Impact of dispersion on dynamics of a discrete metapopulation model

Consider a discrete time model for metapopulation on two patches with local logistic dynamics. We analyze the impact of the dispersion on the positive equilibrium. Our results show that the abscissa and ordinate of the positive equilibrium are monotone functions of the dispersion rate under some conditions. Moreover, we prove that the positive equilibrium is saddle point when the dispersion rate is either small or large enough.