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Stability and Hopf bifurcations in a delayed Leslie-Gower predator-prey system. (English) Zbl 1170.34051

The delayed Leslie - Gower (LG) predator - prey system

${x}^{\text{'}}\left(t\right)={r}_{1}x\left(t\right)\left(1-\frac{x\left(t-\tau \right)}{K}\right)-mx\left(t\right)y\left(t\right),$
${y}^{\text{'}}\left(t\right)={r}_{2}y\left(t\right)\left(1-\frac{y\left(t\right)}{\gamma x\left(t\right)}\right)$

is studied. The delay $\tau$ is considered as the bifurcation parameter and the characteristic equation of the linearized system of the original system at the positive equilibrium is analysed. It is shown that Hopf bifurcations can occur as the delay crosses some critical values. The main contribution of this paper is that the linear stability of the system is investigated and Hopf bifurcations are demonstrated. Conditions ensuring the existence of global Hopf bifurcation are given, i.e., when ${r}_{1}>2mK\gamma ,$ LG system has at least $j$ periodic solutions for $\tau >{\tau }_{j}^{+}\left(j\ge 1\right)·$ The formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. The numerical simulations are also included. Basing on the global Hopf bifurcation result by J. Wu [Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034)] for functional differential equations, the authors demonstrate the global existence of periodic solutions.

##### MSC:
 34K18 Bifurcation theory of functional differential equations 34K60 Qualitative investigation and simulation of models 92D25 Population dynamics (general) 34K13 Periodic solutions of functional differential equations 34K20 Stability theory of functional-differential equations