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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
\[ \begin{aligned} \frac {dN(t)}{dt} & = r_1N(t)-\varepsilon P(t)N(t), \\ \frac{dP(t)}{dt} & = P(t)\left(r_2-\theta \frac {P(t-\tau)}{N(t)}\right), \end{aligned} \] which describes the dynamics of a ratio-dependent predator (\(P\))-prey (\(N\)) system. First, the local stability of the positive equilibrium point \(E^\ast=(\frac {r_1\theta}{r_2\varepsilon}, \frac{r_1}{\varepsilon})\) is studied. \(E^ast\) is stable for \(\tau\in [0,\tau_0)\) and Hopf bifurcation occurs for \(\tau=\tau_k\), where \(\tau_k=\frac {(2k+1)\pi}{r_2+\sqrt{r_2^2+4r_1r_2}}\) for \(k=0,1,\dots\). 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)].

MSC:
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
34K19 Invariant manifolds of functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
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References:
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