一个微分生态系统的极限环分析

Limit cycle analysis of a differential ecosystem

在已有功能性反应的生态系统的基础上,应用数学生态学理论建立了一个具有功能性反应的微分生态系统,其中食饵种群具有非密度制约,且食饵种群密度的变化与常数相比对捕食种群的影响更为明显.应用微分方程定性理论,讨论了该微分生态系统,研究了系统的平衡点,对中心焦点的阶数和稳定性做出分析,并给出了系统的环域构成图.在给定参数满足一定条件时,利用Bendixson环域定理和张芷芬唯一性定理,证明了该系统极限环的存在性和唯一性.结果表明,两种群的密度或产生周期性变化,或都稳定在一组定值的附近,可以保持一种稳定状态.

Based on the ecosystem with functional response, a differential ecosystem with functional response is established with the application of mathematic ecology. The system is discussed by using differential equations qualities theory. The system＇s equilibrium points are studied. The degree and the stability of the center focus are analyzed. The existence zone of the system＇s limit cycle is given. By using Bendixson theorem and Zhang Zhi-fen theorem, the existence and uniqueness of the system＇s limit cycle are proved if the given parameters satisfy certain conditions. It shows that the densities of preys and predators are stable under some conditions.