  We develop a time and space dependent predator-prey model. The predators'
equation is a non local hyperbolic balance law, while the diffusion of prey
obeys a parabolic equation, so that predators "hunt" for prey, while prey
diffuse. A control term allows to describe the use of predators as parasitoids
to limit the growth of prey-parasites. The general well posedness and stability
results here obtained ensure the existence of optimal pest control strategies,
as discussed through some numerical integrations. The specific example we have
in mind is that of Trichopria drosophilae used to fight against the spreading
of Drosophila suzukii.
