  We investigate in the present paper the maximization problem for the L1 norm
of the unique positive solution of an heterogeneous Fisher-KPP equation with
respect to the growth rate. It is already known that the BV norms of maximizers
of this functional blow up when the diffusivity tends to zero. Here, we first
show that the maximizers are always BV. Next, we completely characterize the
limit of the maximas of this functional as the diffusivity tends to zero, and
we show that one can construct a quasi-maximizer which is periodic, in a sense,
and with a BV norm behaving like the inverse of the square root of the
diffusivity. Lastly, we prove that along a subsequence of diffusivities, any
maximizer is periodic, in a sense.
