On Reconciling Patchy Microspatial Distributions with Competition Models

We propose a model, focused on competition among periodical cicadas, which partitions the conventional competition coefficients into three components: how the nymphs affect each other, precisely where they are located, and the disparity between fecundities. The second component depends on having measured the microspatial distribution and determining the degree of patchiness, gij, both between and within species. In a patchy distribution, a nymph of species i encounters more nymphs of its own species, and either more or fewer nymphs of species j, than it would encounter in a random distribution. From the nymph's point of view it is as though the environment contained, not nj nymphs of species j, but gijnj nymphs, where gij is the measure of interspecies patchiness given by Lloyd (1967). Making that substitution enables the second component to be defined. A further consequence is that the carrying capacity for nymphs of species i is seen to be, not ki, but something lower, namely ki/gii, because the patchy microspatial distribution means that the environment is being underutilized. So far, this procedure treats the habitat as though it were uniform. If local favorability of the habitat is patchy, and if local density is influenced by local favorability, then this needs to be measured and incorporated into the partitioning. On the other hand, if local density is determined by some other process only partially correlated with local favorability (in our case, the oviposition preferences of female cicadas), then it may not be necessary to measure local favorability. This is especially likely to be so if the excess population is eliminated by aggressive interactions between individuals. The effect of incorporating the patchy microspatial distribution of individuals into a competition model will be greater the more variable are the gij. If all gij are equal there is no effect except a uniform lowering of carrying capacities. A useful way to summarize the information contained in the (partitioned) competition coefficients is to diagram the set of carrying capacities for which all species are K-compatible versus sets where only one, two, etc. species would be expected to persist. In our example of periodical cicadas we are able to conclude that all three species should not be able to come to a neighborhood-stable equilibrium, even in disturbed habitats, unless the carrying capacities of Magicicada septendecim and M. septendecula are both substantially lower than that for M. cassini. This is a robust conclusion, because it does not depend on knowing the carrying capacity for any species, or knowing the precise ecological machinery whereby competition operates. It is a conclusion about equilibrium, and so long as there are no higher-order interactions all models have the same equilibrium. By computer simulation one can also determine global stability for each combination of possible carrying capacities and incorporate this information into the same diagram. This becomes expedient with systems of three or more species. We believe the idea of substituting gijnj for nj is one that should have wide applicability as a way of incorporating patchy microspatial pattern into a competition model. However, the model should be built from the ground up, specifying all assumptions. Of crucial importance is the question whether gij's measured in one place would apply in another place ("Is overlap habitat-dependent?"). This will have to be determined empirically in each case.