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.


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)
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