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.     

Dynamic behaviors of the periodic predator–prey system with distributed time delays and impulsive effect

In this paper, a periodic predator–prey system with distributed time delays and impulsive effect is investigated. By using the Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We improve some results in Guo and Chen (2009) [1].     

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.     

Multiple positive almost periodic solutions to an impulsive non-autonomous Lotka–Volterra predator–prey system with harvesting terms

In this paper, by using Mawhin’s continuation theorem of coincidence degree theory, we study an impulsive non-autonomous Lotka–Volterra predator–prey system with harvesting terms and obtain some sufficient conditions for the existence of multiple positive almost periodic solutions for the system under consideration. Our results of this paper are completely new and our method used in this paper can be used to study the existence of multiple positive almost periodic solutions to other types of population systems.     

Dispersal permanence of periodic predator–prey model with Ivlev-type functional response and impulsive effects

In this paper, we study the permanence of a periodic Ivlev-type predator–prey system where the prey disperses in patchy environment with two patches. We assume the Ivlev-type functional response within-patch dynamics and provide a sufficient condition to guarantee the predator and prey species to be permanent. Furthermore, we give numerical analysis to confirm our theoretical results. It will be useful to ecosystem control.     

Multiple periodic solutions of an impulsive predator–prey model with Holling-type IV functional response

An impulsive periodic predator–prey model with Holling-type IV functional response is considered. Using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. It is important to point out that we establish a better estimation on the difference between the supremum and infimum of a differentiable piecewise continuous periodic function. As illustrated in this paper, with the help of this estimation, many existing results can be improved.     

A predator–prey system with stage-structure for predator and nonlocal delay

This paper deals with the behavior of solutions to the reaction–diffusion system under homogeneous Neumann boundary condition, which describes a prey–predator model with nonlocal delay. Sufficient conditions for the global stability of each equilibrium are derived by the Lyapunov functional and the results show that the introduction of stage-structure into predator positively affects the coexistence of prey and predator. Numerical simulations are performed to illustrate the results.     

Multiple periodic solutions of a delayed predator–prey system with stage structure for the predator

By using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a delayed predator–prey system with stage structure for the predator is established.     

An impulsive periodic predator-prey Lotka–Volterra type dispersal system with mixed functional responses

An impulsive two species periodic predator-prey Lotka–Volterra type dispersal system with mixed functional responses is presented and studied in this paper. 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.     

Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response

An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end.     

Dynamics of cooperative predator–prey system with impulsive effects and Beddington–DeAngelis functional response

In this paper, cooperative predator–prey system with impulsive effects and Beddington–DeAngelis functional response is studied. By using comparison theorem and some analysis techniques as well as the coincidence degree theory, sufficient conditions are obtained for the permanence, extinction and the existence of positive periodic solution.     

The study of a predator–prey system with group defense and impulsive control strategy

A predator–prey system with group defense and impulsive control strategy is established. By using Floquet theorem and small amplitude perturbation skills, a locally asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than some critical value. Otherwise, if the impulsive period is larger than the critical value, the system is permanent. By using bifurcation theory, we show the existence and stability of positive periodic solution when the pest-eradication lost its stability. Further, numerical examples show that the system considered has more complicated dynamics, such as: (1) quasi-periodic oscillating, (2) period-doubling bifurcation, (3) period-halving bifurcation, (4) non-unique dynamics (meaning that several attractors coexist), (5) attractor crisis, etc. Finally, the biological implications of the results and the impulsive control strategy are discussed.     

The effects of impulsive harvest on a predator–prey system with distributed time delay

A kind of predator–prey system with distributed time delay and impulsive harvest is firstly presented and then the effects of impulsive harvest on the system are discussed by means of chain transform. By using the Floquet’s theory and the comparison theorem of impulsive differential equation, the thresholds between permanence and extinction of each species are obtained as functions of model parameters. It is proved that the impulsive period and the proportion of the impulsive harvest will ultimately affect the fate of each species. Finally, the theoretical results obtained in this paper are confirmed by numerical simulations.     

Traveling waves for a periodic Lotka–Volterra predator–prey system

This paper is concerned with the time periodic traveling wave solutions for a periodic Lotka–Volterra predator–prey system, which formulates that both species synchronously invade a new habitat. We first establish the existence of periodic traveling wave solutions by combining the upper and lower solutions with contracting mapping principle and Schauder’s fixed point theorem. The asymptotic behavior of nontrivial solution is given precisely by the stability of the corresponding kinetic system that has been widely investigated. Then, the nonexistence of periodic traveling wave solutions is confirmed by applying the theory of asymptotic spreading. We show the conclusion for all positive wave speed and obtain the minimal wave speed.     

Existence and stability of periodic solution of a Lotka–Volterra predator–prey model with state dependent impulsive effects

According to biological and chemical control strategy for pest, we investigate the dynamic behavior of a Lotka–Volterra predator–prey state-dependent impulsive system by releasing natural enemies and spraying pesticide at different thresholds. By using Poincaré map and the properties of the Lambert W$W$ function, we prove that the sufficient conditions for the existence and stability of semi-trivial solution and positive periodic solution. Numerical simulations are carried out to illustrate the feasibility of our main results.     

Permanence of periodic predator–prey system with two predators and stage structure for prey

A stage-structured three-species predator–prey system with Beddington–DeAngelis and Holling IV functional response is proposed and analyzed. Based on the comparison theorem, some sufficient and necessary conditions are derived for permanence of the system. Finally, two examples are presented to illustrate the application of our main results.     

Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects

In this paper, we propose a stochastic non-autonomous Lotka–Volterra predator–prey model with impulsive effects and investigate its stochastic dynamics. We first prove that the subsystem of the system has a unique periodic solution which is globally attractive. Furthermore, we obtain the threshold value in the mean which governs the stochastic persistence and the extinction of the prey–predator system. Our results show that the stochastic noises and impulsive perturbations have crucial effects on the persistence and extinction of each species. Finally, we use the different stochastic noises and impulsive effects parameters to provide a series of numerical simulations to illustrate the analytical results.