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 \(b(x)\) and \(e(x)\), respectively, in the steady-state predator-prey model
\[
\begin{matrix} -d_1(x)\Delta u=\lambda a_1(x)u-b(x)u^2-c(x)uv,\\ -d_2(x)\Delta v=\mu a_2(x)c-e(x) v^2+d(x)uv, \end{matrix} \quad u| _{\partial \Omega}=v| _{\partial \Omega}=0,
\]
where all other coefficient functions are strictly positive over the bounded domain \(\Omega\) in \(\mathbb R^{N}\). 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 | Variational methods for second-order elliptic equations |
35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |
35B45 | A priori estimates in context of PDEs |
35B32 | Bifurcations in context of PDEs |
47J10 | Nonlinear spectral theory, nonlinear eigenvalue problems |
92D25 | Population dynamics (general) |