一类具时滞和收获的捕食模型的稳定性与Hopf分支

Stability and Hopf Bifurcation of a Predator-prey Model with Time Delay and Harvesting

本文研究一类具有常数收获率和时滞的捕食模型,其中时滞描述了捕食种群的妊娠期。通过分析特征方程,得到了正平衡点局部稳定的条件。当时滞τ增加时,正平衡点失去稳定性,当τ跨过临界值时系统将出现Hopf分支。应用中心流形定理和规范型理论,得到了确定Hopf分支方向和分支周期解的稳定性的计算公式。最后对所得理论结果进行了数值模拟。

A predator-prey model with harvesting and a time delay describing the gestation period of the predator is considered. We investigate the local stability of a positive equilibrium by analyzing the corresponding characteristic equation. It is proved that when the time delay τ increases, the positive equilibrium looses its stability and a Hopf bifurcation occurs when τ passes through a critical value. Formula are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and the center manifold theorem. Numerical simulations are carried out to illustrate theoretical results.