Emergence of homogamy in a two-loci stochastic population model

Abstract

Birth and death processes with interactions, multitype branching processes, large population limits, mating preferences. The authors thank the CNRS for its financial support through its competitive funding programs on interdisciplinary research. This work was partially funded by the Chair "Modélisation Mathématique et Biodiversité" of VEOLIA-Ecole Polytechnique-MNHN-F.X. H.L. acknowledges support from CONACyT-MEXICO and the foundation Sofía Kovalevskaia of SMM. The authors are grateful to András Tóbiás and an anonymous referee for their careful reading of the paper and useful comments.

Invasion and fixation of microbial dormancy traits under competitive pressure

“…Note that for any four events A 1 , A 2 , A 3 , A 4 , one has (as observed e.g. in [CCLLS19]) P(A 1 ∩ A 2 ∩ A 3 ∩ A 4 ) ≥ P(A 3 ∩ A 4 ) − P(A c 1 ∩ A 4 ) − P(A c 2 ∩ A 4 ). Applying this to the previous chain of inequalities, we obtain…”

“…2.2, and 2.3 will be carried out in Section 4. In multiple parts of the proof, we are able to employ arguments that are similar to the ones used in [CCLLS19,CCLS17] for the three phases of invasion in individual-based models in the context of emergence of homogamy respectively speciation. A particular additional difficulty of our setting lies in guaranteeing convergence of the underlying dynamical system (2.6) to its stable equilibrium (0, xa , xd ), in other words, in verifying certain global attractor properties of this equilibrium.…”

“…With high probability, a successful mutation follows the three classical phases exhibited in basic adaptive dynamics models (see e.g. [B19, Section 4.1] for a slightly more general picture, but in particular [CCLS17,CCLLS19] for work in a closely related context that inspired our analysis and provides many of the necessary tools): (1) mutant growth until reaching a population size comparable to the carrying capacity, while during the same time period the resident population stays close to its equilibrium size, (2) a phase where all sub-populations are large and the dynamics of the frequency process can be approximated by a deterministic dynamical system, (3) extinction of the resident population, while the mutant population remains close to its equilibrium size.…”

“…Further, we explicitly determine the limiting probability of invasion and the asymptotic time to fixation of mutants in the case of a successful invasion. In the proofs, we observe the three classical phases of invasion dynamics in the guise of Coron et al (2017Coron et al ( , 2019.MSC 2010. 60J85, 92D25.…”

The interplay of dormancy and transfer in bacterial populations: Invasion, fixation and coexistence regimes

“…During the analysis of the invasion (and, if applicable, fixation) of trait 1 in a trait 2 population, we are able to use multiple arguments of our recent paper [BT20] where we investigated competitioninduced dormancy without HGT. The analysis of phases (1) and (if applicable) (3) is a relatively straightforward adaptation of the methods of [BT20], which rely on a number of proof techniques from the papers [CCLS17,CCLLS19]. Here, the main task is to take into account the additional effect of HGT and the arising coexistence equilibrium, handling also cases that were not included in [BT20] (eg., the case where the probability of mutant extinction tends to 1, the one where there is stable coexistence, and in general the one where a mutant lacking dormancy tries to invade a resident population exhibiting dormancy and not the other way around).…”