一类具有两个时滞的捕食模型的稳定性与Hopf分支

ON THE STABILITY AND HOPF BIFURCATION OF A PREDATOR-PREY MODEL WITH TWO DELAYS

研究了一类具有两个时滞和Holling-Ⅲ型功能性反应的捕食系统.首先通过对特征方程的分析得到系统平衡点的局部稳定的充分条件以及在其周围出现Hopf分支的条件,其次,在以τ_1=τ_2=τ作为分支参数,利用规范型方法和中心流形定理,得到确定周期解的分支方向,分支周期解的稳定性等显式算法.最后通过数值模拟验证了所得结论的正确性.

In this paper, a predator-prey system with Holling type-Ⅲ functional response and two delays is investigated. By analyzing the characteristic equations, the local stability of each of feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. The delay τ- is used as the bifurcation parameter, we show that Hopf bifurcations can occur as τ crosses some critical values. Applying the normal form theory and center manifold argument, we obtain explicit formulas to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Finally, some numerical simulations are carried out to illustrate the main theoretical results.