A nonlocal dispersal equation arising from a selection–migration model in genetics

This paper is concerned with the existence, uniqueness and asymptotic stability of positive steady-states for a nonlocal dispersal equation arising from selection–migration models in genetics. Due to the lack of compactness and regularity of the nonlocal operators, many classical methods cannot be used directly to the nonlocal dispersal problems. This motivates us to find new techniques. We first establish a criterion on the stability and instability of steady-states. This result is effective to get a necessary condition to guarantee a positive steady-state, it also gives the uniqueness. Then we prove the existence of nontrivial solutions by the corresponding auxiliary equations and maximum principle. Finally, we consider the dynamic behavior of the initial value problem.