Behavior analyses of stochastic predator-prey systems under fish predation

Many spatio-temporal patterns are observed in various fields of engineering and the analysis of such patterns has been one of challenging problems in pattern formation problems. So far, many excellent works have been done in the analysis of the spatio-temporal pattern formations, however, most of such works are the deterministic analysis. Since physical, chemical and social phenomena contain some kind or another of uncertainties to a greater or lesser extent, the analysis of influence of uncertainties on phenomena is very important as a basic problem in a wide range of fields including design and planning of controlled systems in control engineering and option pricing in economics. Among many phenomena, we especially focus on biological phenomena and study the influence of the random uncertainties on the spatio-temporal patterns generated by the predator-prey system with diffusion and uncertainties. In this paper, as the predator-prey system, we consider the phytoplankton-zooplankton-fish system and analyze the influence of the random uncertainties on the spatio-temporal patterns generated by the stochastic plankton-fish system with diffusion by the numerical simulations.