具有Holling Ⅲ功能反应和阶段结构的Gompertz生态系统的持久性

以生态学与微分方程的理论和方法为基础,建立了一类具有HollingⅢ功能反应和阶段结构的生态Gompertz模型.利用频闪映射,获得了捕食者灭绝周期解,分析了此周期解的全局吸引性.在对食饵进行脉冲收获和捕食者具有成长期时滞条件下,运用脉冲微分方程比较定理和小振幅扰动技巧,获得了系统一致持续生存的条件.     

一类非自治无时滞捕食食饵模型的持久性

主要针对一类非自治食饵具有阶段结构的捕食者非密度制约的捕食食饵模型进行了分析讨论,得到了种群灭绝以及持久的积分形式的充分条件,把捕食者密度制约的一些重要结论推广到捕食者非密度制约的情形,并且通过构造Lyapunov函数得到了系统的全局吸引性,最后利用数值模拟得到了当系统持久时周期模型的全局吸引性.     

Stability and bifurcation of a stage-structured predator–prey model with both discrete and distributed delays

This paper concerns with a new delayed predator–prey model with stage structure on prey, in which the immature prey and the mature prey are preyed by predator and the delay is the length of the immature stage. Mathematical analysis of the model equations is given with regard to invariance of non-negativity, boundedness of solutions, permanence and global stability and nature of equilibria. Our work shows that the stage structure on the prey is one of the important factors that affect the extinction of the predator, and the predation on immature prey is a cause of periodic oscillation of population and can make the behaviors of the system more complex. The predation on the immature and mature prey brings both positive and negative effects on the permanence of the predator, if ignore the predation on immature prey in the system, the stage-structure on prey brings only negative effect on the permanence of the predator.     

Permanence,extinction and periodic solution of the predator–prey system with Beddington–DeAngelis functional response and stage structure for prey

In this paper, we study the permanence, extinction and periodic solution of the periodic predator–prey system with Beddington–DeAngelis functional response and stage structure for prey. A set of sufficient and necessary conditions which guarantee the predator and prey species to be permanent are obtained. In addition, sufficient conditions are derived for the existence of positive periodic solutions to the system. Numeric simulations show the feasibility of the main results.     

A predator–prey system with a stage structure for the prey

This paper considers a periodic predator–prey system where the prey has a life history that takes the prey through two stages: immature and mature. We provide a sufficient and necessary condition to guarantee permanence of the system. It is shown that the system is permanent if and only if the growth of the predator by foraging the prey minus its death rate is positive on average during the period.     

Global dynamics of a predator-prey model

This paper deals with the dynamics of a predator-prey model with Hassell-Varley-Holling functional response. First, we show that the predator coexists with prey if and only if predator's growth ability is greater than its death rate. Second, using a blow-up technique, we prove that the origin equilibrium point is repelling and extinction of both predator and prey populations is impossible. Third, the local and global stability of the positive steady state coincide when the predator interference is large. Finally, for a typical biological case, we show instability of the positive equilibrium implies global stability of the limit cycle. Numerical simulations are carried out for a hypothetical set of parameter values to substantiate our analytical findings.     

The dynamics of an age structured predator–prey model with disturbing pulse and time delays

Many of the existing predator–prey models on stage structured populations are some ordinary differential equations (ODE) or models without a disturbing effect of human behavior. In reality, death of the juvenile during its immature stage and catching or poisoning for the prey or predator occur continuously. From this basic standpoint, we formulate a general and robust prey-dependent consumption predator–prey model with periodic harvesting (catching or poisoning) for the prey and stage structure for the predator with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and ecological study. We show that the conditions for global attractivity of the ‘predator-extinction’ (‘predator-eradication’) periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. We also show that constant maturation time delay and impulsive catching or poisoning for the prey can bring great effects on the dynamics of system by numerical analysis. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy–pest) model with age structure, exhibit a new modeling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

Stability and Hopf bifurcation in a predator–prey model with stage structure for the predator

A predator–prey system with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. Further, the existence of a Hopf bifurcation at the positive equilibrium is also studied. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. As a result, the threshold is obtained for the permanence and extinction of the system. Numerical simulations are carried out to illustrate the main results.     

Dynamics of a Stochastic Predator–Prey Model with Stage Structure for Predator and Holling Type II Functional Response

In this paper, we develop and study a stochastic predator–prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results.     

Rich dynamic of a stage-structured prey–predator model with cannibalism and periodic attacking rate

The dynamic behavior of a stage-structure prey–predator model with cannibalism for prey and periodic attacking rate for predator is investigated. Firstly, the permanence, locally and globally asymptotic stability analyses of the model with constant attacking rate are explored. After that, sufficient conditions for the permanence of the corresponding nonautonomous system with periodic attacking rate are obtained. Furthermore, numerical simulations are presented to illustrate the effects of periodic attacking rate. Simulation results show that the system with periodic attacking rate shows a rich behaviors, including period-doubling and period-having bifurcations, chaos and windows of periodicity.     

一个具有时滞和阶段结构的捕食-被捕食模型

研究一个具有时滞和阶段结构的捕食者-食饵模型.通过构造适当的Lyapunov泛函,讨论了该模型的正平衡点和非负边界平衡点的全局吸引性,从而得到了保证该生态系统永久持续生存与绝灭的充分性条件.     

一类疾病在食饵中传播的生态-传染病模型分析

分析并建立疾病在食饵中传播的生态-传染病模型,且考虑易感食饵具有常数输入,捕食者种群以Logistic模型增长,讨论了系统解的有界性和各平衡点的存在性,以及局部渐近稳定性,通过构造适当的Lyapunov函数分析了各平衡点的全局渐近稳定性,并运用比较定理证明了系统的持久性.     

Analysis of a nonautonomous eco‐epidemic diffusive model with disease in the prey

This paper deals with the behavior of positive solutions to a nonautonomous reaction‐diffusion system with homogeneous Neumann boundary conditions, which describes a two‐species predator‐prey system in which there is an infectious disease in prey. The sufficient condition on the permanence of the prey and the predator is established by combining the comparison principle with the results related to the corresponding ODE system. Some sufficient conditions for the spreading and vanishing of the disease are obtained. The global attractivity is also discussed by constructing a Lyapunov functional. Our results show that the disease is spreading if the transmission rate is suitably large, while if the transmission rate is small, the disease must be vanishing.     

A nonautonomous predator–prey system with stage structure and double time delays

In the present paper we study a nonautonomous predator–prey model with stage structure and double time delays due to maturation time for both prey and predator. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the immature prey. Based on some comparison arguments we discuss the permanence of the species. By virtue of the continuation theorem of coincidence degree theory, we prove the existence of positive periodic solution. By means of constructing an appropriate Lyapunov functional, we obtain sufficient conditions for the uniqueness and the global stability of positive periodic solution. Two examples are given to illustrate the feasibility of our main results.     

一个具有时滞和功能反应的非自治阶段结构的捕食模型

研究一个具有时滞和功能反应的非自治阶段结构的捕食模型,其中食饵种群分两个阶段——幼年和成年,且捕食种群只捕成年食饵.通过利用微分不等式、比较定理和构造适当的Lyapunov泛函,得到该模型的永久持续生存和绝灭的充分条件,同时也得到了周期系统存在唯一全局渐近稳定的周期解的充分条件.     

连续收获捕食者与脉冲存放食饵的阶段结构捕食-食饵模型的全局吸引和一致持久

研究了一个捕食者具连续收获与食饵具脉冲存放的阶段结构时滞捕食-食饵模型．根据生物资源管理的实际,改进了捕食者具阶段结构的捕食-食饵模型,即原来假设每个捕食者个体都具有相同的捕食食饵的能力．假设捕食者按年龄分为两个阶段,即幼体和成体,而且幼体无能力捕食食饵．得到了捕食者灭绝周期解全局吸引和系统持久的充分条件．结论说明了脉冲存放食饵对系统的持久起了重要的作用,并且为生物资源管理提供了策略基础．数值分析也进一步说明了系统的动力学性质．     

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.     

Periodic solutions of a nonautonomous predator–prey system with stage structure and time delays

A nonautonomous Lotka–Volterra type predator–prey model with stage structure and time delays is investigated. It is assumed in the model that the individuals in each species may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators do not feed on prey and do not have the ability to reproduce. By some comparison arguments we first discuss the permanence of the model. By using the continuation theorem of coincidence degree theory, sufficient conditions are derived for the existence of positive periodic solutions to the model. By means of a suitable Lyapunov functional, sufficient conditions are obtained for the uniqueness and global stability of the positive periodic solutions to the model.     

具有收获和阶段结构的时滞脉冲的捕食-食饵模型

研究了食饵分布在不同斑块,捕食者具有阶段结构和收获的时滞脉冲的捕食-食饵模型.利用离散动力系统的频闪映射,得到了捕食者灭绝周期解的存在性和它的精确表达式.使用比较原理,得到了捕食者灭绝周期解全局渐近稳定的充分条件和系统的持久性.最后,用Matlab软件进行数值仿真验证了获得的结果.     

一类带时滞的阶段结构捕食模型的定性分析

研究了一类具有时滞和阶段结构的捕食模型系统,给出了系统持续生存的充分条件.利用比较定理和构造适当的Lyapunov泛函得到了该系统正平衡态全局渐近稳定的充分条件.