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.

Antigen loss is a key mechanism how tumor cells escape from T-cell immunotherapy. Using a mouse model of melanoma we directly compared antigen downregulation by phenotypic adaptation with genetically hardwired antigen loss. Unexpectedly, genetic ablation of Pmel, the melanocyte differentiation antigen targeted by adoptively transferred CD8 + T-cells, impaired melanoma cell growth in untreated tumors due to competitive pressure exerted by the bulk wild-type population.This established an evolutionary scenario, where T-cell immunotherapy imposed a dynamic fitness switch on wild-type melanoma cells and antigen loss variants, which resulted in highly variable enrichment of the latter in recurrent tumors. Stochastic simulations by an individual-based continuous-time Markov process suggested variable fitness of subclones within the antigen loss variant population as the most likely cause, which was validated experimentally. In summary, we provide a framework to better understand how subclone heterogeneity in tumors influences immune selection of genetic antigen loss variants through stochastic events.

We consider a stochastic individual-based model for the evolution of a haploid, asexually reproducing population. The space of possible traits is given by the vertices of a (possibly directed) finite graph G = (V, E). The evolution of the population is driven by births, deaths, competition, and mutations along the edges of G. We are interested in the large population limit under a mutation rate µK given by a negative power of the carrying capacity K of the system: µK = K −1/α , α > 0. This results in several mutant traits being present at the same time and competing for invading the resident population. We describe the time evolution of the orders of magnitude of each sub-population on the log K time scale, as K tends to infinity. 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.

We consider a stochastic individual-based model for the evolution of a haploid, asexually reproducing population. The space of possible traits is given by the vertices of a (possibly directed) finite graph G = (V, E). The evolution of the population is driven by births, deaths, competition, and mutations along the edges of G. We are interested in the large population limit under a mutation rate µ K given by a negative power of the carrying capacity K of the system: µ K = K −1/α , α > 0. This results in several mutant traits being present at the same time and competing for invading the resident population. We describe the time evolution of the orders of magnitude of each sub-population on the log K time scale, as K tends to infinity. Using techniques developed in [10], 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 [24] 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.

We consider a stochastic individual-based model of adaptive dynamics on a finite trait graph G = (V, E). The evolution is driven by a linear birth rate, a density dependent logistic death rate an the possibility of mutations along the (possibly directed) edges in E. We study the limit of small mutation rates for a simultaneously diverging population size. Closing the gap between [8] and [14] we give a precise description of transitions between evolutionary stable conditions (ESC), where multiple mutations are needed to cross a valley in the fitness landscape. The system shows a metastable behaviour on several divergent time scales associated to a degree of stability. We develop the framework of a meta graph that is constituted of ESCs and possible metastable transitions between those. This allows for a concise description of the multiscale jump chain arising from concatenating several jumps. Finally, for each of the various time scale, we prove the convergence of the population process to a Markov jump process visiting only ESCs of sufficiently high stability.