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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

$\begin{array}{cc}\hfill \frac{dN\left(t\right)}{dt}& ={r}_{1}N\left(t\right)-\epsilon P\left(t\right)N\left(t\right),\hfill \\ \hfill \frac{dP\left(t\right)}{dt}& =P\left(t\right)\left({r}_{2}-\theta \frac{P\left(t-\tau \right)}{N\left(t\right)}\right),\hfill \end{array}$

which describes the dynamics of a ratio-dependent predator ($P$)-prey ($N$) system. First, the local stability of the positive equilibrium point ${E}^{*}=\left(\frac{{r}_{1}\theta }{{r}_{2}\epsilon },\frac{{r}_{1}}{\epsilon }\right)$ is studied. ${E}^{a}st$ is stable for $\tau \in \left[0,{\tau }_{0}\right)$ and Hopf bifurcation occurs for $\tau ={\tau }_{k}$, where ${\tau }_{k}=\frac{\left(2k+1\right)\pi }{{r}_{2}+\sqrt{{r}_{2}^{2}+4{r}_{1}{r}_{2}}}$ for $k=0,1,\cdots$. 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 34K18 Bifurcation theory of functional differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general) 34K19 Invariant manifolds (functional-differential equations) 34K17 Transformation and reduction of functional-differential equations and systems; normal forms