  We consider a space structured population model generated by two point
clouds: a homogeneous Poisson process $M$ with intensity $n\to\infty$ as a
model for a parent generation together with a Cox point process $N$ as
offspring generation, with conditional intensity given by the convolution of
$M$ with a scaled dispersal density $\sigma^{-1}f(\cdot/\sigma)$. Based on a
realisation of $M$ and $N$, we study the nonparametric estimation of $f$ and
the estimation of the physical scale parameter $\sigma>0$ simultaneously for
all regimes $\sigma=\sigma_n$. We establish that the optimal rates of
convergence do not depend monotonously on the scale and we construct minimax
estimators accordingly whether $\sigma$ is known or considered as a nuisance,
in which case we can estimate it and achieve asymptotic minimaxity by plug-in.
The statistical reconstruction exhibits a competition between a direct and a
deconvolution problem. Our study reveals in particular the existence of a least
favourable intermediate inference scale, a phenomenon that seems to be new.
