Abstract. Mechanisms leading to speciation are a major focus in evolutionary biology. In this paper, we present and study a stochastic model of population where individuals, with type a or A, are equivalent from ecological, demographical and spatial points of view, and differ only by their mating preference: two individuals with the same genotype have a higher probability to mate and produce a viable offspring. The population is subdivided in several patches and individuals may migrate between them. We show that mating preferences by themselves, even if they are very small, are enough to entail reproductive isolation between patches, and we provide the time needed for this isolation to occur as a function of the population size. Our results rely on a fine study of the stochastic process and of its deterministic limit in large population, which is given by a system of coupled nonlinear differential equations. Besides, we propose several generalisations of our model, and prove that our findings are robust for those generalisations.

To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on type structured populations, to models of populations structured by type and space, considering diffusion and nonlocal competition between individuals?We study a nonlocal competitive Lotka-Volterra type system, describing a spatially structured population which can be either monomorphic or dimorphic. Considering spatial diffusion, intrinsic death and birth rates, together with death rates due to intraspecific and interspecific competition between the individuals, leading to some integral terms, we analyze the long time behavior of the solutions. We first prove existence of steady states and next determine the long time limits, depending on the competition rates and the principal eigenvalues of some operators, corresponding somehow to the strength of traits. Numerical computations illustrate that the introduction of a new mutant population can lead to the long time evolution of the spatial niche.Spatially structured Lotka-Volterra systems resources available. The paper by Philipps and Co [26] shows the strong impact of the morphologic parameters of the cane toads on the expansion of their invasion.In this context, the study of space-related traits, such as dispersal speed or sensibility to heterogeneously distributed resources, is fundamental and has been object of mathematical developments. In Champagnat-Méléard [12], a stochastic individual-based model is introduced where individuals are characterized both by their location and one or several phenotypic and heritable traits. The individuals move, reproduce with possible mutation and die of natural death or because of competition for resources. The spatial motion is modeled as a diffusion and the spatial interaction between individuals is modeled by a convolution kernel in some spatial range. In a large population scale, it is shown that this microscopic stochastic model can be approximated by a nonlinear nonlocal reaction-diffusion equation defined on the space of traits and space. The latter has been studied in Ferrière-Desvillettes-Prévost [16] and Arnold-Desvillettes-Prévost [2] and existence and uniqueness of the solution, numerical simulations and steady states are studied. Propagation phenomena and existence of traveling waves are explored numerically and theoretically for different variants of such models in [1], [3], [4], [6]. This problem has also been studied from an asymptotic point of view using Hamilton-Jacobi equations [7], [8].

The decomposition of soil organic matter (SOM) is a critical process in global terrestrial ecosystems. SOM decomposition is driven by micro-organisms that cooperate by secreting costly extracellular (exo-)enzymes. This raises a fundamental puzzle: the stability of microbial decomposition in spite of its evolutionary vulnerability to “cheaters”—mutant strains that reap the benefits of cooperation while paying a lower cost. Resolving this puzzle requires a multi-scale eco-evolutionary model that captures the spatio-temporal dynamics of molecule-molecule, molecule-cell, and cell-cell interactions. The analysis of such a model reveals local extinctions, microbial dispersal, and limited soil diffusivity as key factors of the evolutionary stability of microbial decomposition. At the scale of whole-ecosystem function, soil diffusivity influences the evolution of exo-enzyme production, which feeds back to the average SOM decomposition rate and stock. Microbial adaptive evolution may thus be an important factor in the response of soil carbon fluxes to global environmental change.