一个带有Holling-IV功能性反应的捕食与被捕食模型的稳定性和分支

Stability and Hopf Bifurcation of a Predator-prey System with Holling type IV Functional Response

研究了一个带Holling-IV型功能反应的捕食与被捕食模型,讨论了系统解的有界性和各平衡点的存在性,使用Routh-Hurwitz定理得到了平衡点局部渐近稳定的充分条件.引入两个离散时滞,得出了重要的结果:边界平衡点的稳定性随着τ1的增加,由稳定变为不稳定,并且会发生Hopf分支.对正平衡点的稳定性变化,考虑了两个时滞相等的情况,结果是随着分支参数的增加,不仅稳定性会发生变化,产生Hopf分支,甚至可能出现小范围周期解.

We analyze and formulate a predator-prey model of Holling type IV functional response, study the boundedness of solutions and the existence of the equilibrias, and obtain the sufficient conditions of locally asymptotic stability of the equilibrias by the Routh-Hurwitz criterion. We incorporate two discrete delays into the system. The important observation is ： as the delay r1 is increased the originally asymptotic stable boundary equilibrium point loses its stability and a Hopf bifurcation takes place. As for the stability switch of the positive equilibrium point, we study the model with two equal delays. The observation is that as the delay is increased the originally asymptotic stable equilibrium E. loses its stability. Furthermore at a certain critical value a Hopf bifurcation takes place： small amplitude periodic solutions arise.