A POPULATION DYNAMICS MODEL WITH NONAUTONOMOUS PAST

摘要

In this paper we study a two-phase population model, which distinguishes the population by two different stages {u'(t) = -du(t) + F(u(t)) + integral(r)(0) b(a)v(t, a)da + integral(r)(0) b(a) partial derivative/partial derivative a v(t, a)da,t >= 0,v'(t, a) = -partial derivative/partial derivative a v(t, a) - dv(t, a) - b(a)v(t, a), t >= 0,0 <= a <= r,v(t, 0) = beta u(t), t >= 0,u(0) = u(0), v(0, a)= v(0)(a), 0 <= a <= r.By the standard technique of characteristics, this population equation is transformed as the ordinary differential equation with nonautonomous past{x'(t) = Bx(t) + Phi((x) over tilde (t)), t >= 0,x(0) = x(0) is an element of E, x(0) = f is an element of L-p(I, E),where 1 <= p < infinity and I = [-r, 0] (finite delay) or I = (-infinity, 0] (infinite delay), E a Banach space, Phi : W-1,W-p(I, E) -> E a linear delay operator and B a nonlinear operator on E. The main result of this paper is the well-posedness of this delay equation by using the (right) multiplicative perturbation result of Desch and Schappacher in [8].