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).


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