Asymptotic profile in selection-mutation equations: Gauss versus Cauchy distributions.

In this paper, we study the asymptotic (large time) behaviour of a selection-mutation-competition model for a population structured with respect to a phenotypic trait when the rate of mutation is very small. We assume that the reproduction is asexual, and that the mutations can be described by a linear integral operator. We are interested in the interplay between the time variable t and the rate ε of mutations. We show that depending on α > 0 , the limit ε → 0 with t = ε - α can lead to population number densities which are either Gaussian-like (when α is small) or Cauchy-like (when α is large).

Dissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments.

An enduring puzzle in evolutionary biology is to understand how individuals and populations adapt to fluctuating environments. Here we present an integro-differential model of adaptive dynamics in a phenotype-structured population whose fitness landscape evolves in time due to periodic environmental oscillations. The analytical tractability of our model allows for a systematic investigation of the relative contributions of heritable variations in gene expression, environmental changes and natural selection as drivers of phenotypic adaptation. We show that environmental fluctuations can induce the population to enter an unstable and fluctuation-driven epigenetic state. We demonstrate that this can trigger the emergence of oscillations in the size of the population, and we establish a full characterisation of such oscillations. Moreover, the results of our analyses provide a formal basis for the claim that higher rates of epimutations can bring about higher levels of intrapopulation heterogeneity, whilst intense selection pressures can deplete variation in the phenotypic pool of asexual populations. Finally, our work illustrates how the dynamics of the population size is led by a strong synergism between the rate of phenotypic variation and the frequency of environmental oscillations, and identifies possible ecological conditions that promote the maximisation of the population size in fluctuating environments.