一类具有收获的共享资源型捕食-食饵模型的建立与分析

Establishment and analysis of a shared resource predator-preymodel with harvest

本文建立了一类具有线性收获率及Holling Ⅱ型功能反应函数的共享资源型捕食-食饵模型,分析了该系统平衡点的存在性与稳定性。系统可能存在无食饵种群的平衡点,无捕食者种群的平衡点以及正平衡点,通过定性分析得到无食饵种群的平衡点在不同的条件下可以是稳定的平衡点、鞍点或者鞍结点,而无捕食者种群的平衡点始终是鞍点,并给出了正平衡点稳定的条件。最后,对系统进行数值模拟,发现收获率的不同取值可以改变系统的稳定状态。因此,利用理论分析结果可得到在保证生态稳定的前提下人类捕捞的最优收获。

In this paper,a shared resource predator-prey model with linear harvesting rate and Holling type II functional response function is established,and the existence and stability of the equilibria of the system are analyzed.There may exist the prey-free equilibrium,the predator-free equilibrium,and the positive equilibrium in the system.Through qualitative analysis,we give the conditions that the prey-free equilibrium is stable,saddle,or saddle-node one,respectively,and find that the predator-free equilibrium always be saddle one.Furthermore,we obtain the conditions that the positive equilibrium is stable.Finally,some numerical simulations are carried out.It was found that different values of harvesting rate can change the stable state of the system.Hence,the optimal harvest for human fishing can be obtained while ensuring ecological stability using theoretical analysis.