High dimension analysis of the symbiotic two-species contact process model

In the symbiotic contact process model each node in a graph can be vacant, occupied by one or two particles of different species. The symbiotic interaction is represented by the reduced death rate $\mu$ $(0< \mu < 1)$ for a pair of particles in the same node. From the analytical analysis of the mean-field theory, we show that the model undergoes a discontinuous phase transition at the critical creation rate $\lambda_c(\mu)$, which is a decreasing function of $\mu$.Moreover, we determine a region of bistability, where the system has two stable states, namely, both populations persist or are extincted depending on the initial state of the system. By performing Monte Carlo simulations, we have analyzed two different methods in order to maintain the system out of the absorbing state. The first method introduces a small perturbation on the absorbing state replacing one particle of each species. In the second method, once the system reaches the absorbing state, this state is replaced by a visited active one. Our results show that the bistable phase can be observed just with the first method. In this case, a hysteresis loop appears either in the Erd?s-R?nyi graph or in the complete one. Finally, this method can be applied to low-dimensional systems, where it is possible to find a hysteresis loop which has its length depending on the simulation time.