Diffusion model of population genetics incorporating group selection, with special reference to an altruistic trait

In order to investigate under what conditions an altruistic trait evolves through group selection, the following diffusion model is formulated. Consider a species consisting of an infinite number of competing groups (demes) each having a constant number of reproducing members and in which mating is at random. Then consider a gene locus and assume a pair of alleles A and A', where A' is the "altruistic allele." Let x be the relative frequency of A' within a deme, and let φ = φ(x; t) be the density function of x at time t such that φ(x; t)Δx represents the fraction of demes whose frequency of A' lies in the range (x, x + Δx). Then, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\partial \phi }}{{\partial t}} = \frac{1}{2}\frac{{\partial ^2 }}{{\partial x^2 }}\{ V_{\delta x} \phi \} - \frac{\partial }{{\partial x}}\{ M_{\delta x} \phi \} + c(x - \bar x)\phi ,$$\end{document} where Mδx and Vδx stand for the mean and variance of the change in x per generation (due to mutation, migration, individual selection and random sampling of gametes) within demes, and c is a positive constant (called the coefficient of interdeme competition) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar x$$\end{document} is the mean of x over the species, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar x = \int_0^1 {x\phi } dx$$\end{document}. By studying the above diffusion equation at steady state (∂φ/∂t = 0), a condition is obtained for group selection to prevail over individual selection in the evolution of an altruistic trait.