Comparison and Stability Theorems for Reaction-Diffusion Systems

This paper extends a comparison technique of Conway and Smoller [Comm. Part. Duff. Egns., 2 (1977) pp. 679–697] for systems of n reaction-diffusion equations. By altering the definition of the comparison system we obtain 2ⁿ (rather than two) spatially homogeneous comparison vectors. The existence of additional comparison vectors is useful in obtaining a more precise description of the asymptotic behaviorof solutions. In particular, we study a few examples in which the above extension enables us to give a description of (1), the domains of attraction of rest points of a system arising in mathematical ecology, and (2), a threshold effect for a system arising in chemical reactor theory. The second part of this paper relates the (diffusion-independent) domains of attraction (R) of constant rest states (P) which are obtained via the above comparison technique, to the diffusion-dependent stability results obtainable by energy estimates, for the Neumann problem on a bounded domain. In particular, suppose that λ is the measure of the set of values of x for which the initial data lie outside R, and that d is the minimum diffusion rate; if the space average of the initial data lies in R, and if (roughly) λ - 4 < K, where K is a positive constant, then the solution of the reaction-diffusion system must tend uniformly to P as t approaches infinity. Applications to mathematical ecology and mathematical neurophysiology are discussed.