On the non-existence of an optimal migration rate

We show that an optimal migration rate may not exist in a population distributed over an infinite number of individual living sites if empty sites occur. This is the case when the mean number of offspring per individual mu is finite. We make the assumption of uniform migration to other sites whose rate is determined by the parent's genotype or the offspring's genotype at a single locus in a diploid hermaphrodite population undergoing random mating. In both cases, for mu small enough, any population at fixation would go to extinction. Moreover, in the latter case, for intermediate values of mu, the only fixation state that could resist the invasion of any mutant would lead the population to extinction. These are the two conditions for the non-existence of an optimal migration rate. They become less stringent as the cost for migration expressed by a coefficient of selection 1-beta becomes larger, that is closer to 1. The results are obtained assuming that the allele at fixation is either nondominant or dominant. Although the optimal migration rate is the same in both cases when it exists, the optimality properties may differ.