A fixed interface boundary value problem for differential equations: a problem arising from population genetics

Diffusion type fixed interface conditions are formulated to describe the transport of diffusing materials across porous thin barriers embedded in media supporting the diffusion process. We consider a Neumann boundary value problem with fixed interface conditions for general diffusion-reaction differential equations that models the gene dispersal in a population under natural selection in a finite habitat with embedded narrow barriers. We establish for the problem a new comparison principle, the global existence of solutions, and sufficient conditions of stability and instability of equilibria. We show that the stability of equilibrium changes as the barrier permeability changes through a critical value. Also, the nonconstant stable equilibria for the problem can arise due to the interaction of the selection force and the barrier. Results in this work are applicable to general situations of materials or heat diffusing through permeable barriers.

Keywords

diffusion, fixed interface, gene, barrier, permeability

2010 Mathematics Subject Classification

35B35, 35B40, 35Dxx, 35K20, 92D25

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Published 1 January 2006