A long range sexual reproduction process

We describe a long range growth model with sexual reproduction on εZ in which particles die at rate 1 and pairs of adjacent particles produce offspring at rate λ. The offspring is sent to a site chosen at random from the neighborhood of the parent particles. If the site is already occupied, the birth is suppressed, that is, we allow at most one particle per site. The size of the neighborhood increases as ε tends to 0. We investigate the behavior for small ε. In the limit as ε → 0, the particle density evolves according to an integro-differential equation which has traveling wave solutions whose wave speed is a nondecreasing function of λ. We compare the system for small ε with the limiting system and discuss phase transition. We show that the behavior of the particle system for sufficiently small ε is similar to the behavior of the limiting system. That is, if λ is sufficiently small so that the wave speed of the traveling wave of the limiting equation is negative, then, for small enough ε, the pointmass at the all-empty configuration is the only stable equilibrium. If λ is large enough, so that the wave speed of the traveling wave of the limiting equation is positive, them, for small enough ε, the system has a positive probability of survival, that is, in addition to the pointmass at the all-empty configuration, there is a nontrivial equilibrium. Not predicted by the limiting system, there is a range of values of λ for which the all-empty configuration is the only stable equilibrium but for which the particle system exhibits metastable bahavior.