Dispersal and Total Population Size in Discrete-Time Metapopulation Models

We investigate the impact of dispersal on the long-term total population size in a discrete-time model of a population distributed across a patchy environment. Dispersal is allowed to be asymmetric, and each isolated patch exhibits a stable carrying capacity in the absence of movement. The central question is whether dispersal increases or decreases the total equilibrium population, measured by comparing the sum of local carrying capacities to the population size under dispersal. We establish the global asymptotic stability of the equilibrium when dispersal rates are constant. In scenarios where dispersal and demographic processes occur on separate time scales, we derive simple analytical conditions to assess the effect of dispersal. These conditions are explicitly characterized for the two-patch case with Beverton–Holt local dynamics. In the fast-dispersal limit, when the local growth rates differ, there is a nonempty interval of dispersal asymmetries for which the total equilibrium population exceeds the sum of the local carrying capacities, and this interval widens as the relative difference between growth rates increases. We also use this limiting criterion to construct dispersal rates that produce the same type of increase in the model without explicit time-scale separation.