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The stability and Hopf bifurcation for a predator-prey system with time delay. (English) Zbl 1152.34059

The author incorporates delay into an ODE considered by S.-R. Zhou et al. [Theor. Popul. Biol. 67, 23–31 (2005; Zbl 1072.92060)] to obtain

dN(t) dt=r 1 N(t)-εP(t)N(t),dP(t) dt=P(t)r 2 -θP(t-τ) N(t),

which describes the dynamics of a ratio-dependent predator (P)-prey (N) system. First, the local stability of the positive equilibrium point E * =(r 1 θ r 2 ε,r 1 ε) is studied. E a st is stable for τ[0,τ 0 ) and Hopf bifurcation occurs for τ=τ k , where τ k =(2k+1)π r 2 +r 2 2 +4r 1 r 2 for k=0,1,. Then the stability and direction of bifurcating periodic solutions is discussed using the normal form theory and center manifold theorem due to [B. D. Hassard and N. D. Kazarinoff, Theory and applications of Hopf bifurcation. Moskva: Mir (1985; Zbl 0662.34001)].

34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
34K19Invariant manifolds (functional-differential equations)
34K17Transformation and reduction of functional-differential equations and systems; normal forms