Superprocesses for the Population of Rabbits on Grassland

Motivated by the control of rabbits on grassland, a model of a population with branching dynamics in a random environment is constructed. The system is described as the solution to a conditional martingale problem given the random environment which satisfies a stochastic partial differential equation (SPDE). The weak uniqueness of the solution to the system is established by characterizing its conditional log-Laplace transform through the solution to a related nonlinear SPDE.