Rare mutations in evolutionary dynamics

Abstract

In this paper we study the effect of rare mutations, driven by a marked point process, on the evolutionary behavior of a population. We derive a Kolmogorov equation describing the expected values of the different frequencies and prove some rigorous analytical results about their behavior. Finally, in a simple case of two different quasispecies, we are able to prove that the rarity of mutations increases the survival opportunity of the low fitness species.2000 Mathematics Subject Classification: 92D25 (35R09, 60G55)

A rare mutation model in a spatial heterogeneous environment

“…In [2], mutations are described by means of a pure point process that alters replicator dynamics and the Brownian motion term is zero (b = 0). Any mutation has a fixed progenitor (type i) and a unique descendant (type j): this gives 2 d 2 = d(d − 1) different mutations, precisely all those that transform a type i in a type j as…”

“…In the same paper [2], a Kolmogorov integro-differential equation describing the expected frequencies is derived and investigated analytically, with particular attention to the long term equilibrium. Analytical investigation is satisfactory in the case of constant fitness (quasispecies equation), but there are some gaps concerning variable fitness, that has been tackled by a numeric approach in the subsequent paper [3].…”

“…Since then many more refined models have been proposed to put into the right light randomness; we refer for instance to [9] showing that one single stochastic microscopic process can generate different macroscopic models of adaptive evolution. More recently, in [2], it has been proposed a macroscopic stochastic model where mutations occur at a different time scale than selection. This approach goes into the direction of adaptive dynamics, but differentiates from trait substitution sequence because it is not assumed that there is complete adaptation (namely invasion or extinction of the mutant trait) between subsequent mutations.…”

“…In this paper we take a step further and address our attention to the environment, seen as a place where individuals can evolve but also as a factor that can influence the dynamics of interaction between strategists. The model presented in [2] is then expanded to take into account how the natural environment can modify the interactions between individuals, changing selective pressures; we add a new variable y ∈ R N to the variable x, in the simplex, so that the status of the population is described by the pair (x, y). The new variable y stands for the position of the population or, more widely, for an external parameter that affects the results of the interplay between strategies.…”

“…In the following section 2 we recall the stochastic model for replicator dynamics with point-type mutations introduced in [2]. With the aim of performing Monte-Carlo simulations, we give an alternative (but equivalent) description of the process by using a single Poisson random measure.…”