Direct competition results from strong competition for limited resource.

We study a model of competition for resource through a chemostat-type model where species consume the common resource that is constantly supplied. We assume that the species and resources are characterized by a continuous trait. As already proved, this model, although more complicated than the usual Lotka-Volterra direct competition model, describes competitive interactions leading to concentrated distributions of species in continuous trait space. Here we assume a very fast dynamics for the supply of the resource and a fast dynamics for death and uptake rates. In this regime we show that factors that are independent of the resource competition become as important as the competition efficiency and that the direct competition model is a good approximation of the chemostat. Assuming these two timescales allows us to establish a mathematically rigorous proof showing that our resource-competition model with continuous traits converges to a direct competition model. We also show that the two timescales assumption is required to mathematically justify the corresponding classic result on a model consisting of only finite number of species and resources (MacArthur in, Theor Popul Biol 1:1-11, 1970). This is performed through asymptotic analysis, introducing different scales for the resource renewal rate and the uptake rate. The mathematical difficulty relies in a possible initial layer for the resource dynamics. The chemostat model comes with a global convex Lyapunov functional. We show that the particular form of the competition kernel derived from the uptake kernel, satisfies a positivity property which is known to be necessary for the direct competition model to enjoy the related Lyapunov functional.

Dynamics of sexual populations structured by a space variable and a phenotypical trait.

We study sexual populations structured by a phenotypic trait and a space variable, in a non-homogeneous environment. Departing from an infinitesimal model, we perform an asymptotic limit to derive the system introduced in Kirkpatrick and Barton (1997). We then perform a further simplification to obtain a simple model. Thanks to this simpler equation, we can describe rigorously the dynamics of the population. In particular, we provide an explicit estimate of the invasion speed, or extinction speed of the species. Numerical computations show that this simple model provides a good approximation of the original infinitesimal model, and in particular describes quite well the evolution of the species' range.

Evolution of species trait through resource competition.

To understand the evolution of diverse species, theoretical studies using a Lotka-Volterra type direct competition model had shown that concentrated distributions of species in continuous trait space often occurs. However, a more mechanistic approach is preferred because the competitive interaction of species usually occurs not directly but through competition for resource. We consider a chemostat-type model where species consume resource that are constantly supplied. Continuous traits in both consumer species and resource are incorporated. Consumers utilize resource whose trait values are similar with their own. We show that, even when resource-supply has a continuous distribution in trait space, a positive continuous distribution of consumer trait is impossible. Self-organized generation of distinct species occurs. We also prove global convergence to the evolutionarily stable distribution.

Evolution of Specialization in Heterogeneous Environments: Equilibrium Between Selection, Mutation and Migration.

Adaptation in spatially heterogeneous environments results from the balance between local selection, mutation, and migration. We study the interplay among these different evolutionary forces and demography in a classical two-habitat scenario with asexual reproduction. We develop a new theoretical approach that goes beyond the Adaptive Dynamics framework, and allows us to explore the effect of high mutation rates on the stationary phenotypic distribution. We show that this approach improves the classical Gaussian approximation, and captures accurately the shape of this equilibrium phenotypic distribution in one- and two-population scenarios. We examine the evolutionary equilibrium under general conditions where demography and selection may be nonsymmetric between the two habitats. In particular, we show how migration may increase differentiation in a source-sink scenario. We discuss the implications of these analytic results for the adaptation of organisms with large mutation rates, such as RNA viruses.

A Fokker-Planck approach to the study of robustness in gene expression.

We study several Fokker-Planck equations arising from a stochastic chemical kinetic system modeling a gene regulatory network in biology. The densities solving the Fokker-Planck equations describe the joint distribution of the mRNA and µRNA content in a cell. We provide theoretical and numerical evidence that the robustness of the gene expression is increased in the presence of µRNA. At the mathematical level, increased robustness shows in a smaller coefficient of variation of the marginal density of the mRNA in the presence of µRNA. These results follow from explicit formulas for solutions. Moreover, thanks to dimensional analyses and numerical simulations we provide qualitative insight into the role of each parameter in the model. As the increase of gene expression level comes from the underlying stochasticity in the models, we eventually discuss the choice of noise in our models and its influence on our results.