一个具有恐惧效应的捕食者-食饵系统的无穷远平衡点

Equilibria at Infinity of a Predator-Prey System with a Fear Effect

对于一个具有恐惧效应的捕食者-食饵系统,在有限远处定性性质的基础上进一步研究无穷远平衡点附近的轨道走向,从而获得系统的全局结构.首先利用Poincaré变换证明该系统有两个退化的无穷远平衡点,其中之一在其两个特殊方向的任意角邻域中都存在极坐标半径随时间变化率为零的点,因此无法构造正常区域以满足无转的条件.通过构造广义正常区域,给出无穷远平衡点附近轨道的走势,最后利用Briot-Bouquet变换确定有几条轨线连接平衡点.当两物种种群数量较大时,种群数量都不稳定,并且食饵因感受到捕食风险而产生恐惧进而表现出的反捕食反应对种群生态系统没有影响.

Based on analysis of the qualitative properties of equilibria at finity,the orbit direction on the vicinity of equilibria at infinity in a predator-prey system with a fear effect are further examined so as to obtain the global structure.Firstly,the system is proved by Poincarétransformation to have two degenerative equilibria at infinity,one of which possesses in the neighborhood of any angle at its two exceptional directions a point whose polar coordinates change at the rate of zero with time,and therefore it is impossible to construct normal sectors.Then,the trend of orbits near equilibria at infinity is given by constructing generalized normal sectors,and the intersecting equilibria of a few orbits are further determined by using Briot-Bouquet transformation.Consequently,when the numbers of two populations are large,the population number of both species become unstable,and the anti-predation reaction triggered by the fear felt by the risk of predation will exert no effect on the population ecosystem.