On the Bifurcation of Radially Symmetric Steady-State Solutions Arising in Population Genetics

Abstract

This paper considers a semilinear elliptic equation which arises in a selection-migration model in population genetics, involving two alleles $A_1$ and $A_2$ such that $A_1$ is at an advantage over $A_2$ in certain subregions and at a disadvantage in others. The system is studied on all of $R^n$ and is assumed to possess radial symmetry. Existence and asymptotic properties of solutions of the corresponding ordinary differential equation are investigated and, by using shooting method type arguments, results are obtained on the bifurcation of solutions from the trivial solutions corresponding to the cases where $A_1$ or $A_2$ is extinct. The nature of the results obtained varies according to whether $A_1$ or $A_2$ has an overall advantage.