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Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey. (English) Zbl 1391.35221
The authors consider the system of two reaction-diffusion equations $u_t - d_1 \Delta u = u (1 - u) (u/b - 1) - \beta u v, \quad v_t - d_2 \Delta v = \mu v (1 - v / u), \qquad (x,t)\in\Omega\times(0,\infty),$ where $$\Omega\subset\mathbb{R}^N$$ is a smooth and bounded domain, with the homogeneous Neumann boundary conditions on $$\partial\Omega$$. Assuming that $$b\in(0,1)$$ and $$\beta$$, $$\mu$$, $$d_1$$ and $$d_2$$ are positive constants, they look for nonnegative stationary solutions to this problem. First, they describe all nonnegative constant solutions and analyze their stability. Then, they prove several theorems concerned with the existence and non-existence of nonconstant positive solutions for different choices of the system parameters.

##### MSC:
 35K57 Reaction-diffusion equations 35J47 Second-order elliptic systems 35B09 Positive solutions to PDEs 35B36 Pattern formations in context of PDEs
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##### References:
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