From adaptive dynamics to adaptive walks

Abstract

We consider an asexually reproducing population on a finite trait space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event. On the individual-based level this population can be modeled as a measure-valued Markov process. Multiple variations of this system have been studied in the limit of large populations and rare mutations, where the regime is chosen such that mutations are separated. We consider the deterministic system, resulting from the large population limit, and let the mutation probability tend to zero. This corresponds to a much higher frequency of mutations, where multiple subcritical traits are present at the same time. The limiting process is a deterministic adaptive walk that jumps between different equilibria of coexisting traits. The graph structure on the trait space, determined by the possibilities to mutate, plays an important role in defining the adaptive walk. In a variation of the above model, where the radius in which mutants can be spread is limited, we study the possibility of crossing valleys in the fitness landscape and derive different kinds of limiting walks.2010 Mathematics Subject Classification. 37N25, 60J27, 92D15, 92D25.

Punctuated equilibrium as the default mode of evolution of large populations on fitness landscapes dominated by saddle points in the weak-mutation limit

“…Results similar to ours have been reported in the mathematical biology literature (26)(27)(28). Specifically, it…”

Stochastic individual-based models with power law mutation rate on a general finite trait space

“…We show that the limiting process is deterministic, given by piecewise affine continuous functions, which are determined by an algorithm describing the changes in the fitness landscape due to the succession of new resident or emergent types. This work constitutes an extension of the paper by Kraut and Bovier [25] to the stochastic setting. They consider the deterministic system resulting from the large population limit of the individual-based model (K → ∞), and let the mutation probability µ tend to zero.…”

“…Moreover, comparisons of these models with actual empirical fitness landscapes have been performed in [41]. As Kraut and Bovier showed [25], adaptive walks and flights arise as the limit of individualbased models of adaptive dynamics, when the large population followed by the rare mutations limit is taken. They also conjecture, and this will be proved in the present article, that similar results hold in the stochastic setting under the mutation rate (1.2), as we detail below.…”

“…Using techniques developed in [8], we show that these are piecewise affine continuous functions, whose slopes are given by an algorithm describing the changes in the fitness landscape due to the succession of new resident or emergent types. This work generalises [25] to the stochastic setting, and Theorem 3.2 of [6] to any finite mutation graph. We illustrate our theorem by a series of examples describing surprising phenomena arising from the geometry of the graph and/or the rate of mutations.…”

Deterministic approaches for head lice infestations and treatments

“…5 b where, though the infestation could be considered eradicated, in the long run the lice population increases again. In a further study we plan to improve the modeling approach proposed here by considering a mixed approach of deterministic and stochastic processes, as it has been proposed in other fields of biology ( Kraut and Bovier, 2019 ).…”