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Effects of a degeneracy in the competition model. I: Classical and generalized steady-state solutions. (English) Zbl 1042.35016

This paper is devoted to how heterogeneous environmental effects can greatly change the dynamical behaviour of a population model. The author considers the competition model given by

$\left\{\begin{array}{c}\frac{\partial u}{\partial t}-{d}_{1}\left(x\right){\Delta }u=\lambda {a}_{1}\left(x\right)u-b\left(x\right){u}^{2}-c\left(x\right)uv\hfill \\ \frac{\partial v}{\partial t}-{d}_{2}\left(x\right){\Delta }v=\mu {a}_{2}\left(x\right)v-e\left(x\right){v}^{2}-d\left(x\right)uv\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $x\in {\Omega }$ and $t\ge 0$, ${\Omega }$ denotes a smooth bounded domain in ${ℝ}^{N}$ $\left(N\ge 2\right)$, ${d}_{1},{d}_{2},{a}_{1},{a}_{2},b,c,d,e$ are nonnegative functions over ${\Omega }$, and $\lambda ,\mu$ are positive constants. Throughout this paper the author supposes that $u$ and $v$ satisfy homogeneous Dirichlet boundary conditions. The author shows that there exists a critical number ${\lambda }_{*}$ such that (1) behaves as if $b\left(x\right)$ is a positive constant if $\lambda <{\lambda }_{*}$, but if $\lambda >{\lambda }_{*}$, then interesting new phenomena occur. This shows that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well.

MSC:
 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35B40 Asymptotic behavior of solutions of PDE 35K57 Reaction-diffusion equations 92D25 Population dynamics (general)