In this talk, I will address the following problem : assuming a species spreads and reproduces in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? The mathematical formulation of this problem relies on the logistic diffusive equation, also called the heterogeneous Fisher-KPP equation. This gives rise to a nonconvex optimisation problem. I will present the following results. 1. For large diffusivity rates, the optimal distribution of resources is not fragmented, in a sense. 2. The optimizers are always of "bang-bang" type. 3. For small diffusivity rates, fragmentation occurs and we could estimate the blow-up rate of the BV norm of the optimizer. (Joint works with I. Mazari and Y. Privat)
