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.     

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 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.     

Permanence and extinction of periodic predator–prey systems in a patchy environment with delay

This paper studies two species predator–prey Lotka–Volterra type dispersal systems with periodic coefficients and infinite delays, in which the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. Sufficient and necessary conditions of integrable form for the permanence, extinction and the existence of positive periodic solutions are established, respectively. Some well-known results on the nondelayed periodic predator–prey Lotka–Volterra type dispersal systems are improved and extended to the delayed case.     

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.     

PERMANENCE OF PERIODIC BEDDINGTON-DEANGELIS PREDATOR-PREY SYSTEM IN A TWO-PATCH ENVIRONMENT WITH DELAY

In this paper, we study a two-species periodic Beddington-DeAngelis predator-prey model with delay in a two-patch environment, in which the prey species can disperse between two patches, but the predator species cannot disperse. On the basis of the comparison theorem of differential equations, we establish sufficient conditions for the permanence and extinction of the system.     

Permanence of dispersal population model with time delays

Permanence of a dispersal single-species population model is considered where environment is partitioned into several patches and the species requires some time to disperse between the patches. The model is described by delay differential equations. The existence of food-rich patches and small dispersions among the patches are proved to be sufficient to ensure partial permanence of the model. It is also shown that partial permanence ensures permanence if each food-poor patch is connected to at least one food-rich patch and if each pair in food-rich patches is connected. Furthermore, it is proved that partial persistence is ensured even under large dispersion among food-rich patches if the dispersion time is relatively small.     

Permanence and extinction for dispersal population systems

This paper studies the effect of dispersal on the permanence of population models in poor patchy environment. We first consider the logistic system with dispersal for single species and obtain the conditions for its permanence. On the basis of the conditions, we then consider a periodic predator-prey system where the prey can disperse among several patches. A necessary and sufficient condition is obtained for the permanence of the periodic predator-prey system. We discuss the biological implications of the main results.     

具有连续时滞和功能性反应的非自治竞争系统的持续生存

本文利用Lyapunov-Razumikhin理论讨论了具有连续时滞和Ⅱ类功能性反应的非自治扩散竞争系统．此系统有两个种群n个斑块，其中一个种群可以在n个斑块中自由扩散，另一种群被限定在一斑块中不能扩散．当系数满足一定的条件时，证明了系统是持续生存的，此外，给出了该系统的一周期解全局吸引的充分条件．     

Symmetrization in neumann problems

This paper is concerned with a model of a predator–prey system, where both populations disperse among n patches forming their habitat. Criteria are given tor both survival and extinction of the predator population. In case the predator survives, conditions are derived which guarantee a globally asymptotically stable positive equilibrium     

具有扩散和比率依赖的三种群混合模型的分析

本文讨论了捕食者具有比率依赖的功能性反应,食饵与另一种群竞争且自身可以 扩散的混合模型.证明了系统一致持久与扩散有关,而且得到了系统存在全局吸引周期 解的充分条件.     

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.     

Aggregation and heterogeneity from the nonlinear dynamic interaction of birth,maturation and spatial migration

We consider a single species structured population distributed in two identical patches connected by spatial dispersal. Assuming that the maturation time for each individual is a random variable with a gamma distribution and that the spatial dispersal rate is constant, we obtain from a hyperbolic differential equation a system of six ordinary differential equations for the matured populations and their moments. Our qualitative analysis and numerical simulations show that the nonlinear interaction of birth process, the maturation delay and the spatial dispersal can lead to a new mechanism for individual aggregation in the form of the existence of multiple stable heterogeneous equilibria, even though the spatial dispersal is assumed to be proportional to the population gradients with a constant rate.     

Curves and Surfaces Construction Based on New Basis with Exponential Functions

By incorporating two exponential functions into the cubic Bernstein basis functions, a new class of λμ-Bernstein basis functions is constructed. Based on these λμ-Bernstein basis functions, a kind of λμ-Bézier-like curve with two shape parameters, which include the cubic Bernstein-Bézier curve, is proposed. The C ¹ and C ² continuous conditions for joining two λμ-Bézier-like curves are given. By using tensor product method, a class of rectangular Bézier-like patches with four shape parameters is shown. The G ¹ and G ² continuous conditions for joining two rectangular Bézier-like patches are derived. By incorporating three exponential functions into the cubic Bernstein basis functions over triangular domain, a new class of λμη-Bernstein basis functions over triangular domain is also constructed. Based on the λμη-Bernstein basis functions, a kind of triangular λμη-Bézier-like patch with three shape parameters, which include the triangular Bernstein-Bézier cubic patch, is presented. The conditions for G ¹ continuous smooth joining two triangular λμη-Bézier-like patches are discussed. The shape parameters serve as tension parameters and have a predictable adjusting role on the curves and patches.     

Spread of disease in a patchy environment

For a two patches SIR model, it is shown that its dynamic behavior is determined by several quantities. We have shown that if R₀ < 1, then the disease-free equilibrium is globally asymptotically stable, otherwise it is unstable. Some sufficient conditions for the local stability of boundary equilibria are obtained. Numerical simulations indicate that travel between patches can reduces oscillations in both patches; may enhances oscillations in both patches; or travel switches oscillations from one patch to another.     

PERSISTENCE AND PERIODIC ORBITS FOR TWO-SPECIES NON-AUTONOMOUS DIFFUSION LOTKA-VOLTERRA MODELS

A non-autonomous competing system is investigated in this paper,where the species x can diffuse between two patches of a heterogeneous environment with barriers between patches,but for species y,the diffusion does not involve a barrier between patches,further it is assumed that all the parameters are time dependent. It is shown that the system can be made persistent under some appropriate conditions. Moreover,sufficient conditions that guarantee the existence of a unique positive periodic orbit which is globally asymptotic stable are derived.     

A model of an age-structured population in a multipatch environment

We propose a model of an age-structured population divided into N geographical patches. We distinguish two time scales, at the fast time scale we have the migration dynamics and at the slow time scale the demographic dynamics. The demographic process is described using the classical McKendrick-von Foerster model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process.Assuming that 0 is a simple strictly dominant eigenvalue for the migration matrix, we transform the model (an e.d.p. problem with N state variables) into a classical McKendrick-von Foerster model (scalar e.d.p. problem) for the global variable: total population density. We prove, under certain assumptions, that the semigroup associated to our problem has the property of positive asynchronous exponential growth and so we compare its asymptotic behaviour to that of the transformed scalar model. This type of study can be included in the so-called aggregation methods, where a large scale dynamical system is approximately described by a reduced system. Aggregation methods have been already developed for systems of ordinary differential equations and for discrete time models.An application of the results to the study of the dynamics of the Sole larvae is also provided.     

The effect of population dispersal on the spread of a disease

The effect of population dispersal among n patches on the spread of a disease is investigated. Population dispersal does not destroy the uniqueness of a disease free equilibrium and its attractivity when the basic reproduction number of a disease R₀<1. When R₀>1, the uniqueness and global attractivity of the endemic equilibrium can be obtained if dispersal rates of susceptible individuals and infective individuals are the same or very close in each patch. However, numerical calculations show that population dispersal may result in multiple endemic equilibria and even multi-stable equilibria among patches, and also may result in the extinction of a disease, even though it cannot be eradicated in each isolated patch, provided the basic reproduction numbers of isolated patches are not very large.     

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.     

具有连续时滞和比率型功能反应的竞争系统的正周期解的存在性

本文利用重合度理论中的延拓定理讨论了具有连续时滞和比率型功能反应的非自治扩散竞争系统的正周期解的存在性,得到了正周期解存在的充分条件。