具有稀疏效应的Predator-Prey模型的分支问题

讨论了具有稀疏效应的捕食－食饵模型的分支问题,并利用Ｈｏｐｆ分支理论和分界线环分支理论,得到了有多个极限环的结果。     

一类具Holling Ⅲ型功能反应的捕食者-食饵模型的定性分析

研究一类具Holling Ⅲ型功能反应的捕食者一食饵模型．应用定性分析和Hopf分支理论,得到了一个正平衡点的全局稳定性、三个正平衡点的局部稳定性和极限环的存在性的充分条件,使用MATLAB软件。本文给出了三个例子来模拟这些结论。     

食饵具有周期强迫的捕食模型

建立并研究了一类具有周期强迫和脉冲扰动的捕食模型,通过理论分析和数值模拟,得到了食饵灭绝周期解全局渐近稳定和系统持久的充分条件,利用分支理论证明了边界周期解附近会分支出正周期解.     

具阶段结构及比率依赖型捕食系统的Hopf分支与稳定性

本文研究了一类食饵具有阶段结构的比率依赖型捕食模型的稳定性和Hopf分支的存在性问题.通过分析相应的特征方程,得到了平衡点局部稳定的充分条件,并指出当时滞穿过某特定值时正平衡点出现Hopf分支.利用比较定理与迭代方法证明了正平衡点的全局渐近稳定性,得到正平衡点全局渐近稳定的充分条件.最后,举例说明所得结果的可行性.     

食饵染病的非自治生态-流行病模型的分析

研究了一类疾病仅在食饵中传播的非自治捕食者-食饵模型.文中假定捕食者受密度制约影响,在捕食时具有甄别能力而只捕食健康的食饵,且具有HollingⅢ类功效函数;对于食饵,将其分为食饵易感者和食饵染病者两类,模型假定疾病发生率为非线性的,且考虑到了环境对食饵种群的容纳量.文中运用Barbalat引理、比较原理等稳定性理论的相关方法,找到了系统的正向不变集,证明了模型解的最终有界性;通过理论分析,分别得到了两种群持续生存、周期解的存在性以及周期解全局渐进稳定的相关条件;最后,通过MATLAB数值模拟,验证了有关结论.     

一类两种群均有收获率捕食系统的细焦点与极限环

研究一类具有HollingⅡ类功能反应且两种群均为非常数收获率的捕食系统,其中食饵种群具有非线性密度制约.利用微分方程定性与稳定性理论及分支理论,得到系统平衡点的性态及极限环存在与否的充分条件,利用Hopf分支理论得到存到多个极限环的充分条件.     

双密度制约的Holling Ⅱ型捕食动力系统的定性分析

研究食饵具有非线性密度制约捕食者具有线性密度制约的HollingⅡ型捕食动力系统．以食饵的环境容纳量为分支参数,由Hopf分支得到小振幅极限环的存在性,同时也得到了正平衡点的全局稳定性和非小振幅极限环的存在唯一性的充分条件．     

一类捕食者具有阶段结构的捕食系统的全局稳定性分析

本文建立了一类捕食者具有阶段结构的捕食系统,计算得到了不存在食饵种群时捕食者种群模型和食饵种群存在时捕食系统的平衡点,并证明了平衡点的存在性.分析和比较了两个模型平衡点的全局稳定性,最终确定了决定模型全局稳定性的捕食者种群基本再生数、食饵灭绝与否的捕食率阈值以及捕食存在时食饵种群的净增长率.     

脉冲捕获捕食者与食饵具阶段结构的捕食-食饵模型

研究脉冲捕获捕食者与食饵具阶段结构的捕食-食饵模型.利用频闪映射理论,得到食饵灭绝的周期解是全局吸引的;运用时滞脉冲微分方程理论,证明了此系统是持久的.本文的结论为生态保护提供了可靠的策略依据.     

一类具时滞和非线性发生率的生态流行病模型的稳定性和Hopf分支

研究一类具有时滞和非线性发生率的生态流行病模型.以滞量为参数,通过分析特征方程,得到了正平衡点局部稳定和Hopf分支存在的条件.同时,应用中心流形定理和规范型理论,得到了分支方向和分支周期解的稳定性计算公式.最后对所得理论结果进行了数值模拟.     

A powerful truncated tail strength method for testing multiple null hypotheses in one dataset

This article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a higher densities they were available to predators. Thus, their functional response was of the Holling type III. They analyzed this model and predicted existence of a limit cycle in predator-prey dynamics. In this article I show that their model is ill posed, because trajectories are not well defined. Using the Filippov method, I define and analyze solutions of the Gause model. I show that depending on parameter values, there are three possibilities: (1) trajectories converge to a limit cycle, as predicted by Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.     

具有捕食正效应的捕食-食饵系统

在经典的捕食食饵系统中考虑到由于捕食效应对食饵种群带来的正向调节作用后,提出了具有捕食正效应的捕食-食饵系统.通过对模型的动力学行为的分析,从理论上说明了正向调节作用对系统的影响,并就第一象限内平衡点存在时的相图解释了捕食正效应的作用.结果表明:(1)捕食系统中适当的正向调节作用会增加系统的稳定性;(2)当捕食正效应达到一定的程度后系统拥有一个不稳定的极限环;(3)当捕食正效应过大时会使系统的稳定性发生变化,使捕食者种群与食饵种群同时趋向无穷,出现了调节放纵现象.这些结果在保护生物学中具有重要的意义.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

Global behaviour of a predator–prey like model with piecewise constant arguments

The present study deals with the analysis of a predator–prey like model consisting of system of differential equations with piecewise constant arguments. A solution of the system with piecewise constant arguments leads to a system of difference equations which is examined to study boundedness, local and global asymptotic behaviour of the positive solutions. Using Schur–Cohn criterion and a Lyapunov function, we derive sufficient conditions under which the positive equilibrium point is local and global asymptotically stable. Moreover, we show numerically that periodic solutions arise as a consequence of Neimark-Sacker bifurcation of a limit cycle.     

The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs

This article studies the effects of adaptive changes in predator and/or prey activities on the Lotka-Volterra predator-prey population dynamics. The model assumes the classical foraging-predation risk trade-offs: increased activity increases population growth rate, but it also increases mortality rate. The model considers three scenarios: prey only are adaptive, predators only are adaptive, and both species are adaptive. Under all these scenarios, the neutral stability of the classical Lotka-Volterra model is partially lost because the amplitude of maximum oscillation in species numbers is bounded, and the bound is independent of the initial population numbers. Moreover, if both prey and predators behave adaptively, the neutral stability can be completely lost, and a globally stable equilibrium would appear. This is because prey and/or predator switching leads to a piecewise constant prey (predator) isocline with a vertical (horizontal) part that limits the amplitude of oscillations in prey and predator numbers, exactly as suggested by Rosenzweig and MacArthur in their seminal work on graphical stability analysis of predator-prey systems. Prey and predator activities in a long-term run are calculated explicitly. This article shows that predictions based on short-term behavioral experiments may not correspond to long-term predictions when population dynamics are considered.     

A predator-prey reaction-diffusion system with nonlocal effects

We consider a predator-prey system in the form of a coupled system of reaction-diffusion equations with an integral term representing a weighted average of the values of the prey density function, both in past time and space. In a limiting case the system reduces to the Lotka Volterra diffusion system with logistic growth of the prey. We investigate the linear stability of the coexistence steady state and bifurcations occurring from it, and expressions for some of the bifurcating solutions are constructed. None of these bifurcations can occur in the degenerate case when the nonlocal term is in fact local.     

Sigmoid functional responses and population stability

Sigmoid functional responses are known to stabilize the differential Lotka-Volterra predator-prey model. However, we have found that they have no such effect in a comparable discrete generation model. The difficulty in stabilizing this model results from the one-generation time delay between changes in predator population density and the level of prey mortality. By contrast, sigmoid functional responses can stabilize the system if the predator population remains relatively constant, as is more likely of generalist predators.     

Modeling and analysis of a predator-prey model with disease in the prey

A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator k=k(0) is constant (independent of delay tau;, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay tau; is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If k=k(0)e(-dtau;) (d is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.     

The subcritical collapse of predator populations in discrete-time predator-prey models.

Many discrete-time predator-prey models possess three equilibria, corresponding to (1) extinction of both species, (2) extinction of the predator and survival of the prey at its carrying capacity, or (3) coexistence of both species. For a variety of such models, the equilibrium corresponding to coexistence may lose stability via a Hopf bifurcation, in which case trajectories approach an invariant circle. Alternatively, the equilibrium may undergo a subcritical flip bifurcation with a concomitant crash in the predator's population. We review a technique for distinguishing between subcritical and supercritical flip bifurcations and provide examples of predator-prey systems with a subcritical flip bifurcation.     

Dispersal delays, predator-prey stability, and the paradox of enrichment

It takes time for individuals to move from place to place. This travel time can be incorporated into metapopulation models via a delay in the interpatch migration term. Such a term has been shown to stabilize the positive equilibrium of the classical Lotka-Volterra predator-prey system with one species (either the predator or the prey) dispersing. We study a more realistic, Rosenzweig-MacArthur, model that includes a carrying capacity for the prey, and saturating functional response for the predator. We show that dispersal delays can stabilize the predator-prey equilibrium point despite the presence of a Type II functional response that is known to be destabilizing. We also show that dispersal delays reduce the amplitude of oscillations when the equilibrium is unstable, and therefore may help resolve the paradox of enrichment.