Abstract
A food chain describes predator-prey interaction between species and behaviors of some species effect a change in the whole ecosystem. Hence, the food chain plays the basic and most important role in the ecosystem. It is very important to consider temporal-spatio change of the population of species from the view point of maintenance, management, protection of the ecosystem. In this paper, we consider the food chain consisting of three populations, fish, zooplankton and phytoplankton, and we study the modeling of such a food chain and analyze their behaviors by numerical simulations. In the food chain, environmental changes cause some kinds of random fluctuations in the growth and the death rates of species. In order to study the impact of these uncertainties on the food chain, we need the analysis by the stochastic predator-prey model. Thus, we propose the stochastic model of the considered food chain. In numerical simulations, we consider the two types of fish-plankton models: one is the model with a partially homogeneous distribution of fish, the other is the model with fish school that move temporally and spatially. We show that spiral waves are formed by deflection of fish density by simulations. In the numerical analyses for the stochastic model with fish school, it is shown that the size, number and motion of fish school can give rise to plankton spiral waves. We find that this spatio-temporal spiral pattern is robust for disturbance and is non-linear phenomenon such as the spiral pattern in the BZ reaction.
