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.     

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.     

Permanence, extinction and periodic solution of predator-prey system with Beddington-DeAngelis functional response

In this paper we study permanence, extinction and periodic solution of periodic predator-prey system with Beddington-DeAngelis functional response. We provide a sufficient and necessary condition to guarantee the predator and prey species to be permanent. In addition, sufficient condition is derived for the existence of positive periodic solution. This paper improves some main results obtained by Fan and Kuang [M. Fan, Y. Kuang, Dynamics of nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl. 295 (2004) 15-39].     

时滞Lotka-Volterra系统的持久性和周期解

本文研究具有时滞周期捕食系统的持久性和周期解,得到了系统永久持续生存的充要条件。在此条件下系统存在正的ω周期解,改进了一些已知结果。     

具有脉冲扰动的比率相关功能性反应捕食者-食饵扩散模型的持续生存和周期解

本文考虑一类具有脉冲扰动的比率相关的捕食者一食饵扩散模型,利用比较原理研究了这类系统的持续生存和灭绝性,通过将脉冲反应扩散方程转化为相应的算子方程,并证明了解在适当空间的紧性,得到了周期解的存在性、唯一性和全局稳定性.最后分析了脉冲效应对系统性态的影响.     

Periodic and almost periodic solution for a non-autonomous epidemic predator-prey system with time-delay

In this paper, we studied a non-autonomous predator-prey system with discrete time-delay, where there is epidemic disease in the predator. By using some techniques of the differential inequalities and delay differential inequalities, we proved that the system is permanent under some appropriate conditions. When all the coefficients of the system is periodic, we obtained the existence and global attractivity of the positive periodic solution by Mawhin’s continuation theorem and constructing a suitable Lyapunov functional. Furthermore, when the coefficients of the system are not absolutely periodic but almost periodic, sufficient conditions are also derived for the existence and asymptotic stability of the almost periodic solution.     

The permanence and extinction of a nonlinear growth rate single-species non-autonomous dispersal models with time delays

In this paper, we consider the effect of diffusion on the permanence and extinction of a non-autonomous nonlinear growth rate single-species dispersal model with time delays. Firstly, the sufficient conditions of the permanence and extinction of the species are established, which shows if the growth rate and dispersal coefficients is suitable, the species is permanent, on the contrary, it is extinction. Secondly, an interesting result is established, that is, if only the species in some patches even in one patch is permanent, then it is also permanent in other patches. Finally, some examples together with their numerical simulations show the feasibility of our main results.     

Permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure

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

Positive periodic solutions for impulsive Gause-type predator-prey systems

This paper is devoted to impulsive periodic Gause-type predator-prey systems with monotonic or non-monotonic numerical responses. With the help of a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions. As corollaries, some applications are listed. In particular, our results improve and generalize some known ones.     

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

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 for non-autonomous food chain systems with delay

In this paper we first establish a new general criterion for the permanence of Kolmogorov-type systems of nonautonomous functional differential equations. Then, as applications of this criterion we study the permanence of a class of n-species general nonautonomous food chain systems with delay and new sufficient condition are established.     

Positive periodic solutions in delayed Gause-type predator-prey systems

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions in delayed Gause-type predator-prey systems. Some known results are shown to be special cases of the presented paper.     

Permanence and extinction of a stochastic hybrid model for tumor growth

A tumor growth model perturbed by both white noise and Markov switching is formulated and explored. The threshold between permanence and extinction is obtained. Some effects of environmental stochasticity on permanence and extinction of the model are revealed.     

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 and extinction for a nonautonomous SIRS epidemic model with time delay

In this paper, a nonautonomous SIRS epidemic model with time delay is studied. We introduce some new threshold values _R∗$R_{\ast }$ and R^∗$R^{\ast }$ and further obtain the disease will be permanent when _R∗>1$R_{\ast }>1$ and the disease extinct when R^∗<1$R^{\ast }<1$. Using the method of Liapunov functional, some sufficient conditions are derived for the global attractivity of the system. The known results are extended.     

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.     

PERMANENCE OF A NONAUTOMONOUS POPULATION MODEL

1.IntroductionPersistenceofpopulationsplaysanimportantroleinmathematicalecology.Inrecentyears)muchattentionhasbeengiventothepersistenceofnonautomonouspopulationmodelsll--6].[1]proposedtheconceptofpersistenceinthemeanofpopulations.Thisisofimportancebecausenotonlyitprovidesawaytocharacterizethepersistenceofpopula-nons,butalsothereareonlythresholdsbetweenpersistenceinthemeanandtheextinctionofpopulationsforgeneralnonautomonouspopulationmodels.In[2]thethresholdwasestablishedforone-dimensionalnonaut…     

Stability and bifurcation analysis of a delayed predator-prey model of prey dispersal in two-patch environments

In this paper, a class of delayed predator-prey model of prey dispersal in two-patch environments is considered. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulation for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

Periodic solution and almost periodic solution for a nonautonomous Lotka-Volterra dispersal system with infinite delay

This paper studies a nonautonomous Lotka-Volterra dispersal systems with infinite time delay which models the diffusion of a single species into n patches by discrete dispersal. Our results show that the system is uniformly persistent under an appropriate condition. The sufficient condition for the global asymptotical stability of the system is also given. By using Mawhin continuation theorem of coincidence degree, we prove that the periodic system has at least one positive periodic solution, further, obtain the uniqueness and globally asymptotical stability for periodic system. By using functional hull theory and directly analyzing the right functional of almost periodic system, we show that the almost periodic system has a unique globally asymptotical stable positive almost periodic solution. We also show that the delays have very important effects on the dynamic behaviors of the system.     

Bifurcation analysis for a three-species predator-prey system with two delays

In this paper, a three-species predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.