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Bifurcation and stability analysis for a delayed Leslie-Gower predator-prey system. (English) Zbl 1201.34132
The authors study the following Leslie-Gower predator-prey system with time delay \aligned \dot x(t)&=r_1x(t)\left(1-\frac{x(t-\tau)}{K}\right)-mx(t)y(t),\\ \dot y(t)&=r_2y(t)\left(1-\frac{y(t-\tau)}{\gamma x(t)}\right), \endaligned\tag1 where $x(t)$ and $y(t)$ represent the densities of the prey and the predator populations, respectively. The parameters $r_1, r_2, m, K$ and $\gamma$ are positive constants, where $r_1$ and $r_2$ are the intrinsic growth rates of the prey and the predator, respectively, $K$ is the carrying capacity of the prey, $\gamma x$ stands for a prey-dependent carrying capacity of the predator, $m$ is the capturing rate of the predator. $\tau>0$ is a constant representing the feedback time delay of the prey and the predator species to their own growth. By analysing the characteristic equation, the local stability of the positive equilibrium of system (1) is addressed and the existence of Hopf bifurcations is established. It is shown that system (1) can exhibit an interesting property: under certain conditions, the positive equilibrium may switch a finite number of times between being stable and being unstable, but always becomes unstable eventually. By using the normal form theory and the center manifold theorem, the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are determined. Using the global bifurcation theory for functional differential equations developed by {\it J. Wu} [Trans. Am. Math. Soc. 350, No. 12, 4799--4838 (1998; Zbl 0905.34034)], the global existence of periodic solutions of system (1) is established.

##### MSC:
 34K60 Qualitative investigation and simulation of models 34K18 Bifurcation theory of functional differential equations 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general) 34K17 Transformation and reduction of functional-differential equations and systems; normal forms 34K19 Invariant manifolds (functional-differential equations)
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