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

A fragmentation phenomenon for a non-energetic optimal control problem: optimisation of the total population size in logistic diffusive models