Periodic Predator-prey System Research Articles

Optimal control for a size-structured predator-prey model in a polluted environment

In this paper, we deal with an optimal harvesting problem for a periodic predator-prey hybrid system dependent on size-structure in a polluted environment. In other words, a size-dependent model in an environment with a small toxicant content has been established. The well-posedness of state system is proved by using the fixed point theorem. The necessary optimality conditions are derived by tangent-normal cone technique in nonlinear functional analysis. The existence of a unique optimal harvesting policy is verified via the Ekeland’s variational principle. The optimal harvesting problem has an optimal harvesting policy, which has a Bang-Bang structure and provides a threshold for the optimal harvesting problem. Using the optimization theories and methods in mathematics to control phenomena of life. The objective function represents the total economic profit from the harvested population. Some theoretical results obtained in this paper provide a scientific theoretical basis for the practical application of the model.

Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays

<p style='text-indent:20px;'>This article is concerned with a generalized almost periodic predator-prey model with impulsive effects and time delays. By utilizing comparison theorem and constructing a feasible Lyapunov functional, we obtain sufficient conditions to guarantee the permanence and global asymptotic stability of the system. By applying Arzelà-Ascoli theorem, we establish the existence and uniqueness of almost-periodic positive solutions. A feasible numerical simulation is provided to explain the suitability of our main criteria.

On the dynamics of one-prey-n-predator impulsive reaction-diffusion predator–prey system with ratio-dependent functional response

ABSTRACTIn this paper, a one-prey-n-predator impulsive reaction-diffusion periodic predator–prey system with ratio-dependent functional response is investigated. On the basis of the upper and lower solution method and comparison theory of differential equation, sufficient conditions on the ultimate boundedness and permanence of the predator–prey system are established. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Examples and numerical simulations are presented to verify the feasibility of our results. A discussion is conducted at the end.

On a periodic predator-prey system with nonlinear diffusion and delays

In this paper, we consider a two species Lotka–Volterra type predator-prey system with nonlinear diffusion and distributed time delays. Based on the continuation theorem of the coincidence degree theory and the Liapunov function method, some new sufficient conditions on the existence and global attractivity of positive periodic solutions are established.

Dynamics of n+1-dimensional impulsive reaction-diffusion predator-prey system with Holling type III functional response

An n+1-dimensional impulsive reaction-diffusion periodic predator-prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions on the ultimate boundedness and permanence of the predator-prey system are established based on the comparison theory of differential equation and upper and lower solution method. By constructing 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 in the end of the paper.

Analysis of a Periodic Impulsive Predator-Prey System with Disease in the Prey

We investigate a periodic predator-prey system subject to impulsive perturbations, in which a disease can be transmitted among the prey species only, in this paper. With the help of the theory of impulsive differential equations and Lyapunov functional method, sufficient conditions for the permanence, global attractivity, and partial extinction of system are established, respectively. It is shown that impulsive perturbations contribute to the above dynamics of the system. Numerical simulations are presented to substantiate the analytical results.

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.

On the Dynamics of an Impulsive Reaction-Diffusion Predator-Prey System with Ratio-Dependent Functional Response

An impulsive reaction-diffusion periodic predator-prey system with ratio-dependent 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 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 in the end of the paper.

Dispersal permanence of a periodic predator–prey system with Holling type-IV functional response

Abstract In this paper, we study the permanence of a class of periodic predator–prey system with Holling type-IV functional response where the prey disperses in patchy environment with two patches, and provide a sufficient and necessary condition to guarantee permanence of the system. Finally, two examples are presented to illustrate the application of our main results.

Permanence and global attractivity of a periodic predator–prey system with mutual interference and impulses

In this paper, we consider a periodic predator–prey system with mutual interference and impulses. By constructing a suitable Lyapunov function and using the comparison theorem of impulsive differential equation, sufficient conditions which ensure the permanence and global attractivity of the system are obtained. We also present some examples to verify our main 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.

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].

Survival analysis for a periodic predator–prey model with prey impulsively unilateral diffusion in two patches

In this paper, we study a periodic predator–prey system with prey impulsively unilateral diffusion in two patches. Firstly, based on the results in [41], sufficient conditions on the existence, uniqueness and globally attractiveness of periodic solution for predator-free and prey-free systems are presented. Secondly, by using comparison theorem of impulsive differential equation and other analysis methods, sufficient and necessary conditions on the permanence and extinction of prey species x with predator have other food source are established. Finally, the theoretical results both for non-autonomous system and corresponding autonomous system are confirmed by numerical simulations, from which we can see some interesting phenomena happen.

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.

Existence of multiple positive periodic solutions to a periodic predator–prey system with harvesting terms and Holling III type functional response

This paper is concerned with a periodic predator–prey system with Holling III functional response and harvesting terms. By means of the coincidence degree theory, we establish the existence of at least eight positive periodic solutions for the system. An example is given to illustrate the effectiveness of our results.

Periodicity and asymptotic stability of a predator–prey system with infinite delays

By using a fixed point theorem and Lyapunov functional, an especially easily checked criterion is obtained for the global existence and global asymptotic stability of positive periodic solutions of a periodic predator–prey system with infinite delays. Moreover, the global existence theorem is also sufficient and necessary. This result improves and generalizes noticeably some known results.

Permanence and extinction analysis for a delayed periodic predator–prey system with Holling type II response function and diffusion

In this paper, it is studied that two species predator–prey Lotka–Volterra type dispersal system with delay and Holling type II response function, in which the prey species can disperse among n patches, while the density-independent predator species is confined to one of the patches and cannot disperse. Sufficient conditions of integrable form for the boundedness, permanence, extinction and the existence of positive periodic solution are established, respectively.

On the permanence of periodic predator–prey systems with stage structure and pulse action

We establish permanence conditions for a periodic predator-prey system with stage structure, pulse action, and Beddington-DeAngelis functional response.

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.

Permanence of Periodic Predator-Prey System with General Nonlinear Functional Response and Stage Structure for Both Predator and Prey

We study the permanence of periodic predator-prey system with general nonlinear functional responses and stage structure for both predator and prey and obtain that the predator and the prey species are permanent.