A stochastic model for speciation by mating preferences

Invasion and fixation of microbial dormancy traits under competitive pressure

“…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.…”