关于李纳方程及捕食者 - 猎物系统中极限环的唯一性

On Liénard's equation and the uniqueness of limit cycles in predator-prey systems.

作者信息

Moreira H N

机构信息

Departamento de Matemática, Universidade do Amazonas, Manaus-Am, Brazil.

出版信息

J Math Biol. 1990;28(3):341-54. doi: 10.1007/BF00178782.

DOI:10.1007/BF00178782

PMID:2332710

Abstract

We study a system of ODE's modelling the interaction of one predator and one prey dx/dt = xg(x) - yp(x), dy/dt = gamma y[- delta - nu y - alpha y2 + h(x)]. This system defines a two-species community which incorporates competition among prey in the absence of any predators as well as a density-dependent predator specific death rate. This system is investigated under ecologically natural regularity conditions and assumptions on g, p and h to ensure the existence and uniqueness of limit cycles. The proof uses the standard Hopf-Andronov bifurcation theory and the technique of Liénard's equation.

摘要

我们研究一个常微分方程组，它对一种捕食者和一种猎物之间的相互作用进行建模，(\frac{dx}{dt} = xg(x) - yp(x))，(\frac{dy}{dt} = \gamma y[-\delta - \nu y - \alpha y^2 + h(x)])。该系统定义了一个两物种群落，它包含了在没有任何捕食者的情况下猎物之间的竞争以及一个依赖于密度的捕食者特定死亡率。在关于(g)、(p)和(h)的生态自然正则性条件和假设下对该系统进行研究，以确保极限环的存在性和唯一性。证明使用了标准的霍普夫 - 安德罗诺夫分岔理论和李纳方程技术。