Nonlineor Analysis. Theory. Merhods & Applicarions, Vol. 21, No. 5. pp. 389-395, 1993. 0362-546X/93 $6.00+ .N Printed in Great Britain. 0 1993 Pergamon Press Ltd zyxwvutsrq ON A MODIFIED VOLTERRA POPULATION EQUATION WITH DIFFUSION S. A. GouRLEYt and N. F. BRITTON~ t Department of Mathematics, University of Surrey, Guildford, Surrey GU2 5XH; and $ School of Mathematical Sciences, University of Bath, Bath BA2 7AY, U.K. zyxwvutsrqponmlkjihgfed (Received 2 February 1992; received zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON for publication zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR 1 March 1993) Key words and phrases: Reaction-diffusion equations, delay effects, global solution, asymptotic behaviour. 1. INTRODUCTION IN THIS PAPER we study the delay reaction-diffusion equation u, = u + au2 - bu3 - (1 + a - b)uf* u + Au where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA f * u is the Laplace convolution (1) zyxwvutsrq (f * 24)(x, t) = i ':nf - s)u(x, s) ds. This equation describes the evolution of a single diffusing animal species with population density u. The growth rate reaction to population density changes includes the delay term f * u which involves the entire past history of u, and we will assume throughout that the delay kernel f satisfies i m f E CC& a) fl L’(O, =J), fro and o f(s) d.s = 1 (2) so that, asymptotically, f * u is a weighted average of u. We further assume that the population lives in a bounded domain fi c R” and that there is no migration of individuals across the boundary aa. Thus, we impose the boundary and initial conditions Vu * n = 0 on 82 x (0, ~0) and u(x, 0) = uO(x) for x E fi (3) where n is the outer normal to &2. If b = 0 and -1 < a < 0, (1) reduces to the classical Volterra diffusion equation (see for example Schiaffino [l] and Redlinger [2]). Under our assumptions on f it follows from [2] that if a < -l/2 then solutions corresponding to nonnegative initial data u,, f 0 tend uniformly to 1 as t + 00. However, equations of this form with a > 0 are much less common in the literature and in a recent paper, Britton [3] has studied (1) with b = 0 and with delay terms which are also nonlocal in space. With a > 0 the term au2 represents an advantage to the species in grouping together, in that it adds to the growth rate in regions of high population density. The intraspecific competition term -( 1 + a)uf * u, which represents competition for resources, inhibits population growth. Britton’s paper shows that the interaction between these two terms is important and he examines the travelling waves and other spatially structured solutions which can develop near u = 1 as a is increased. 389