S. Harris, State University of New York, Stony Brook, NY

Abstract: A piecewise linear model is used to provide a caricature of the nonlinear equation describing genetic variation due to migration and local natural selection in an inhomogeneous bounded habitat. The conditions for which nontrivial spatially-dependent steady state solutions exist are analytically determined together with these solutions for three distinct scenarios. For the usual case of no flux (Neumann) boundary conditions, explicit solutions require additional information to fix the genetic frequency within the habitat. This difficulty can be bypassed in two of the scenarios considered (this is an artifact of the model), but in the remaining scenario it is necessary to take into account the connection with the initial data to explicitly determine the steady state frequency. Some numerical examples are considered for each of the scenarios to illustrate the analytical results that are the primary focus of the work presented here.

Keywords: clines, genetic variation, model solutions