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On diffusive population models with toxicants and time delays. (English) Zbl 0927.35049

The authors study two generalized reaction-diffusion systems, as logistic systems in mathematical ecology. The first of these is the following:

u(t,x)/t-Au(t,x)=r(x)u(t,x)(K(x)-u(t,x))/(K(x)+c(x)u(t,x))in[0,)×Ω,
B[u](t,x)=0on(0,)×Ω,u(0,x)=u 0 (x)onΩ ¯,

where Ω is a bounded domain in n with smooth boundary Ω; the functions r, c and K are positive in Ω and Hölder continuous on Ω ¯; B[u]=u or B[u]=u/ν+γ(x)u, with γC 1+α (Ω) and γ(x)0 on Ω; the initial function u 0 is a Hölder continuous function on Ω ¯, and the differential operator A is a uniformly strongly elliptic operator.

The main goal of this paper is to show the existence of a unique positive steady-state solution in these models and investigate the asymptotic behavior of the time-dependent solutions, in both models, in relation to such steady solutions.

MSC:
35K60Nonlinear initial value problems for linear parabolic equations
35K57Reaction-diffusion equations
35B40Asymptotic behavior of solutions of PDE
92D40Ecology