Stability and Instability of the Unique Positive Steady State for a Nonlocal Population Equation

Abstract: This paper is concerned with the effect of spatial heterogeneity on the linear stability of unique positive steady states for a nonlocal population equation. The main difficulty is that, because of the nonlocal term, the linearized operator need not be self-adjoint or satisfy the maximum principle, so the usual stability arguments based on the principal eigenvalue do not apply directly. We establish linear stability for sufficiently large diffusion and, under suitable conditions on the maximum set of the local growth ratio, for sufficiently small diffusion. In one space dimension, we also construct unstable positive steady states and obtain Hopf bifurcations for sufficiently small diffusion. Keeping the local coefficients fixed, we obtain either a supercritical or a subcritical Hopf bifurcation by changing the weight in the nonlocal integral. Our results show that uniqueness of a positive steady state does not imply its linear stability.