Some optimisation problems in spatial ecology

Idriss Mazari (Laboratoire Jacques-Louis Lions, Paris-Sorbonne Université)

Abstract: In this talk, we will present some optimisation problems arising in spatial ecology. In a broad way, the main question of this talk is: how can we spread resources in an optimal manner? Assuming our population can be modeled via the Fisher-KPP equation $$-muDelta u-u(m-u)=0$$

with Neumann or Dirichlet boundary conditions, and with a resources distribution $m:Omega to mathbb R$, we investigate two questions:

1) Under natural ($L^1$ and $L^infty$) constraints on the resources distribution $m$, which resources distribution yields the optimal chance of survival? Can we take into account more precise models involving a drift? These questions can be recast as spectral optimisation problems and lead to “concentration” properties.

2) Under the same $L^1$ and $L^infty$ constraints on $m$, which resources distribution yields the maximal population size? Here, we prove the existence of bang-bang (i.e characteristic functions) maximisers for large enough diffusivities and investigate several qualitative properties. A new asymptotic method to obtain the existence result is introduced. An unexpected qualitative result is the influence of the diffusivity on the optimisers, which opens up many questions about “fragmentation” properties for such models and optimisation problems.