  We consider an agro-ecologically motivated coupling between a prey-predator
system and a vector-borne epidemic system. The coupled system contains one ODE,
two reaction-diffusion PDEs and one reaction-diffusionadvection PDE; it has no
complete variational or monotonic structure and coefficients are spatially
heterogeneous. We study the long-time behavior of solutions of the
Cauchy-Neumann problem, which turns out to be largely decided by a linear
stability criterion but still involves nonlinear subtleties. Then we consider
an optimal control problem. Although it remains elusive analytically, we show
numerically the variety of outcomes and the strong impact of the initial
conditions.
