Understanding productivity (i.e., the emergence rate of adults ) in heterogeneous habitats depends upon understanding the relationship between egg laying, carrying capacity (), and crowding.
a) The functional relationship between the rate of egg-laying and productivity depends on the functional response to crowding. In this model, the relationship is sensitive to the power-law scaling relationship ( blue; red; purple). Carrying capacity is given for a single value of egg laying rates, given at the steady state if that pool had existed in isolation. b) In a system with 2 pools linked by egg-laying, where the carrying capacity of pool 1 is approximately 90% of the total (dashed blue line) and pool 2 has the rest (dashed red line), the population totals overall (solid black) are generally below the maximum, unless egg laying is fine-tuned such that the proportion of eggs laid was equal to that pool’s proportion of carrying capacity (vertical grey). c) A comparison of productivity (red) and carrying capacity (black line) for a typical set of heterogeneous aquatic habitats. Productivity equals carrying capacity when the distribution of eggs laid is finely tuned to match the distribution of carrying capacities (i.e. ). d) The ratio of productivity to carrying capacity was computed for 100 sets of heterogeneous aquatic habitat. The green line plots the 1∶1 ratio, when productivity equals carrying capacity. These distributions, plotted here as the median (solid line) and the 10th and 90th quantiles (dashed lines), shows the robust pattern that the habitats with the lowest productivity tend to be under capacity and the few highly productive habitats tend to be over capacity.