Stochastic models for biological evolution

In this work, we deal with the problem of creating a model that describes a population of agents undergoing Darwinian Evolution, which takes into account the basic phenomena of this process. According to the principles of evolutionary biology, Evolution occurs if there is selection and adaptation of phenotypes, mutation of genotypes, presence of physical space. The evolution of a biological population is then described by a system of ordinary stochastic differential equations; the basic model of dynamics represents the trend of a population divided into different types, with relative frequency in a simplex. The law governing this dynamics is called Replicator Dynamics: the growth rate of type k is measured in terms of evolutionary advantage, with its own fitness compared to the average in the population. The replicator dynamics model turns into a stochastic process when we consider random mutations that can transform fractions of individuals into others. The two main forces of Evolution, selection and mutation, act on different layers: the environment acts on the phenotype, selecting the fittest, while the randomness of the mutations affects the genotype. This difference is underlined in the model, where each genotype express a phenotype, and fitness influences emerging traits, not explicitly encoded in genotypes. The presence of a potentially infinite space of available genomes makes sure that variants of individuals with characteristics never seen before can be generated. In conclusion, numerical simulations are provided for some applications of the model, such as a variation of Conway's Game of Life