On Reconciling Patchy Microspatial Distributions with Competition Models

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Abstract
A model focused on competition among periodical cicadas was proposed, which partitions the conventional competition coefficients into 3 components: how the nymphs affect each other, where they are located, and the disparity between fecundities. The 2nd component depends on having measured the microspatial distribution and determining the degree of patchiness, gij, both between and within species [Magicicada septendecim, M. cassini and M. septendecula]. 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. 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. 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. 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 vs. sets where only 1, 2, etc. species would be expected to persist. In this example of periodical cicadas all 3 spp. should not be able to come to a neighborhood-stable equilibrium, even in disturbed habitats, unless the carrying capacities of M. septendecim and M. septendecula are both substantially lower than for M. cassini. By computer simulation global stability can be determined for each combination of possible carrying capacities and this information can be incorporated into the same diagram. The idea of substituting gijnj for nj should have wide applicability as a way of incorporating the patchy microspatial pattern into a competition model.