In this presentation, we consider a population (typically bacteria) structured by both a spatial variable and a phenotypical trait. Our model takes into account the effects of migrations, mutations, growth and competition. When the environment is assumed homogeneous, if the population survives, it spreads to the whole space, and we have a complete picture of the large-time propagation : the solution converges towards a front, which connects a positive steady state to zero, and spreads at a determined speed.
When the environment is heterogeneous, the situation is much more complex. Depending on the profile of heterogeneities, the invasion may be either slowed or completely blocked. In some cases, the population adaption to the local environment is crucial for invasion to occur. We first consider a linear profile of heterogeneities, and then investigate the fully nonlinear case numerically as well as analytically (in a perturbative framework for the latter).
Romain Duboscq, Ariane Trescases