Predation with Indirect Effects in Fluctuating Environments

We investigate the long term dynamics for a predation model of Plankton community with indirect effects, under fluctuating environments. A random version and a stochastic version with multiplicative noise of the model are discussed and compared. We prove that the solutions to both versions are non-negative and bounded given any non-negative positive initial conditions. We also prove that both the random system and the stochastic system possess a unique random attractor under the same set of assumptions, by using the classical theory of random dynamical systems. In addition we provide conditions under which coexistence of species exists for the random system.