Equilibrium Measures of the Stepping Stone Model with Selection in Population Genetics

Let X be a countable set and N be a positive integer. X is a collection of colonies. Consider the process of gene frequencies of an allele A₁, which is subjected to the following changes:(a) mutation occurs from A_l into A₂ and from A₂ into A_l with mutation rates u and v, respectively, (b) for any colonies x and z the genes migrate from z to x with migration rates λ_xz, (c) selection occurs in each colony x, where A_l and A₂ have relative fitness 1 + s_x/2, 1 − s_x/2 respectively, and (d) after having reproduced an infinite numbers of offsprings, N individuals are sampled at random within each colony. The process is called the stepping stone model. The process is a Markov chain with state space describing gene frequencies at colonies. Possible values of gene frequencies at each colony are 0, 1/(2N), ..., (2N)/(2N). We say the process is ergodic if the distribution of the process at the n-th time converges to the unique equilibrium measure independently of the initial distribution as n → ∞. It is shown that if s_x ≧ 0 for all x in X or s_x ≦ 0 for all x in X, and u + v ≦ 1 and u, v > 0, the process is ergodic, but assuming u > 0 and v = 0 and X = Zd the d-dimensional lattice, then there exists a constant s_l < 2 such that the process is ergodic if sup_x∈X s_x < s₁ and is not ergodic if inf_x∈X s_x > s₁. The latter fact shows that under the condition u > 0 = v, the process has a nontrivial equilibrium measure, if s_x is large enough. Assume X = Z1 and consider the case that sign of s_x is different between x ≧ 0 and x < 0. Some monotonicity properties concerning the equilibrium measure and related estimates are obtained under the condition that the process is ergodic.