The effect of diffusion in the one-dimensional long-range contact process is investigated by mean-field calculations. Recent works have shown that diffusion decreases the effectiveness of long-range interactions, affecting the character of the phase transition: for higher values of the diffusion coefficient, stronger long-range interactions are required to enable phase coexistence and first-order behavior. Here we apply a generalized mean-field approximation for the master equation of the model that considers states of an aggregate of L lattice sites. The phase diagram of the model for values of L up to 10 is obtained, and for some values of the diffusion rate extrapolations to infinite-sized systems are given. For low-diffusive systems, approximations with L≥3 are able to reveal the suppression of the phase coexistence induced by diffusion, however, in the high-diffusion regime, larger values of L are necessary to correctly account for the higher range of correlations. We present a very efficient method to study the mean-field equations and determine the nature of the phase transitions that may be of general utility.
We present a simple lattice model that consists of two competitive contact processes with local interactions on a one-dimensional lattice. The sites of the lattice can be empty or occupied by particles of type A or type B. The time evolution of the densities is governed by a master equation, whose transition among the states depends essentially on the spreading and annihilation rates of both particles. This is a competitive model where the stationary states are determined as a function of the spreading and annihilation rates. We employ mean-field approximations, at the level of one and two sites, and we obtain a phase diagram that shows four well distinguished phases, including a mixed one in which both particles coexist. Monte Carlo simulations show that this mixed phase no longer exists in the limit of large one-dimensional lattices. However, simulations of the model in two dimensions show that the mixed phase does not disappear as in the one-dimensional case. We also calculate some static critical exponents of the one-dimensional version of the model and we have shown that they belong to the directed percolation universality class.
We study the prisoner's dilemma model with a noisy imitation evolutionary dynamics on directed out-homogeneous and uncorrelated directed random networks. An heterogeneous pair mean-field approximation is presented showing good agreement with Monte Carlo simulations in the limit of weak selection (high noise) where we obtain analytical predictions for the critical temptations. We discuss the phase diagram as a function of temptation, intensity of noise and coordination number of the networks and we consider both the model with and without self-interaction. We compare our results with available results for non-directed lattices and networks.
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