Inclusive Fitness in Finite Deme-structured and Stepping-stone Populations

Abstract

Very little attention has been given in the literature to the interesting question of how to handle relatedness in finite populations. The main problem is that a finite population is never really ieat equilibriuml. in that it represents just one realization of an infinite assemblage of possible allelic distributions. A recent paper of Rousset and Billiard (manu- script) provides coefficients which, if used in inclusive fitness models under conditions of weak selection, give us a measure of average allele frequency change where the average is taken over all such realizations. Their coefficients are expressed in terms of identity in state, and an alternative formulation (Taylor and Day, manuscript) in terms of coefficients of consanguinity permits the calculation of relatedness in simple cases from pedigree analysis. Here we implement these calculations in a finite asexual haploid population with either a deme structure or a one-dimensional stepping-stone structure and verify our results with numerical simulations in small populations. Our simulations al- low us to investigate the dependence of relatedness on allele frequency, and our results here agree qualitatively with those obtained by Rousset and Billiard. Finally, we examine a model of altruism in a deme-structured population to verify numerically that our relatedness coefficients provide a correct measure of allele frequency change.