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.     

An impulsive periodic predator–prey system with Holling type III functional response and diffusion

This paper studies an impulsive two species periodic predator–prey Lotka–Volterra type dispersal system with Holling type III functional response in a patchy environment, in which the prey species can disperse among n different patches, but the predator species is confined to one patch and cannot disperse. Conditions for the permanence and extinction of the predator–prey system, and for the existence of a unique globally stable periodic solution are established. Numerical examples are shown to verify the validity of our results.     

Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independent

In this paper, we study two species time-delayed predator-prey Lotka-Volterra type dispersal systems with periodic coefficients, in which the prey species can disperse among n patches, while the density-independent predator species is confined to one of patches and cannot disperse. Sufficient conditions on the boundedness, permanence and existence of positive periodic solution for this systems are established. The theoretical results are confirmed by a special example and numerical simulations.     

On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay

In this paper, a nonlinear nonautonomous predator–prey model with diffusion and continuous distributed delay is studied, where all the parameters are time-dependent. The system, which is composed of two patches, has two species: the prey can diffuse between two patches, but the predator is confined to one patch. We first discuss the uniform persistence and global asymptotic stability of the model; after that, by constructing a suitable Lyapunov functional, some sufficient conditions for the existence of a unique almost periodic solution of the system are obtained. An example shows the feasibility of our main results.     

Two-patches prey impulsive diffusion periodic predator–prey model

In this paper, we study a periodic predator–prey system with prey impulsive diffusion in two patches. On the basis of comparison theorem of impulsive differential equation and other analysis methods, sufficient and necessary conditions on the predator–prey system where predator have not other food source are established. Two examples and numerical simulations are presented to illustrate the feasibility of our results. A conclusion is given in the end.     

A predator–prey system with two impulses on the diseased prey and a Beddington–DeAngelis response

A predator–prey system with two impulses on the diseased prey is formulated and analyzed for the purpose of integrated pest management. The local and global stability of the susceptible pest‐eradication periodic solution, as well as the permanence of the system, are obtained under the sufficient conditions by means of Floquet's theory for impulsive differential equations. Finally, we interpret our mathematical results. Copyright © 2009 John Wiley & Sons, Ltd.     

An impulsive predator–prey model with disease in the prey for integrated pest management

In this paper, an impulsive predator–prey model with disease in the prey is investigated for the purpose of integrated pest management. In the first part of the main results, we get the sufficient condition for the global stability of the susceptible pest-eradication periodic solution. This means if the release amount of infective prey and predator satisfy the condition, then the pest will be doomed. In the second part of the main results, we also get the sufficient condition for the permanence of the system. This means if the release amount of infective prey and predator satisfy the condition, then the prey and the predator will coexist. In the last section, we interpret our mathematical results. We also point out some possible future work.     

Chaos in periodically forced Holling type IV predator–prey system with impulsive perturbations

The effect of periodic forcing and impulsive perturbations on predator–prey model with Holling type IV functional response is investigated. The periodic forcing is affected by assuming a periodic variation in the intrinsic growth rate of the prey. The impulsive perturbations are affected by introducing periodic constant impulsive immigration of predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can easily give rise to complex dynamics, including (1) quasi-periodic oscillating, (2) period doubling cascade, (3) chaos, (4) period halfing cascade.     

Bifurcation and complexity of Monod type predator–prey system in a pulsed chemostat

In this paper, we introduce and study a model of a predator–prey system with Monod type functional response under periodic pulsed chemostat conditions, which contains with predator, prey, and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

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.     

The effect of dispersal on the permanence of a predator–prey system with time delay

A predator–prey model with prey dispersal and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of each of the nonnegative equilibria is discussed. By using an iteration technique, a threshold is derived for the permanence and extinction of the proposed model. Numerical simulations are carried out to illustrate the main results.     

Predator–prey harvesting model with fatal disease in prey

A theoretical eco‐epidemiological model of a prey–predator interaction system with disease in prey species is studied. Predator consumes both susceptible and infected prey population, but predator also feeds preferentially on many numerous species, which are over represented in the predator's diet. Equilibrium points of the system are determined, and the dynamic behaviour of the system is investigated around equilibrium points. Death rate of predator species is considered as a bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighbourhood of the coexisting equilibria. Numerical simulations are carried out to support the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Weak average persistence and extinction of a predator–prey system in a polluted environment with impulsive toxicant input

In this paper, we have investigated a predator–prey system in a polluted environment with impulsive toxicant input at fixed moments. We have obtained two thresholds on the impulsive period by assuming the toxicant amount input is fixed to the environment at each pulse moment. If the impulsive period is greater than the big threshold, then both populations are weak average persistent. If the period lies between of the two thresholds, then the prey population will be weak average persistent while the predator population extinct. If the period is less than the small threshold, both populations tend to extinction. Finally, our theoretical results are confirmed by own numerical simulations.     

Dynamics of an impulsive control system which prey species share a common predator

In an ecosystem multiple prey species often share a common predator and the interactions between the preys are neutral. In view of these facts and based on a multiple species prey–predator system with Holling IV and II functional responses, an impulsive differential equation to model the process of periodically releasing natural enemies and spraying pesticides at different fixed times for pest control is proposed and investigated. It is proved that there exists a locally asymptotically stable pest-eradication periodic solution under the assumption that the impulsive period is less than some critical value (or the release amount of the predator is greater than another critical value). Permanence conditions are established when the impulsive period is greater than another critical value (or the release amount of the predator is less than some critical value). Numerical results show that the system we consider has more complex dynamics including period solution, quasi-periodic oscillation, chaos, intermittency and crises.     

A study of predator–prey models with the Beddington–DeAnglis functional response and impulsive effect

In this paper, the predator–prey system with the Beddington–DeAngelis functional response is developed, by introducing a proportional periodic impulsive catching or poisoning for the prey populations and a constant periodic releasing for the predator. The Beddington–DeAngelis functional response is similar to the Holling type II functional response but contains an extra term describing mutual interference by predators. This model has the potential to protect predator from extinction, but under some conditions may also lead to extinction of the prey. That is, the system exists a locally stable prey-eradication periodic solution when the impulsive period satisfies an inequality. The condition for permanence is established via the method of comparison involving multiple Liapunov̀ functions. Further, by numerical simulation method the influences of the impulsive perturbations and mutual interference by predators on the inherent oscillation are investigated. With the increasing of releasing for the predator, the system appears a series of complex phenomenon, which include (1) period-doubling, (2) chaos attractor, (3) period-halfing. (4) non-unique dynamics (meaning that several attractors coexist).     

PERMANENCE OF A NONLINEAR PERIODIC PREDATOR-PREY SYSTEM WITH PREY DISPERSAL AND PREDATOR DENSITY-INDEPENDENCE

In this paper,a set of suffcient conditions which ensure the permanence of a nonlinear periodic predator-prey system with prey dispersal and predator density-independence are obtained,where the prey species can disperse among n patches,while the density-independent predator is confined to one of the patches and cannot disperse. Our results generalize some known results.     

Dynamics of a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects

This paper is concerned with a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects. By using the fixed point theory based on monotone operator, some simple conditions are obtained for the existence of at least one positive periodic solution of the model. The existence result of this paper implies that the functional response on prey does not influence the existence of positive periodic solution of the model, which completes some results given in recent years. Further, by applying the comparison theorem in impulsive differential equations and constructing a suitable Lyapunov functional, the permanence and global attractivity of the model are also investigated. The main results in this paper extend, complement, and improve the previously known result. And some examples and numerical simulations are given to illustrate the feasibility and effectiveness of the main results. Copyright © 2015 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.     

A two-patch prey-predator model with food-gathering activity

In insect ecosystem, the dynamics of prey and predator is regulated by complex interactions between them. Insect pests are spatially aggregated in patches forming a spatial pattern in the environment. An efficient predator dynamically changes its strategies and time for its random search movements to concentrate on higher resource patches based on the benefit of assessment. This food-gathering activity of both prey and predator plays a major role in stabilizing the system by influencing the per unit food consumption. Extending Holling time-budget argument by migration, here we formulate a two patch prey-predator model and show that how several foraging parameters such as handling time, dispersal rate can have important consequences in stability of prey-predator system. Specifically, the ratio between timings that a predator remains mobile in searching and handling their food, is the most important one and simulation on this suggests that the stabilizing effect continues to operate when the dispersal process is modeled more realistically. Thus we conclude that the migration submodel is an important constituent of a spatial predator-prey system. These results are shown to have important implications for possible biological control.     

Existence and global stability of positive periodic solutions for predator–prey system with infinite delay and diffusion

In this paper, we study a periodic predator–prey system with Holling type III functional response, in which the prey species can diffuse among two patches but the predator is confined in one patch. By using the continuation theorem of coincidence degree theory and Lyapunov functional, some sufficient conditions are obtained.