一类带有脉冲的捕食-食饵随机模型的分析

Analysis for a Stochastic Predator-Prey System with Impulses

在自然界中存在着大量的捕食关系,而且两种群的关系要受到环境的干扰和人类或自然灾害对物种数量的脉冲干扰,因此本文研究具有脉冲的随机模型.首先,引入并建立了非自制的脉冲随机模型.然后,引入一个不带脉冲项的等价微分方程组,来证明脉冲模型解的存在唯一性,并构造李雅普诺夫泛函证明了脉冲模型周期解的存在性.接着,通过比较定理分别研究了两种群持久和灭绝情况的条件.最后,进行数值模拟,并将模型与生物入侵的案例相连接,把本模型的脉冲主要看成是人类对物种的捕猎,达到保护生态的目的,以此验证理论的合理性.

There is so much predation in the nature, and the population of the two species is influenced by environmental disturbance and the impulse of human activities or natural disaster. Therefore, we studied a stochastic predator-prey system with impulses. First we draw into and built the stochastic non-autonomous predator-prey system with impulses step by step. Second, we proved the existence and uniqueness of the positive solution by constructing the equivalent system without impulse. Third, we proved the existence of the T-periodic solution by choosing a suitable Lyapunov function. Next, we studied the sufficient conditions for permanence and extinction of the two populations by using the comparison theorem. Finally, we applied the system to the biological invasion by numerical simulations, and the impulse was mainly regarded as the hunt to the species by human. The outcome of the numerical simulations will show whether the conclusion we made is reasonable.