From discrete to continuous evolution models: A unifying approach to drift-diffusion and replicator dynamics.
We study the large population limit of the Moran process, under the assumption of weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral evolution) or natural selection; for one precise scaling, both effects are present. For the scalings that take the genetic-drift into account, the continuous model is given by a singular diffusion equation, together with two conservation laws that are already present at the discrete level. For scalings that take into account only natural selection, we obtain a hyperbolic singular equation that embeds the Replicator Dynamics and satisfies only one conservation law. The derivation is made in two steps: a formal one, where the candidate limit model is obtained, and a rigorous one, where convergence of the probability density is proved. Additional results on the fixation probabilities are also presented.
Chalub
Fabio A C C
FA
Departamento de Matemática and Centro de Matemática e Aplicações, Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal. chalub@fct.unl.pt
Souza
Max O
MO
D005075:Biological Evolution; D040961:Genetic Drift; D008962:Models, Theoretical; D013816:Thermodynamics