Effects of Certain Degeneracies in the Predator-Prey Model

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(x) and e(x), respectively, in the steady-state predator-prey model {l} -d_1(x)\Delta u=\lambda a_1(x)u-b(x)u^2-c(x)uv,\\ -d_2(x)\Delta v=\mu a_2(x)c-e(x) v^2+d(x)uv, \end{array} \quad u|_{\partial \Omega}=v|_{\partial \Omega}=0, where all other coefficient functions are strictly positive over the bounded domain $\Omega$ in RN. 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.