Nonlinear group survival in Kimura's model for the evolution of altruism.
Establishing the conditions that guarantee the spreading or the sustenance of altruistic traits in a population is the main goal of intergroup selection models. Of particular interest is the balance of the parameters associated to group size, migration and group survival against the selective advantage of the non-altruistic individuals. Here we use Kimura's diffusion model of intergroup selection to determine those conditions in the case the group survival rate is a nonlinear non-decreasing function of the proportion of altruists in a group. In the case this function is linear, there are two possible steady states which correspond to the non-altruistic and the altruistic phases. At the discontinuous transition line separating these phases there is a non-ergodic coexistence phase. For a continuous concave survival function, we find an ergodic coexistence phase that occupies a finite region of the parameter space in between the altruistic and the non-altruistic phases, and is separated from these phases by continuous transition lines. For a convex survival function, the coexistence phase disappears altogether but a bistable phase appears for which the choice of the initial condition determines whether the evolutionary dynamics leads to the altruistic or the non-altruistic steady state.
Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970 São Carlos SP, Brazil. Electronic address: fontanari@ifsc.usp.br.
José F
JF
Fontanari
Departamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, RN, Brazil.
Maurizio
M
Serva
D000533:Altruism; D025041:Animal Migration; D000818:Animals; D005075:Biological Evolution; D005828:Genetics, Population; D016014:Linear Models; D055641:Mathematical Concepts; D008954:Models, Biological; D008957:Models, Genetic; D017711:Nonlinear Dynamics; D012641:Selection, Genetic
Diffusion approximation; Evolution of altruism; Group selection; Migration; Population genetics