Solutions with positive components to quasilinear parabolic systems

We obtain sufficient conditions for the existence and uniqueness of solutions with non-negative components to general quasilinear parabolic problems \begin{equation*} \partial_t u^k = \sum_{i,j=1}^n a_{ij} (t,x,u)\partial^2_{x_i x_j}\!u^k + \sum_{i=1}^n b_i (t,x,u, \partial_x u) \partial_{x_i} u^k +\, c^k(t,x,u,\partial_x u), \\ u^k(0,x) = \varphi^k(x), \\ u^k(t,\,\cdot\,) = 0, \quad \text{on } \partial \mathbb F, k=1,2, \dots, m, \quad x\in\mathbb F, \;\; t>0. \end{equation*} Here, $\mathbb F$ is either a bounded domain or $\mathbb R^n$; in the latter case, we disregard the boundary condition. We apply our results to study the existence and asymptotic behavior of componentwise non-negative solutions to the Lotka-Volterra competition model with diffusion. In particular, we show the convergence, as $t\to+\infty$, of the solution for a 2-species Lotka-Volterra model, whose coefficients vary in space and time, to a solution of the associated elliptic problem

The spatio-temporal dynamics of neutral genetic diversity

International audienceThe notions of pulled and pushed solutions of reaction-dispersal equations introduced by Garnier et al. (2012) and Roques et al. (2012) are based on a decomposition of the solutions into several components. In the framework of population dynamics, this decomposition is related to the spatio-temporal evolution of the genetic structure of a population. The pulled solutions describe a rapid erosion of neutral genetic diversity, while the pushed solutions are associated with a maintenance of diversity. This paper is a survey of the most recent applications of these notions to several standard models of population dynamics, including reaction-diffusion equations and systems and integro-differential equations. We describe several counterintuitive results, where unfavorable factors for the persistence and spreading of a population tend to promote diversity in this population. In particular, we show that the Allee effect, the existence of a competitor species, as well as the presence of climate constraints are factors which can promote diversity during a colonization. We also show that long distance dispersal events lead to a higher diversity, whereas the existence of a nonreproductive juvenile stage does not affect the neutral diversity in a range-expanding population

Front-like entire solutions for monostable reaction-diffusion systems

This paper is concerned with front-like entire solutions for monostable reactiondiffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Combining a SIS and traveling fronts with different wave speeds and directions, the existence and various qualitative properties of entire solutions are then established using comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish some comparison arguments for the three systems. The existence of entire solutions is then proved via the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems