Dynamics of three interacting species in single compartment and in spatially extended system by moment equations

Real ecosystems are influenced by random fluctuations of environmental parameters, such as temperature, food resources, migrations, genetic changes. This caused, during last decades, an increasing interest on the role played by the noise in populationdynamics. In systems governed by nonlinear dynamics the presence of noise sources can give rise to counterintuitive phenomena like stochastic resonance, noise enhanced stability, resonant activation, noise delayed extinction. Therefore, the stability of biological systems in the presence of noise sources has become one of the most relevant topics both in experimental and theoretical investigations on complex systems. In this work we consider the dynamics of three interacting species, two preys and one predator, in the presence of two different kinds of noise sources. To describe the spatial distributions of the species we use a model based on Lotka-Volterra (LV) equations. A correlated dichotomous noise acts on $\beta$, the interaction parameter between the two preys, and a multiplicative white noise affects directly the dynamics of each one of the three species. Using low levels of multiplicative noise we analyze the system dynamics for two different values of $\beta$, $\beta_{down}$ and $\beta_{up}$, which determines respectively the coexistence and exclusion regimes for the system. Successively we consider $\beta$ as a stochastic process governed by a periodical driving force in the presence of a dichotomous noise. This causes the interaction parameter $\beta$ to switch quasi-periodically between $\beta_{down}$ and $\beta_{up}$. As a consequence, a dynamical regime appears where coexistence and exclusion alternatively take place. In this condition we study the time behaviour of the three species in a single compartment system, for different values of the multiplicative noise intensity. Afterwards we consider a spatially extended system formed by a two-dimensional spatial domain, i.e. a square lattice with N x N sites, and we write the three species equations adding a diffusion term. Then, by applying a mean field approach, we obtain the corresponding moment equations in Gaussian approximation. Within this formalism we get the time behavior of the first and second order moments for different values of the multiplicative noise intensity, according to the procedure followed in the single compartment case.