Non-local effects in an integro-PDE model from population genetics

In this paper, we study the following non-local problem: This model, proposed by T. Nagylaki, describes the evolution of two alleles under the joint action of selection, migration, and partial panmixia - a non-local term, for the complete dominance case, where g(x) is assumed to change sign at least once to reflect the diversity of the environment. First, properties for general non-local problems are studied. Then, existence of non-trivial steady states, in terms of the diffusion coefficient d and the partial panmixia rate b, is obtained under different signs of the integral ∫Ω g(x)dx. Furthermore, stability and instability properties for non-trivial steady states, as well as the trivial steady states u ≡ 0 and u ≡ 1 are investigated. Our results illustrate how the non-local term - namely, the partial panmixia - helps the migration in this model.