一类具有分段常数变量的三维食饵捕食者系统的稳定性和分支行为

The Stability and Bifurcation Behavior of a Three Dimensional Prey-Predator Model with Piecewise Constant Arguments

本文研究一类具有分段常数变量的三维食饵-捕食者系统的稳定性和分支行为,该系统由一个捕食者和两个食饵构成,其中一个食饵可由捕食者对另一个食饵的捕食行为中获益.首先通过计算得到三维食饵-捕食者系统对应的差分模型,其次通过选择合适的参数讨论边界和正平衡点的存在性,进而利用线性稳定性理论讨论平衡点局部渐近稳定的充分条件.将两个食饵种群的出生率以及最大环境容纳量作为分支参数,使用分支理论研究差分模型在平衡点处产生翻转分支、Neimark-Sacker分支、折-翻转分支和1:2共振分支的充分条件.最后通过数值模拟验证了理论分析的正确性.

In this paper, the stability and bifurcation behavior of a three dimensional prey-predator with piecewise constant arguments are researched. Firstly, through calculation the discrete solution of the model is achieved, which has the same dynamical behavior. Next, by selecting suitable parameters, the existence of the boundary and the positive equilibrium points are discussed. Then applying the linearized stability theorem, some sufficient conditions for the local asymptotic stability of equilibrium are achieved.Secondly choosing the the birth rate and the maximum carrying capacity of the environment of the prey populations as the bifurcation parameter, it is shown that the discrete model can undergo Flip bifurcation,Neimark-Sacker bifurcation, Fold-Flip bifurcation, and 1 ： 2 resonance branch through using the bifurcation theory. Finally, the illustration of the analytic results and the complex dynamical behaviors of the model are shown from numerical simulations.