Effects of a degeneracy in the competition model - part I. Classical and generalized steady-state solutions

We study the competition model {u(t)(x, t)-d(1)(x) Deltau(x, t) = lambdaa(1)(x) u-b(x) u(2)-c(x) uv, {v(t)(x,t)-d(2)(x) Deltav(x,t) = mua(2)(x) v-e(x) v(2)-d(x) uv, where the coefficient functions are strictly positive over the underlying spatial region Omega except b(x), which vanishes in a nontrivial subdomain of Omega, and is positive in the rest of Q. We show that there exists a critical number lambda* such that if lambda < lambda*, then the model behaves similarly to the well-studied classical competition model where all the coefficient functions are positive constants, but when lambda > lambda*, new phenomena occur. Our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well. In part I here, we mainly study two kinds of steady-state solutions which determine the dynamics of the model: one consists of finite functions while the other consists of generalized functions which satisfy (u, v) = (infinity, 0) on the part of the domain that b(x) vanishes, but are positive and finite on the rest of the domain, and are determined by certain boundary blow-up systems. The research is continued in part 11, where these two kinds of steady-state solutions will be used to determine the dynamics of the model. (C) 2002 Elsevier Science (USA).