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.     

Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response

An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end.     

一类具有HollingⅠ型功能反映的半比例依赖捕食-食饵离散系统的持久性

主要研究了一类具有Holling I型功能函数半比例依赖的捕食-食饵离散系统的一般非自治情形,得到了系统的持久性的充分条件.     

A prey-predator model with Holling II functional response and the carrying capacity of predator depending on its prey

One prey-predator model is formulated and the global behavior of its solution is analyzed. In this model, the carrying capacity of predator depends on the amount of its prey, and the Holling II functional response is involved. This model may have four classes of positive equilibriums and limit cycle. The positive equilibriums may be stable, or a saddle-node, or a saddle, or a degenerate singular point. In alpine meadow ecosystem, the dynamics of vegetation and plateau pika can be described by this model. Through simulating with virtual parameters, the cause of alpine meadow degradation and effective recovery strategy is investigated. Increasing grazing rate or decreasing plateau pika mortality may cause alpine meadow degradation. Correspondingly, reducing grazing rate and increasing plateau pika mortality may recover the degraded alpine meadow effectively.     

Hopf bifurcation analysis of a predator-prey system with Holling type IV functional response and time delay

This paper is concerned with a predator-prey system with Holling type IV functional response and time delay. Our aim is to investigate how the time delay affects the dynamics of the predator-prey system. By choosing the delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium and existence of local Hopf bifurcations are analyzed. Based on the normal form and the center manifold theory, the formulaes for determining the properties of Hopf bifurcation of the predator-prey system are derived. Finally, to support these theoretical results, some numerical simulations are given to illustrate the results.     

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.     

Bifurcation Analysis of a Predator–Prey System with Generalised Holling Type III Functional Response

We consider a generalised Gause predator–prey system with a generalised Holling response function of type III: . We study the cases where b is positive or negative. We make a complete study of the bifurcation of the singular points including: the Hopf bifurcation of codimensions 1 and 2, the Bogdanov–Takens bifurcation of codimensions 2 and 3. Numerical simulations are given to calculate the homoclinic orbit of the system. Based on the results obtained, a bifurcation diagram is conjectured and a biological interpretation is given.      

Global stability of a periodic Holling‐Tanner predator‐prey model

This paper is concerned with the global dynamics of a Holling‐Tanner predator‐prey model with periodic coefficients. We establish sufficient conditions for the existence of a positive solution and its global asymptotic stability. The stability conditions are first given in average form and afterward as pointwise estimates. In the autonomous case, the previous criteria lead to a known result.     

Graph‐theoretic method on the periodicity of multipatch dispersal predator‐prey system with Holling type‐II functional response

We study the periodicity of multipatch dispersal predator‐prey system with Holling type‐II functional response in this paper. By providing a new method, we overcome the difficulty to get the priori bounds estimation of unknown solutions of operator equation Lu=λNu. Graph theory with coincidence degree theory is used, and a sufficient criterion for the periodicity of the system is obtained. The criterion presented in this paper is closely related with topological structure of dispersal network and can be verified easily. Finally, a numerical example is also provided to verify the effectiveness of theoretical results.     

具Ricker增长率的Holling Ⅰ型捕食系统的性态分析

研究食饵具有Ricker增长率的Holling I型捕食系统.得到该系统存在两个极限环以及正平衡点全局渐近稳定的条件.