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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(x) and e(x), respectively, in the steady-state predator-prey model

-d 1 (x)Δu=λa 1 (x)u-b(x)u 2 -c(x)uv,-d 2 (x)Δv=μa 2 (x)c-e(x)v 2 +d(x)uv,u| Ω =v| Ω =0,

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


MSC:
35J60Nonlinear elliptic equations
35J20Second order elliptic equations, variational methods
35J55Systems of elliptic equations, boundary value problems (MSC2000)
35B45A priori estimates for solutions of PDE
35B32Bifurcation (PDE)
47J10Nonlinear spectral theory, nonlinear eigenvalue problems
92D25Population dynamics (general)