  In this article, we give an in-depth analysis of the problem of optimising
the total population size for a standard logistic-diffusive model. This
optimisation problem stems from the study of spatial ecology and amounts to the
following question: assuming a species evolves in a domain, what is the best
way to spread resources in order to ensure a maximal population size at
equilibrium? {In recent years, many authors contributed to this topic.} We
settle here the proof of two fundamental properties of optimisers: the
bang-bang one which had so far only been proved under several strong
assumptions, and the other one is the fragmentation of maximisers. Here, we
prove the bang-bang property in all generality using a new spectral method. The
technique introduced to demonstrate the bang-bang character of optimizers can
be adapted and generalized to many optimization problems with other classes of
bilinear optimal control problems where the state equation is semilinear and
elliptic. We comment on it in a conclusion section.Regarding the geometry of
maximisers, we exhibit a blow-up rate for the $BV$-norm of maximisers as the
diffusivity gets smaller: if $\O$ is an orthotope and if $m_\mu$ is an optimal
control, then $\Vert m_\mu\Vert_{BV}\gtrsim \sqrt{\mu}$. The proof of this
results relies on a very fine energy argument.
