| Abstract: | Population genetics is a scientific discipline that has extensively benefitted from mathematical modelling; since the Hardy‐Weinberg law (1908) to date, many mathematical models have been designed to describe the genotype frequencies evolution in a population. Existing models differ in adopted hypothesis on evolutionary forces (such as, for example, mutation, selection, and migration) acting in the population. Mathematical analysis of population genetics models help to understand if the genetic population admits an equilibrium, ie, genotype frequencies that will not change over time. Nevertheless, the existence of an equilibrium is only an aspect of a more complex issue concerning the conditions that would allow or prevent populations to reach the equilibrium. This latter matter, much more complex, has been only partially investigated in population genetics studies. We here propose a new mathematical model to analyse the genotype frequencies distribution in a population over time and under two major evolutionary forces, namely, mutation and selection; the model allows for both infinite and finite populations. In this paper, we present our model and we analyse its convergence properties to the equilibrium genotype frequency; we also derive conditions allowing convergence. Moreover, we show that our model is a generalisation of the Hardy‐Weinberg law and of subsequent models that allow for selection or mutation. Some examples of applications are reported at the end of the paper, and the code that simulates our model is available online at https://www.ding.unisannio.it/persone/docenti/del-vecchio for free use and testing. |
