J. Math. Biol. (1996) 34:297-333 ,/eurnl of Mathematical 81ology © Springer-Verlag 1996 A predator-prey reaction-diffusion system with nonlocal effects S. A. Gourley 1, N. F. Britton 2 1Department of Mathematical and Computing Sciences, University of Surrey, Guildford, Surrey, GU2 5XH, UK e-mail: s.gourley@uk.ac.surrey.mcs 2 School of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK Received 21 September 1993; received in revised form 13 April 1995 Abstract. Weconsider apredator-prey systemin theform of a coupled system of reaction-diffusion equations with an integral term representing a weighted average of the values of the prey density function, both in past time and space. In a limiting case the system reduces to the Lotka Volterra diffusion system with logistic growth of the prey. We investigate the linear stability of the coexistence steady state and bifurcations occurring from it, and expressions for some of the bifurcating solutions are constructed. None of these bifurca- tions can occur in the degenerate case when the nonlocal term is in fact local. Key words: Predator-prey - Reaction-diffusion - Time delay - Bifurcation - Pattern formation 1 Introduction This paper is devoted to a study of the predator prey system ut = u[1 + ~u- (1 + ~)G**u] - uv + DAu v, = av(u -- b) + Av, (1.1) for (x, t) ~ R" × (0, ~), with G ** u defined by In this system u and v are, respectively, prey and predator population densities and the quantities a, b and D are positive constants. We give a description of the various terms in this model below, but note that a special case of our model (with v - O) is the scalar equation ut = u[1 + ~u--(1 + ~)G**u] + DAu (1.3)