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Effects of certain degeneracies in the predator-prey model. (English) Zbl 1055.35046

Summary: To demonstrate the influence of spatial heterogeneity on the predator-prey model, we study the effects of the partial vanishing of the nonnegative coefficient functions $b\left(x\right)$ and $e\left(x\right)$, respectively, in the steady-state predator-prey model

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

where all other coefficient functions are strictly positive over the bounded domain ${\Omega }$ in ${ℝ}^{N}$. Critical values of the parameter $\lambda$ are obtained to show that, in each case, the vanishing has little effect on the behavior of the model when $\lambda$ is below the critical value, while essential changes occur once $\lambda$ is beyond the critical value.

##### MSC:
 35J60 Nonlinear elliptic equations 35J20 Second order elliptic equations, variational methods 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35B45 A priori estimates for solutions of PDE 35B32 Bifurcation (PDE) 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems 92D25 Population dynamics (general)