Analysis of the Dynamics of a Predator-prey Model with Holling Functional Response

A diffusive predator-prey system with Holling functional response is considered. Firstly, existence of positive equilibrium of this reaction diffusion model under Neumann boundary condition is obtained. Meanwhile, the existence conditions for Turing instability and Hopf bifurcations of a system with Holling \uppercase\expandafter{\romannumeral2} functional response are established. Next, the existence of the hydra effect is demonstrated, when the system is undergoing non-homogeneous steady-state solutions. Finally, numerical simulations are illustrated to support our theory results.     

Analyses of Bifurcations and Stability in a Predator-prey System with Holling Type-Ⅳ Functional Response

In this paper the dynamical behaviors of a predator-prey system with Holling Type-Ⅳfunctionalresponse are investigated in detail by using the analyses of qualitative method,bifurcation theory,and numericalsimulation.The qualitative analyses and numerical simulation for the model indicate that it has a unique stablelimit cycle.The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddle-node bifurcation,Hopf bifurcation,homoclinic bifurcation and bifurcation of cusp-type with codimension two(ie,the Bogdanov-Takens bifurcation),and we show the existence of codimension three degenerated equilibriumand the existence of homoclinic orbit by using numerical simulation.     

Spatial Dynamics of a Diffusive Predator-prey Model with Leslie-Gower Functional Response and Strong Allee Effect

In this paper, spatial dynamics of a diffusive predator-prey model with Leslie-Gower functional response and strong Allee effect is studied. Firstly, we obtain the critical condition of Hopf bifurcation and Turing bifurcation of the PDE model. Secondly, taking self-diffusion coefficient of the prey as bi- furcation parameter, the amplitude equations are derived by using multi-scale analysis methods. Finally, numerical simulations are carried out to verify our theoretical results. The simulations show that with the decrease of self- diffusion coefficient of the prey, the preys present three pattern structures: spot pattern, mixed pattern, and stripe pattern. We also observe the transi- tion from spot patterns to stripe patterns of the prey by changing the intrinsic growth rate of the predator. Our results reveal that both diffusion and the intrinsic growth rate play important roles in the spatial distribution of species.     

一类具有功能性反应的中立型捕-食系统全局正周期解的存在性

首次研究一类具有HollingII型功能性反应中立型捕食者-食饵系统(即Rosenzweig-MacArthur模型),通过发展一些分析技巧,利用重合度理论中的延拓定理讨论了其全局正周期解的存在性,得到了保证周期解存在的充分条件. 最后举例说明该文定理条件是可行的.     

具有捕食正效应及Holling Ⅱ功能性反应的捕食-食饵系统

本文研究一类具有捕食正效应的Holling Ⅱ功能性反应的食饵-捕食者系统:dx/dt=rx(1-x/k+εy/k)-αxy/1+ωx,dy/dt=kαxy/1+ωx-dy,讨论其平衡点的性态.     

具有HollingIV类功能性反应时滞扩散捕食模型的多个周期解

考虑一类具有HollingIV类功能性反应时滞扩散捕食模型.该模型的系数为周期函数,这和环境的周期变化相一致.作者应用重合度定理,建立了该模型具有至少两个正周期解的充分条件.     

一类带功能反应食饵患病的食饵捕食者模型的稳定性分析

研究了一类食饵患病且有人为干预和Holling IV功能反应的食饵捕食者模型,讨论了系统平衡点的稳定性,解的有界性及极限环的存在和分岔情况.此外,还分析了文中重要参数对系统的影响,给出了数值模拟结果.     

Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

A discrete predator-prey system with Holling type-IV functional response obtained by the Euler method is first investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-two bifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximate expressions for saddle-node, Hopfand homoclinic bifurcation sets near the Bogdanov-Takens bifurcation point. We also show the existence of degenerated fixed point with codimension three at least. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors such as the attracting invariant circle, period-doubling bifurcation from period-2,3,4 orbits.interior crisis, intermittency mechanic, and sudden disappearance of chaotic dynamic.     

一类具有功能性反应的中立型捕食者-食饵系统全局正周期解的存在性

首次研究一类具有HollingⅡ型功能性反应中立型捕食者-食饵系统(即Rosenzweig- MacArthur模型),通过发展一些分析技巧,利用重合度理论中的延拓定理讨论了其全局正周 期解的存在性,得到了保证周期解存在的充分条件．最后举例说明该文定理条件是可行的．     

Periodic Solution for a Stochastic Non-autonomous Predator-Prey Model with Holling II Functional Response

A biological population may be subjected to stochastic disturbance and exhibit periodicity. In this paper, a stochastic non-autonomous predator-prey system with Holling II functional response is proposed, and the existence of a unique positive solution is derived. We give sufficient conditions for extinction and strong persistence in the mean by analyzing a corresponding one-dimensional stochastic system. Also we establish the existence of positive periodic solutions for this stochastic non-autonomous predator-prey system. Finally, we use numerical simulations to illustrate our results and we present some conclusions and future directions. The results of this paper provide methods for other stochastic population models, which we hope to analyze in the future.

     

捕食者具有阶段结构Holling II类功能性反应的捕食模型

利用重合度理论中的延拓定理讨论了捕食者具有阶段结构Holling II类功能性反应的捕食模型的正周期解的存在性,得到了保证周期解存在的充分性条件,推广了已知的相关结果. 同时通过构造Lyapunov函数得到了保证周期解稳定性的充分性条件.     

一类具有Allee效应的捕食系统的稳定性分析及模拟

建立了食饵具有Allee效应的捕食模型,讨论了系统的有界性和平衡点的存在性.并证明了平衡点的局部渐近稳定性,进而通过构造Lyapunov函数分析了正平衡点的全局渐近稳定性,利用数值模拟讨论了Allee效应对系统的影响:Allee效应是系统的不稳定因素.     

一类具有Watt型功能性反应的捕食系统的极限环与稳定性

研究一类具有Watt型功能性反应的捕食模型.讨论了该系统正平衡点的存在性以及非负平衡点的性态,应用Poincare-Bendixson定理和张芷芬定理,证明了极限环的存在性和唯一性,并采用构造Dulac函数的方法,获得了正平衡点全局渐近稳定性的一个充分条件.     

Stability Analysis of an Eco-epidemiological Model with Time Delay and Holling Type-III Functional Response

In this paper, an eco-epidemiological model with diseases in the predator and Holling type-III functional response is analyzed. A time delay due to the gestation of the predator is considered in this model. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium and the endemic-coexistence equilibrium are established respectively. By using Lyapunov functionals and LaSalle''s invariance principle, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium, the disease-free equilibrium and the endemic-coexistence equilibrium respectively. Finally, numerical simulations are performed to illustrate the theoretical results.     

具有Holling-II型功能反应和交叉扩散的捕食-食饵模型正平衡解的存在性和不存在性

本文主要研究一类在齐次Dirichlet边界条件下带交叉扩散的Holling-II型捕食者-食饵模型正平衡解的存在性, 其中两个交叉扩散系数分别代表食饵远离捕食者的趋势和捕食者追逐食饵的趋势. 应用不动点指标理论得到了正平衡解存在的充分条件, 并进一步研究了正平衡解不存在的条件.     

一类具有时滞和Holling Ⅲ型功能性反应的捕食模型的稳定性和Hopf分支

研究一类具有时滞和Holling Ⅲ型功能性反应的捕食模型的稳定性和Hopf分支.以滞量为参数,得到了系统正平衡点的稳定性和Hopf分支存在的充分条件,给出了确定Hopf分支方向和分支周期解的稳定性的计算公式.     

一类具有Holling-Ⅱ型功能性反应的捕食者-食饵系统全局周期解的存在性

利用重合度理论中的延拓定理讨论了一类具有Ｈｏｌｌｉｎｇ Ⅱ 型功能性反应的捕食者-食饵系统（即Ｒｏｓｅｎｚｗｅｉｇ ＭａｃＡｒｔｈｕｒ模型）全局周期解的存在性,得到了保证周期解存在的充分条件,推广了某些已知的相关结果．     

Investigating the Turing Conditions for Diffusion-driven Instability in Predator-prey System with Hunting Cooperation Functional Response

In this paper, we focus on stability analysis of steady-state solutions of a predator-prey system with hunting cooperation functional response. The results show that the Turing instability can be affected not only the existence of hunting cooperation, but also the diffusion coefficients: (1) in the absence of predator diffusion, diffusion-driven instability can be induced by hunting cooperation, but no stable patterns appear; (2) the system can occur diffusion-driven instability and Turing patterns, when both predator and prey have diffusion, and the diffusion coefficient of prey is greater than that of the predator. The numerical simulations of two cases are presented to verify the validity of our theoretical results.     

具有Holling Ⅱ类功能反应且周期系数的非自治捕食扩散系统的持久与全局渐近稳定性

研究了一类具有扩散系数和 Holling 类功能反应的一捕两食三种群非自治捕食系统 ,得到了系统持久生存和周期系统存在唯一全局渐近稳定的周期解的条件 .