  Following some recent works, we investigate the problem of optimising the
total population size for logistic diffusive models with respect to resources
distributions. Using the spatially heterogeneous Fisher-KPP equation, we obtain
a surprising fragmentation phenomenon: depending on the scale of diffusivity
(i.e the dispersal rate), it is better to either concentrate or fragment
resources. Our main result is that, the smaller the dispersal rate of the
species in the domain, the more optimal resources distributions tend to
oscillate. This is in sharp contrast with other criteria in population
dynamics, such as the classical problem of optimising the survival ability of a
species, where concentrating resources is always favourable, regardless of the
diffusivity. Our study is completed by numerous numerical simulations that
confirm our results.
