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This paper is devoted to study measure solutions $\mu_t^h$ to perturbednonlinear structured population models where $t$ denotes time and $h$ controllsthe size of perturbation. We address differentiability of the map $h \mapsto\mu_t^h$. After showing that this type of results cannot be expected in thespace of bounded Radon measures $\mathcal{M}(\mathbb{R}^+)$ equipped with theflat metric, we move to the slightly bigger spaces $Z =\overline{\mathcal{M}(\mathbb{R}^+)}^{(C^{1+\alpha})^*}$. We prove that when$\alpha > \frac{1}{2}$, the map $h \mapsto \mu_t^h$ is differentiable in $Z$.The proof exploits approximation scheme of a nonlinear problem from previousstudies and is based on the iteration of an implicit integral equationsobtained from study of the linear equation. The result shows that space $Z$ isa promising setting for optimal control of phenomena governed by such type ofmodels.

Measure solutions to perturbed structured population models - differentiability with respect to perturbation parameter