Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models

Abstract

We present two approaches to study invasion in growth-fragmentation-death models. The first one is based on a stochastic individual based model, which is a piecewise deterministic branching process with a continuum of types, and the second one is based on an integro-differential model. The invasion of the population is described by the survival probability for the former model and by an eigenproblem for the latter one. We study these two notions of invasion fitness, giving different characterizations of the growth of the population, and we make links between these two complementary points of view. In particular we prove that the two approaches lead to the same criterion of possible invasion. Based on Krein-Rutman theory, we also give a proof of the existence of a solution to the eigenproblem, which satisfies the conditions needed for our study of the stochastic model, hence providing a set of assumptions under which both approaches can be carried out. Finally, we motivate our work in the context of adaptive dynamics in a chemostat model.

Eigensolutions and spectral analysis of a model for vertical gene transfer of plasmids

“…(a) Note that if µ is constant, then Assumption (A5) coincides with Assumption 3-4 of [10] and can therefore be interpreted in the same way: the flow Z(t, z) has to move away from 0 or to be more exact out of the interval [0, m] sufficiently fast (cf. [10,Remark 7] and Example 4(b)). See Example 4 for cases in which Assumption (A5) holds.…”

“…by a logistic plasmid reproduction rate. The model we consider is given by a hyperbolic partial integro-differential equation and is similar to the model considered by [10,16,24,9]. However, to the best of our knowledge there are no existence or stability results for this model with non-constant cell death rate and logistic plasmid reproduction rate.…”

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

“…This section details a numerical approach, based on common adaptive dynamics methods as well as mathematical results of Campillo and Fritsch (2015) and Campillo et al (2016a), which allows one to analyze evolutionary dynamics in a relatively complex chemostat model.…”

“…In Section 2, we introduce the models which are considered for the numerical simulations as well as a model reduction approach, based on Campillo and Fritsch (2015) and Campillo et al (2016a). In Section 2.1, we introduce the mass-structured chemostat model where mutations affect the division parameters, namely the minimal mass for division and the mean proportion of the smallest daughter cell.…”

“…We present two definitions of the invasion fitness: the principal eigenvalue of a growth-fragmentation-death operator in the deterministic case, and the survival probability of the mutant population in the stochastic case. We also present the link between the invasion criteria derived from these two definitions, established by Campillo et al (2016a). Section 3 presents the numerical methods that we use for the simulations.…”