JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 123, 30&323 (1987) Existence of Steady-State Solutions to Predator-Prey Equations in a Heterogeneous Environment CLAYTON KELLER* Department of Mathematics, Holy Cross College, Worcester, Massachusetts 01610 AND ROGER LUI Department qf Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 Submitted by E. Stanley Lee Received July 17, 1985 1. INTRODUCTION In recent years, there has been much interest in reaction-diffusion systems, partly because they arise frequently in modelling chemical and biological processes, and also because the theory itself is rich and only beginning to be explored. Our interest began with work [4] of Conway, Hoff, and Smaller, in which the parabolic system ;= Ddu + i A,(“~,u) 2 +.l‘(u) ,= I ‘, / (1.1) is considered. Here (x, t) E 52 x R +, where 52 is a bounded domain in KY’, u = (u,, u* )..., urn) is an m-tuple of functions, D is a constant positive definite matrix, and the A, are continuous matrix-valued functions. In [4] it is shown that if this system admits a bounded invariant region in R” and a certain quantity 0 is positive, then the solution of the initial-boundary value problem, with initial condition u(x, 0) = u,,(x) and homogeneous Neumann boundary condition, will converge to its own spatial average exponentially in L* or L" as t + co. This result implies that when the dif- fusion is sufficiently strong, any spatially nonconstant steady-state solution of the parabolic system above is unstable. * Present address: CSP, Inc., 40 Lime11 Circle, Billerica, MA 01821. 306 0022-247X/87 $3.00 Copyright c) 1987 by Academic Press, Inc. All righls of reproduction m any form reserved