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 nonnegative coefficient functions b (x) and e (x), respectively, in the steady-state predator-prey model -d(1) (x) Deltau = lambdaa(1) (x) u-b (x) u(2)-c (x) uv, -d(2) (x) Deltav = mua(2) (x) c-e (x) v(2) + d (x) uv, u|partial derivativeOmega = v|partial derivativeOmega = 0, where all other coefficient functions are strictly positive over the bounded domain Omega in 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.