Dancer, E. N.; Du, Yihong Effects of certain degeneracies in the predator-prey model. (English) Zbl 1055.35046 SIAM J. Math. Anal. 34, No. 2, 292-314 (2002). 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. Cited in 1 ReviewCited in 59 Documents 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) Keywords:global bifurcation; a priori estimates; predator-prey model PDFBibTeX XMLCite \textit{E. N. Dancer} and \textit{Y. Du}, SIAM J. Math. Anal. 34, No. 2, 292--314 (2002; Zbl 1055.35046) Full Text: DOI