带有时滞的合作系统稳定性和Hopf分支分析(英文)

Stability and Hopf Bifurcation Analysis in a Mutualistic System with Delay

本文研究了带有时滞的两个物种的合作系统{x(t)=r1x(t)[1-a1x(t-τ)+a2y(t)]y(t)=r2y(t)[1+a3x(t)-a4y(t)]的稳定性和分支分析,通过分析特征根的分布得出系统在正平衡点(x*,y*),当τ=τ～时存在Hopf分支,进一步应用规范型和中心流形的方法给出了计算分支周期解稳定性和方向的计算公式,最后通过数值模拟验证了理论结果的正确性。

In this paper, a two-species Lotka-Volterra mutualistie system with delay （HL（1：1,ZAKx·（t）=r1x（t）[1-a1x（t-τ）＋a2y（t）]Ky·（t）=r2y（t）[1＋a3x（t）-a4y（t）]is considered. The existence of Hopf bifurcations at the positive equilibrium（x. ,y. ） when r=r, is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived, by using the normal form theory and center manifold argu- ment. Finally, some numerical simulations are carried out for supporting the analytic results.