Abstract
We consider a predator-prey system with Michaelis-Menten type functional response and two delays. We focus on the case with two unequal and non-zero delays present in the model, study the local stability of the equilibria and the existence of Hopf bifurcation, and then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delaysτ1≠τ2.
Highlights
IntroductionIn [1], Xu and Chaplain studied the following delayed predator-prey model with Michaelis-Menten type functional response: dx dt
In [1], Xu and Chaplain studied the following delayed predator-prey model with Michaelis-Menten type functional response: dx1 dt = x1 (t) [a1− a11x1 (t − τ11) −a12x2 (t) m1 + x1 (t) ]dx2 dt x2 (t) [−a2 +a21x1 (t − τ21) m1 + x1 (t − τ21) −a22x2
Suppose the Hopf bifurcation: if μ2 > 0(< 0), the Hopf bifurcation is supercritical and the bifurcation exists for τ > τ1(< τ1); β2 determines the stability of the bifurcation periodic solutions: the bifurcating periodic solutions are stable if β2 < 0(> 0); and T2 determines the period of the bifurcating periodic solutions: the period increases if T2 > 0(< 0)
Summary
In [1], Xu and Chaplain studied the following delayed predator-prey model with Michaelis-Menten type functional response: dx dt. It is well known that periodic solutions can arise through the Hopf bifurcation in delay differential equations. Let τ11 = τ22 = τ33 = 0, τ21 = τ1, τ32 = τ2 in (1); we consider Hopf bifurcation and global periodic solutions of the following system with two unequal and nonzero delays:.
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