎａｈｅｔｅｒｏｇｅｎｅｏｕｓｅｎｖｉｒｏ...     

Permanence,extinction and periodic solution of the predator–prey system with Beddington–DeAngelis functional response and stage structure for prey

In this paper, we study the permanence, extinction and periodic solution of the periodic predator–prey system with Beddington–DeAngelis functional response and stage structure for prey. A set of sufficient and necessary conditions which guarantee the predator and prey species to be permanent are obtained. In addition, sufficient conditions are derived for the existence of positive periodic solutions to the system. Numeric simulations show the feasibility of the main results.     

Dynamics of a predator–prey model with disease in the predator

The present investigation deals with a predator–prey model with disease that spreads among the predator species only. The predator species is split out into two groups—the susceptible predator and the infected predator both of which feeds on prey species. The stability and bifurcation analyses are carried out and discussed at length. On the basis of the normal form theory and center manifold reduction, the explicit formulae are derived to determine stability and direction of Hopf bifurcating periodic solution. An extensive quantitative analysis has been performed in order to validate the applicability of our model under consideration. Copyright © 2013 John Wiley & Sons, Ltd.     

Stochastic persistence and stability analysis of a modified Holling–Tanner model

The article aims to study the basic dynamical features of a modified Holling–Tanner prey–predator model with ratio‐dependent functional response. We have proved the global existence of the solution for the deterministic model. The parametric restriction for persistence of both species is also obtained along with the proof of local asymptotic stability of the interior equilibrium point(s). Conditions for local bifurcations of interior equilibrium points are provided. The global dynamic behavior is examined thoroughly with supportive numerical simulation results. Next, we have formulated the stochastic model by perturbing the intrinsic growth rates of prey and predator populations with white noise terms. The existence uniqueness of solutions for stochastic model is established. Further, we have derived the parametric restrictions required for the persistence of the stochastic model. Finally, we have discussed the stochastic stability results in terms of the first and second order moments. Numerical simulation results are provided to support the analytical findings. Copyright © 2012 John Wiley & Sons, Ltd.     

A modified Leslie–Gower predator–prey model with ratio‐dependent functional response and alternative food for the predator

In this work, a modified Leslie–Gower predator–prey model is analyzed, considering an alternative food for the predator and a ratio‐dependent functional response to express the species interaction. The system is well defined in the entire first quadrant except at the origin ( 0 , 0 ) . Given the importance of the origin ( 0 , 0 ) as it represents the extinction of both populations, it is convenient to provide a continuous extension of the system to the origin. By changing variables and a time rescaling, we obtain a polynomial differential equations system, which is topologically equivalent to the original one, obtaining that the non‐hyperbolic equilibrium point ( 0 , 0 ) in the new system is a repellor for all parameter values. Therefore, our novel model presents a remarkable difference with other models using ratio‐dependent functional response. We establish conditions on the parameter values for the existence of up to two positive equilibrium points; when this happen, one of them is always a hyperbolic saddle point, and the other can be either an attractor or a repellor surrounded by at least one limit cycle. We also show the existence of a separatrix curve dividing the behavior of the trajectories in the phase plane. Moreover, we establish parameter sets for which a homoclinic curve exits, and we show the existence of saddle‐node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation, and homoclinic bifurcation. An important feature in this model is that the prey population can go to extinction; meanwhile, population of predators can survive because of the consumption of alternative food in the absence of prey. In addition, the prey population can attain their carrying capacity level when predators go to extinction. We demonstrate that the solutions are non‐negatives and bounded (dissipativity and permanence of population in many other works). Furthermore, some simulations to reinforce our mathematical results are shown, and we further discuss their ecological meanings. Copyright © 2017 John Wiley & Sons, Ltd.     

Dynamics and pattern formations in diffusive predator‐prey models with two prey‐taxis

A reaction‐diffusion two‐predator‐one‐prey system with prey‐taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing preys. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any small prey‐taxis sensitivity coefficient are investigated by the semigroup theory. The spatial pattern formation induced by the prey‐taxis is characterized by the Turing type linear instability of homogeneous state; it is shown that prey‐taxis can both compress and prompt the spatial patterns produced through diffusion‐induced instability in two‐predator‐one‐prey systems.     

Bifurcation analysis of a diffusive predator–prey system with a herd behavior and quadratic mortality

In this paper, a diffusive predator–prey system, in which the prey species exhibits herd behavior and the predator species with quadratic mortality, has been studied. The stability of positive constant equilibrium, Hopf bifurcations, and diffusion‐driven Turing instability are investigated under the Neumann boundary condition. The explicit condition for the occurrence of the diffusion‐driven Turing instability is derived, which is determined by the relationship of the diffusion rates of two species. The formulas determining the direction and the stability of Hopf bifurcations depending on the parameters of the system are derived. Finally, numerical simulations are carried out to verify and extend the theoretical results and show the existence of spatially homogeneous periodic solutions and nonconstant steady states. Copyright © 2014 John Wiley & Sons, Ltd.     

Qualitative analysis of Holling type II predator–prey systems with prey refuges and predator restricts

This paper is devoted to investigation of Holling type II predator–prey systems with prey refuges and predator restricts. Using a transformation technique, we change the system into a generalized Liénard system and give sufficient conditions to ensure the global stability of the positive equilibrium and existence and uniqueness of a stable limit cycle. We also find the property of alternation for phase structure of the system.     

Permanence and periodicity of a delayed ratio-dependent predator–prey model with Holling type functional response and stage structure

A periodic and delayed ratio-dependent predator–prey system with Holling type III functional response and stage structure for both prey and predator is investigated. It is assumed that immature predator and mature individuals of each species are divided by a fixed age, and immature predator do not have the ability to attack prey. Sufficient conditions are derived for the permanence and existence of positive periodic solution of the model. Numerical simulations are presented to illustrate the feasibility of our main results.     

Permanence for a class of periodic time-dependent predator–prey system with dispersal in a patchy-environment

In this paper, we study two species predator–prey Lotka–Volterra type dispersal system with periodic coefficients in two patches, in which both the prey and predator species can disperse between two patches. By utilizing analytic method, sufficient and realistic conditions on permanence and the existence of periodic solution are established. The theoretical results are confirmed by a special example and numerical simulations.     

Persistence in nonautonomous predator–prey systems with infinite delays

This paper studies the general nonautonomous predator–prey Lotka–Volterra systems with infinite delays. The sufficient and necessary conditions of integrable form on the permanence and persistence of species are established. A very interesting and important property of two-species predator–prey systems is discovered, that is, the permanence of species and the existence of a persistent solution are each other equivalent. Particularly, for the periodic system with delays, applying these results, the sufficient and necessary conditions on the permanence and the existence of positive periodic solutions are obtained. Some well-known results on the nondelayed periodic predator–prey Lotka–Volterra systems are strongly improved and extended to the delayed case.     

Extinction and permanence of the predator–prey system with stocking of prey and harvesting of predator impulsively

In this paper, a predator–prey system with stocking of prey and harvesting of predator impulsively is studied. Here, the prey population is stocked with a constant quantity and the predator population is harvested at a rate proportional to the species itself at fixed moments. Under some conditions, the existence and global asymptotic stability of the boundary periodic solution are proved, which implies that the system will be extinct; and given some different restrictions, ultimate positive upper and lower bounds of all solutions are obtained, showing the system being permanent. At last, two examples are given to illustrate our results. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation in a predator–prey model with stage structure for the predator

A predator–prey system with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. Further, the existence of a Hopf bifurcation at the positive equilibrium is also studied. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. As a result, the threshold is obtained for the permanence and extinction of the system. Numerical simulations are carried out to illustrate the main results.     

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.     

Periodic solutions for predator–prey systems with Beddington–DeAngelis functional response on time scales

This paper deals with the question of existence of periodic solutions of nonautonomous predator–prey dynamical systems with Beddington–DeAngelis functional response. We explore the periodicity of this system on time scales. New sufficient conditions are derived for the existence of periodic solutions. These conditions extend previous results presented in [M. Bohner, M. Fan, J. Zhang, Existence of periodic solutions in predator–prey and competition dynamic systems, Nonlinear. Anal.: Real World Appl. 7 (2006) 1193–1204; M. Fan, Y. Kuang, Dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelies functional response, J. Math. Anal. Appl. 295 (2004) 15–39; J. Zhang, J. Wang, Periodic solutions for discrete predator–prey systems with the Beddington–DeAngelis functional response, Appl. Math. Lett. 19 (2006) 1361–1366].     

Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay

In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. 26 We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.     

Ergodicity and threshold behaviors of a predator‐prey model in stochastic chemostat driven by regime switching

This paper deals with a stochastic predator‐prey model in chemostat which is driven by Markov regime switching. For the asymptotic behaviors of this stochastic system, we establish the sufficient conditions for the existence of the stationary distribution. Then, we investigate, respectively, the extinction of the prey and predator populations. We explore the new critical numbers between survival and extinction for species of the dual‐threshold chemostat model. Numerical simulations are accomplished to confirm our analytical conclusions.     

Influence of prey reserve in a prey–predator fishery

The excessive and unsustainable exploitation of marine resources has to led to the promotion of marine reserve as a fisheries management tool. In this paper we study a prey–predator system in a two-patch environment: one accessible to both prey and predators (patch 1) and the other one being a refuge for the prey (patch 2). The prey refuge (patch 2) constitutes a reserve zone of prey and fishing is not permitted, while the unreserved zone area is an open-access fishery zone. The existence of possible steady states, along with their local and global stability, is discussed. We then examine the possibilities of the existence of bionomic equilibrium. An optimal harvesting policy is given using Pontryagin’s maximum principle.