Mean field approach and role of the coloured noise in the dynamics of three interacting species

We study the effects of the coloured noise on the dynamics of three interacting species, namely two preys and one predator, in a two-dimensional lattice with N sites. The three species are affected by multiplicative time correlated noise, which accounts for the effects of environment on the species evolution. Moreover, the interaction parameter between the two preys is a dichotomous stochastic process, which determines two dynamical regimes corresponding to different biological conditions. Preliminarily, we study the noise effect on the three species dynamics in single site. Then, we use a mean field approach to obtain, in Gaussian approximation, the moment equations for the species densities. We find that the multiplicative noise does not affect the time behaviour of the 1st order moments. Conversely, the 2nd order moments are strongly dependent both on the intensity and correlation time of the multiplicative noise. Finally, we compare our results with those obtained from a discrete time approach based on a model of coupled map lattice.