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Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment. (English) Zbl 1180.35290
The authors deal with the following elliptic system: $\begin{cases} -\Delta u=\lambda u-a(x)u^2-\frac{buv}{1+\gamma(x)u}, & x\in\Omega\\ -\Delta v=\mu v\left(1-\frac vu\right), & x\in\Omega\\ \frac{\partial u} {\partial\nu}=\frac{\partial v}{\partial\nu}=0, & x\in\partial\Omega \end{cases}\tag{1}$ where $$\Omega\subset\mathbb{R}^n$$ is a bounded domain with smooth boundary $$\partial\Omega$$, the parameters $$\lambda,\mu$$, and $$b$$ are positive constants and $$a(x)$$ and $$\gamma(x)$$ are non-negative continuous functions on $$\overline\Omega$$. The main purpose of this paper is to study the effect of the degeneracies of $$a(x)$$ and $$(\gamma(x)$$ on the steady-state behaviour in heterogeneous environment, that is on positive solutions of (1).

##### MSC:
 35K57 Reaction-diffusion equations 92D25 Population dynamics (general)
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