Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems

Theoretical models of populations on a system of two connected patches previously have shown that when the two patches differ in maximum growth rate and carrying capacity, and in the limit of high diffusion, conditions exist for which the total population size at equilibrium exceeds that of the ideal free distribution, which predicts that the total population would equal the total carrying capacity of the two patches. However, this result has only been shown for the Pearl-Verhulst growth function on two patches and for a single-parameter growth function in continuous space. Here, we provide a general criterion for total population size to exceed total carrying capacity for three commonly used population growth rates for both heterogeneous continuous and multi-patch heterogeneous landscapes with high population diffusion. We show that a sufficient condition for this situation is that there is a convex positive relationship between the maximum growth rate and the parameter that, by itself or together with the maximum growth rate, determines the carrying capacity, as both vary across a spatial region. This relationship occurs in some biological populations, though not in others, so the result has ecological implications.