Kraut Anna, Bovier Anton
Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität, Endenicher Allee 60, 53115, Bonn, Germany.
J Math Biol. 2019 Oct;79(5):1699-1747. doi: 10.1007/s00285-019-01408-6. Epub 2019 Jul 26.
We consider an asexually reproducing population on a finite type 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 modelled as a measure-valued Markov process. Multiple variations of this system have been studied in the simultaneous 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 then let the mutation probability tend to zero. This corresponds to a much higher frequency of mutations, where multiple microscopic types are present at the same time. The limiting process resembles an adaptive walk or flight and jumps between different equilibria of coexisting types. The graph structure on the type space, determined by the possibilities to mutate, plays an important role in defining this jump process. 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.
我们考虑一个在有限类型空间上进行无性繁殖的种群，其进化由指数出生、死亡和竞争率驱动，同时在出生事件中存在突变的可能性。在个体层面，这个种群可以被建模为一个测度值马尔可夫过程。在大种群和稀有突变的同时极限情况下，已经研究了这个系统的多种变体，其中选择的 regime 使得突变是分离的。我们考虑由大种群极限产生的确定性系统，然后让突变概率趋于零。这对应于更高频率的突变，此时同时存在多种微观类型。极限过程类似于自适应行走或飞行，并在共存类型的不同平衡之间跳跃。由突变可能性决定的类型空间上的图结构，在定义这个跳跃过程中起着重要作用。在上述模型的一个变体中，突变体可以传播的半径是有限的，我们研究了在适应度景观中穿越山谷的可能性，并推导了不同类型的极限行走。
