The Baldwin effect describes how phenotypic variations such as the learned behaviors among clones (individuals with identical genes, i.e., isogenic individuals) can generally affect the course of evolution. We investigate the effects of phenotypic variation via learning on the evolutionary dynamics of a population. In a population, we assume the individuals to be haploid with two genotypes: one genotype shows phenotypic variation via learning and the other does not. We use an individual-based Moran model in which the evolutionary dynamics is formulated in terms of a master equation and is approximated as the Fokker-Planck equation (FPE) and the coupled nonlinear stochastic differential equations (SDEs) with multiplicative noise. In the infinite population limit, we analyze the deterministic part of the SDEs to obtain the fixed points and determine the stability of each fixed point. We find discrete phase transitions in the population distribution when the set of probabilities of reproducing the fitter and the unfit isogenic individuals is equal to the set of critical values determined by the stability of the fixed points. We also carry out numerical simulations in the form of the Gillespie algorithm and find that the results of the simulations are consistent with the analytic predictions.
YGK was supported by a Hanshin University research grant. JMP was supported by the Catholic University of Korea research fund 2019 and by the Basic Science Research Program through the National Research Foundation of Korea (Grant No. NRF-2018R1A2B6007007).