Stability of Real Interacting Populations in Space and Time: Implications, Alternatives and the Negative Binomial kc

Abstract

(1) Data are examined that relate to two-population interaction models based on the negative binomial parameter, k, and the resulting 'empty-cell' probability (e.g. probability of host escaping parasitoid attack). As predicted (Taylor, Woiwod & Perry 1979) estimates of k were non-linearly related to population density for samples of Drosophila melanogaster eggs, predator Phytoseiulus persimilis adults (among its prey, Tetranychus urticae), parasitoid Ooencyrtus kuvanae adults (among eggs of its host, Lymantria dispar) and parasitoid A laptusfusculus eggs (in eggs of its host Mesopsocus immunis). (2) Contagion did not increase with k, in contrast to conventional expectations, and k was highly variable as found by Elliot (1982) for the parasitoid Agriotypus armatus. There was no constant (kc) value. Hence, the criterion required for stability in May's (1978) phenomenological model did not exist. (3) This and similar two-population interaction models dependent on the 'empty-cell' probability, p0, will overestimate p0 unless they incorporate density dependence of k. They will also overestimate equilibrium densities. Stability analyses for May's phenomenological and related models, performed about these equilibrium densities, are not reliable. (4) An alternative simple host-parasitoid model is examined in which heterogeneity is built into the basic Nicholson-Bailey model by allowing parasitoid and host to respond to one another by movement; this model also applies if the host has a partial refuge from the parasitoid. (5) Models involving two interacting species should not depend upon the premise that the two species are distributed independently of one another.