  We interpret a class of nonlinear Fokker-Planck equations with reaction as
gradient flows over the space of Radon measures equipped with the recently
introduced Hellinger-Kantorovich distance. The driving entropy of the gradient
flow is not assumed to be geodesically convex or semi-convex. We prove new
generalized dissipation inequalities, which allow us to control the relative
entropy by its production. We establish the entropic exponential convergence of
the trajectories of the flow to the equilibrium. Along with other applications,
this result has an ecological interpretation as a trend to the ideal free
distribution for a class of fitness-driven models of population dynamics. Our
existence theorem for weak solutions under mild assumptions on the nonlinearity
is new even in the absence of the reaction term.
