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 non negative coefficient functions b(x) and e(x), respectively, in the steady-state predator-prey model -d₁(x)∆u=λa₁(x)u-b(x)u²-c(x)UV, d₂(x)∆v-µa²(x)c-e(x) vˆ²+d(x)UV, u|∂Ω=v|_∂Ω=0, where all other coefficient functions are strictly positive over the bounded domain Ω in RN. 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.