Mathematical analysis of the optimal habitat configurations for species persistence.

We study a reaction-diffusion model in a binary environment made of habitat and non-habitat regions. Environmental heterogeneity is expressed through the species intrinsic growth rate coefficient. It was known that, for a fixed habitat abundance, species survival depends on habitat arrangements. Our goal is to describe the spatial configurations of habitat that maximise the chances of survival. Through numerical computations, we find that they are of two main types - ball-shaped or stripe-shaped. We formally prove that these optimal shapes depend on the habitat abundance and on the amplitude of the growth rate coefficient. We deduce from these observations that the optimal shape of the habitat realises a compromise between reducing the detrimental habitat edge effects and taking advantage of the domain boundary effects. In the case of an infinite-periodic environment, we prove that the optimal habitat shapes can be deduced from those in the case of a bounded domain.