Global convergence in a reaction-diffusion equation with piecewise constant argument

In this paper, we consider the reaction-diffusion equation with piecewise constant argument ∂u/∂t = r u(x, t) (1 - u(x,t)) - Eu(x, [t])u(x, t) + D∇²u on a finite domain, with r, E, D > 0. By employing the method of sub- and super-solutions we prove that, under the condition E < r(1 - exp(-r)), all solutions with positive initial data converge to the positive uniform state. © 2001 Elsevier Science Ltd.