Existence and uniqueness of solutions for a diffusion model of intergroup selection

Abstract

In this memoir, we consider an initial boundary value problem proposed by M. Kimura [1] as a diffusion model of intergroup selection in population genetics. The purpose of the present paper is to prove the existence of solutions and, under a stronger assumption, the uniqueness of the solution.

The unknown function $U=U(t, x)$ is the distribution function of a random variable $x$ running over the interval [0,1]. $x$ is the frequency of an allele of “altruistic” character and $t$ is the time variable representing the generation. Themain equation for $U$ is a partial differential equation of parabolic type degenerated at $x=0$ and $x=1$. This equation is non-linear but the non-linearity is not so strong that we can treat it as a linear one.

To prove the uniqueness of the solutions, we show the continuity of the dependence of solutions on the initial values, not with respect to the strong topology in $L^{1}$-space but with respect to the topology in the space of the moment sequences of solutions. In the proof of existence, we discuss the approximative solutions also in the latter sense of convergence. Therefore, the use of moment sequences is essential in our reasoning. The main theorem of the present paper will be stated in §3.