Stefano Berti (University of Lille)

Title: Reaction-transport systems in heterogeneous environments and population dynamics

Abstract: Understanding the conditions ensuring the persistence of a population or determining the efficiency of its invasion into a new territory are issues of primary importance in population biology. A convenient conceptual framework to study these problems from a theoretical point of view is offered by reaction-transport equations. Taking this modeling approach, in the first part of the talk I present a study on the evolution of a population in a heterogeneous environment in the presence of advection. The habitat consists of a periodic alternation of favorable and unfavorable patches where the population grows according to logistic dynamics and dies at a fixed rate, respectively. I show that the population can survive and invade the habitat only above a critical size of the favorable patches and below a critical velocity of the medium. I characterize the survival/extinction transition, and I present some analytical and numerical results on the invasion speed. Then, I discuss some limitations of this description, extending the analysis to a model with a growth term accounting for decreased reproductive power at low population densities. The phenomenology of this model is found to be richer when varying the advection intensity.
In the second part of the talk I discuss some effects of discreteness and demographic stochasticity on the extinction dynamics of a population confined in a favorable patch (an oasis) surrounded by a deadly environment. For this purpose, I introduce an individual-based reaction-diffusion model and investigate its behavior focusing on the average time to extinction and on the quasistationary distribution of the number of individuals that sets in for large enough patch sizes.