A Nonlocal Eigenvalue Problem and the Stability of Spikes for Reaction-diffusion Systems with fractional Reaction Rates

We consider a nonlocal eigenvalue problem which arises in the study of stability of spike solutions for reaction-diffusion systems with fractional reaction rates such as the Sel’kov model, the Gray-Scott system, the hypercycle Eigen and Schuster, angiogenesis, and the generalized Gierer-Meinhardt system. We give some sufficient and explicit conditions for stability by studying the corresponding nonlocal eigenvalue problem in a new range of parameters. 1. Motivation: The Sel’kov Model We consider a nonlocal eigenvalue problem which arises in the study of spike solutions for reaction-diffusion systems in many areas of applied science. We begin by considering the so-called Sel’kov model [Sel’kov, 1968] which was derived by Sel’kov to describe the enzyme reaction of glycolysis. Starting from a simple kinetic scheme with substrate inhibition and product activation, Sel’kov derived this reaction-diffusion system according to the law of mass action and the law of mass conservation. The model and some modifications of it have also been used in the study of morphogenesis and population dynamics (see [Hunding & Sorensen, 1988] or [Murray, 1989], respectively). In its simplified and 1991 Mathematics Subject Classification. Primary 35B40, 35B45; Secondary 35J40.