A nonlocal eigenvalue problem and the stability of spikes for reaction-diffusion systems with fractional reaction rates

Abstract

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.