A numerical approach to determine mutant invasion fitness and evolutionary singular strategies

Abstract

We propose a numerical approach to study the invasion fitness of a mutant and to determine evolutionary singular strategies in evolutionary structured models in which the competitive exclusion principle holds. Our approach is based on a dual representation, which consists of the modeling of the small size mutant population by a stochastic model and the computation of its corresponding deterministic model. The use of the deterministic model greatly facilitates the numerical determination of the feasibility of invasion as well as the convergence-stability of the evolutionary singular strategy. Our approach combines standard adaptive dynamics with the link between the mutant survival criterion in the stochastic model and the sign of the eigenvalue in the corresponding deterministic model. We present our method in the context of a mass-structured individual-based chemostat model. We exploit a previously derived mathematical relationship between stochastic and deterministic representations of the mutant population in the chemostat model to derive a general numerical method for analyzing the invasion fitness in the stochastic models. Our method can be applied to the broad class of evolutionary models for which a link between the stochastic and deterministic invasion fitnesses can be established.

Stability of two competing populations in chemostat where one of the population changes its average mass of division in response to changes of its population

“…We believe that this as well as the argument in the previous paragraph make the case in favour of extending classical problems through the used of more complex mathematical tools as a lot of the phenomena observed in real life can only be captured by accounting for the underlying complexity and multi-scale interactions. Including adaptation mechanisms and changes in the division of bacterial cells has been done in a very similar setting exploring the fitness of mutant bacterial populations in a chemostat [12].…”

Stability of the chemostat system with a mutation factor

“…It turns out that mutation will modify the behavior of the system leading to coexistence. There exist various approaches to model this phenomenon: each species may convert into other species with a mutation rate depending on various parameters such as the kinetics (see, e.g., [29] or [21]). Throughout this paper, we shall assume that the dispersion is such that each species i converts into neighbor species i + 1 and i − 1 with a constant mutation rate.…”

On the variations of the principal eigenvalue with respect to a parameter in growth-fragmentation models

“…This parameter can, for example, represent an external resource or the influence of other populations supposed to be at equilibrium. The study of the influence of this parameter on the growth of the population is a question of biological interest for a better understanding of the model, but also of numerical interest, for example, for the study of mutant invasions in adaptive dynamics problems (Fritsch et al, 2016).…”

“…However, the stochastic invasion fitness is numerically less straightforward to compute than the deterministic one. The mutual variations of both invasion fitnesses established in this article allow to considerably simplify the numerical analysis of a mutant invasion since the problem is reduced to the computation of a single eigenvalue in order to characterize the possibility of invasion of the mutant population (Fritsch et al, 2016).…”