Nonlinear group survival in Kimura's model for the evolution of altruism

TL;DR

This paper analyzes how nonlinear group survival functions influence the evolution of altruism using Kimura's diffusion model, revealing different phase behaviors including coexistence, bistability, and phase transitions.

Contribution

It extends Kimura's model by incorporating nonlinear group survival functions, characterizing the resulting phase diagrams and transition types in altruistic evolution.

Findings

Linear survival functions lead to two steady states: altruistic and non-altruistic.

Concave survival functions produce an ergodic coexistence phase.

Convex survival functions eliminate coexistence but introduce bistability.

Abstract

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 probability 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.