一类具有弱Allee效应离散捕食系统的1∶4共振

1∶4 Resonance of Discrete Predator-Prey System With Weak Allee Effect

研究一类具有弱Allee效应的离散捕食系统。当参数满足一定条件时,该系统有一个正不动点,其线性算子的特征值为±i,这对应于1∶4共振。利用Picard迭代及时间1映射,将差分系统转化为常微分方程系统,讨论常微分方程系统退化平衡点附近的性质,得到该差分系统不动点附近的性质,并用数学软件模拟其局部相图。结果表明:随着参数值和扰动的变化,系统会产生稳定的焦点、“方形”异宿环、“叶形”异宿环及Neimark-Sacker分支等。

Study on a class of discrete predator prey system with weak Allee effect.If parameters meet certain conditions,the studied system has a positive fixed point and its linear operator has eigenvalues±i,which corresponds to 1∶4 resonance.Using Picard iteration and the time one mapping,this discrete system is transformed into an ordinary differential equation system,and the properties near the degenerate equilibrium of the ordinary differential equation system are discussed,the properties near the fixed point of this discrete system are obtained,and their local phase portraits are simulated by mathematical software.The results show that with the change of parameter values and disturbances,the system can produce stable focal points,“square”heteroclinic cycles,“clover”heteroclinic cycles and Neimark Sacker bifurcations and so on.