Stochastic model for coexistence of ant colonies

We will present a stochastic model defined on a lattice to describe the coexistence and the oscillations of two types of ant colonies that use different resources of the environment. This model has four states per site, which denote occupation by elements of niche 1 (state 1); occupation by individuals of colony 1 (state 2); occupation by elements of niche 2 (state 3); occupation by individuals of colony 2 (state 4). If a site is in state 1 (or in state 3) it means that it has resources for colony 1 (or colony 2). The transition probability of state 1(3) to state 2(4) is given by the probability a(c) times the fraction of first neighbors in states 2(4). Therefore, the processes of birth of individuals of the colonies are given by autocatalytic reactions. And colony 1(2) develops in niche 2(1) by means of a spontaneous process whose probability is b(d). From the perspective of biology, this is interesting because in Amazon, it is observed the coexistence of several genres of ant colonies that use different resources, and there are studies with experimental data about that. From the viewpoint of the study of stochastic processes out of equilibrium, the exploration the model itself is interesting. We will show results obtained with mean field approximation (simple and pair approximations) and Monte Carlo simulations. Thanks to the financial support from FAPESP.