具时滞的非自治扩散捕食系统的概周期解

在本文中,我们考虑具时滞的扩散概周期捕食系统,其中被捕食者可在两个缀块间迁移,而捕食者被限制在其中一个缀块内,并证明了该系统存在唯一的全局吸引的正概周期解．     

具有扩散的非自治两种群Lotka-Volterra模型的概周期问题

研究非自治两种群竞争系统,其中一种群可以在两个斑块之间扩散。而另一种群在一个斑块中,不能扩散。本文结合运用Liapunov函数,得到该系统唯一存在全局渐近稳定的正概周期解的条件．     

Persistence and Periodic Solutions of a System of Two Competing Species with Functional Response

1IntroductionOneofthemOStnit~tingquestionsinrnathernaticalbiologyconcernsthesurvivalofspeCiesinecologiCalmodels.Perslstenceisanimportantconceptindabingwiththeseproblems.Therearemanyliteraturesaboutthedy'ndricsofdiffuSivecompetingspeCies,butthefunctionalresPOnseofthisfOITnhasnotbeenst'Udiedtoomuchyet.Inthispaper,weconsiderthepersistenceproblemforanonautonomoussystemoftwOcompetingspecieswithfunctionalreSPOnse,themodelweconsiderinthispaperishereallri(t),ail(t),D,(t)anda(t)areassumedtobecon…     

Conflict between the need to forage and the need to avoid competition: persistence of two-species model

We consider a model in which the need to forage and the need to avoid a competitor are in conflict. The model is composed of two Lotka-Volterra patches. The system has two competitors; one can diffuse between two patches, but the other is confined to one of the patches and cannot diffuse. It is proved that the system can be made persistent under appropriate diffusion conditions that ensure the instability of boundary equilibria, even if the competitive patch is not persistent without diffusion. Further it is shown that the system is globally stable for any diffusion rate if the competition between the two species is weak.     

具有分离扩散的两种群Lotka—Volterra模型的持久性

本文考虑具有分离扩散的捕食-被捕食系统的持续性。此模型由两种群组成，其中被捕食种群可在两个生态环境中生存，而捕食种群仅能在一个生态环境中生存，两种群的动态行为都用Lotka－Volterra模型来描述。得到了系统强持续的充分必要条件，并证明了无论无扩散时系统是共存的，还是主导的都可以适当选择分离扩散系数使整个系统强持续。     

基于比率的具有反馈控制的捕食系统的持久性与全局渐近稳定性

研究了一类基于比率的具有反馈控制的非自治捕食模型,得到了系统一致持久和其周期系统存在唯一全局渐近稳定的周期解的充分性条件．     

Diffusion-mediated persistence in three-species competition models with heteroclinic cycles.

We consider a model composed of two patches. One patch has three competing species forming a heteroclinic cycle within the path. The other is a refuge for one of the three species, which can diffuse between the two patches. The remaining two competitors are confined to the competitive patch and cannot diffuse. A new heteroclinic cycle can exist in the model, and the underlying cycle in the competitive patch cannot appear with a positive diffusion rate. It is proved that the model can be made persistent under appropriate diffusion conditions even if the underlying heteroclinic cycle is an attractor in the competitive patch and the patch is not persistent without the refuge. Further it is shown that the model with a specific structure is globally stable if the underlying cycle is a repeller.     

一类带有扩散和时滞的非自台Lotka-Volterra系统

本文讨论有时滞的扩散系统,此系统有两个种群两个斑块,其中一种种群可以在两斑块中自由扩散,另一种群被限定在斑块中不能扩散,当系数数满足一定的条件时,得到系统有持续生存和全局稳定的解。     

Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays

This paper studies a non-autonomous Lotka-Volterra almost periodic predator-prey dispersal system with discrete and continuous time delays which consists of n-patches, the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. By using comparison theorem and delay differential equation basic theory, we prove the system is uniformly persistent under some appropriate conditions. Further, by constructing suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. By using almost periodic functional hull theory, we show that the almost periodic system has a unique globally asymptotical stable strictly positive almost periodic solution. The conditions for the permanence, global stability of system and the existence, uniqueness of positive almost periodic solution depend on delays, so, time delays are "profitless". Finally, conclusions and two particular cases are given. These results are basically an extension of the known results for non-autonomous Lotka-Volterra systems.     

具Ⅱ类功能反应的非自治捕食扩散系统的全局稳定性

研究了一类具有扩散率和Ⅱ类功能性反应的非自治捕食系统,证明了在适当条件下,系统是持久的。进一步如果系统是周期系统,则在一定条件下存在唯一严格正的全局稳定的周期解。     

Mathematical analysis of a delayed stage-structured predator–prey model with impulsive diffusion between two predators territories

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in reality, many species diffuse only during a single period, and diffusion often occurs in regular pulses. Further, in forest habitats, the highest-level predator species are restricted to a specific territory, but prey can impulsively move between territories. Therefore, in this paper, we consider a delayed stage-structured predator–prey model with impulsively diffusive prey between two patches; in the model, patches represent the territories of two different predator populations. Here, we analytically obtain the global attractivity condition of predator-extinction periodic solutions for the system by using the concepts of Hui and Chen (2005); a numerical simulation is also included to illustrate this result. Further, we establish permanence conditions for the coexistence of the species using the theory of impulsive delayed differential equations. Finally, we explore the possibilities of the permanence of the system by using the growth rates of immature predators and the impulse period as critical parameters, and we also obtain the parameters’ threshold limits using numerical experimentation.     

Periodic solution of a chemostat model with Monod growth rate and impulsive state feedback control

In this paper, we develop a mathematical model concerning a chemostat with impulsive state feedback control to investigate the periodicity of bioprocess. By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions under which the model has a periodic solution of order one are obtained. Furthermore, we estimate the position of the periodic solution of order one and discuss the existence of periodic solution of order two. The theoretical results and numerical simulations indicate that the chemostat system with impulsive state feedback control either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the dilution rate and the initial concentrations of microorganisms and substrate.     

State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences

A state-dependent impulsive model is proposed for integrated pest management (IPM). IPM involves combining biological, mechanical, and chemical tactics to reduce pest numbers to tolerable levels after a pest population has reached its economic threshold (ET). The complete expression of an orbitally asymptotically stable periodic solution to the model with a maximum value no larger than the given ET is presented, the existence of which implies that pests can be controlled at or below their ET levels. We also prove that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the LambertW function and Poincare map. Moreover, we show that the existence of an order two periodic solution implies the existence of an order one periodic solution. Various positive invariant sets and attractors of this impulsive semi-dynamical system are described and discussed. In particular, several horseshoe-like attractors, whose interiors can simultaneously contain stable order 1 periodic solutions and order 2 periodic solutions, are found and the interior structure of the horseshoe-like attractors is discussed. Finally, the largest invariant set and the sufficient conditions which guarantee the global orbital and asymptotic stability of the order 1 periodic solution in the meaningful domain for the system are given using the Lyapunov function. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. Moreover, our theoretical work suggests how IPM strategies could be used to alter the levels of the ET in the farmers' favour.     

Persistence and periodic orbits of a three-competitor model with refuges.

We consider a model composed of four patches. One patch has three competing species forming a heteroclinic cycle within the patch. The remaining patches are refuges for the three competitors, and each species can diffuse between the competitive patch and its refuge. It is proved that the model can be made persistent by the introduction of the refuges for the competitors even if the isolated competitive patch has an attracting heteroclinic cycle. Further it is shown that Hopf bifurcation is possible when we change the value of the diffusion constant and periodic orbits may exist in a specific case.     

单种群非自治缀块扩散的竞争模型的概周期解

本文考虑单种群非自治缀块扩散的竞争系统,利用微分不等式,证明了系统存在唯一的正概周期解,它在壳的扰动下是稳定的．     

The dynamics of two diffusively coupled predator-prey populations

I analyze the dynamics of predator and prey populations living in two patches. Within a patch the prey grow logistically and the predators have a Holling type II functional response. The two patches are coupled through predator migration. The system can be interpreted as a simple predator-prey metapopulation or as a spatially explicit predator-prey system. Asynchronous local dynamics are presumed by metapopulation theory. The main question I address is when synchronous and when asynchronous dynamics arise. Contrary to biological intuition, for very small migration rates the oscillations always synchronize. For intermediate migration rates the synchronous oscillations are unstable and I found periodic, quasi-periodic, and intermittently chaotic attractors with asynchronous dynamics. For large predator migration rates, attractors in the form of equilibria or limit cycles exist in which one of the patches contains no prey. The dynamical behavior of the system is described using bifurcation diagrams. The model shows that spatial predator-prey populations can be regulated through the interplay of local dynamics and migration.     

非自治Lotka-Volterra型捕食扩散系统中有害时滞对该系统持久性的影响

研究了时滞对一类非自治Lotka-Volterra型捕食扩散系统的影响,该系统由n个斑块组成,食饵种群可以在斑块间迁移,而摘食者限制在某一个斑块不能扩散．我们假设密度制约项系数并不总是严格正的．通过运用比较定理及时滞泛函微分方程的基本原理,分两种情况表明了在一定条件下系统是一致持久的．两种情况的结果表明时滞的引入和变化即可能是“有害”,也可能是”无害”．进一步还说明了系统在一致持久性的条件下至少存在一个正周期解．这些结果是对已知的非自治Lotka-Volterra系统的一些结果的推广与改进．     

有毒物影响和Beddington-DeAngelis型功能性反应的捕食系统的全局吸引性

利用微分方程比较原理,重合度理论中Mawhin’s延拓定理,Lya．punov泛函和Barbalat引理,研究了一类有毒物影响和Beddington—DeAngelis型功能性反应的时滞两种群捕食者-食饵系统．我们得到了该系统一致持久性和其周期系统存在唯一全局渐近稳定的周期解的充分条件．改进了范猛和唐贵坚的相关结果．     

具Ⅲ类功能反应的非自治捕食系统的持续性和周期解

本文讨论了一类具Holling Ⅲ类功能反应的非自治捕食系统,得到该系统的持续性和存在唯一全局稳定周期解的充分条件。     

具有脉冲效应的时滞捕食系统的正周期解

提出了一个具有脉冲效应的周期时滞捕食系统,运用叠合的方法研究了该系统的周期解的存在性．结论表明在一定的条件下捕食者和食饵周期变化．