3. Of Mites and Models

The nonlinear behavior of a particular Kolmogorov-type exploitation ordinary differential equation system assembled by May (Stability and complexity in model ecosystems. Princeton University Press, Princeton, 1973) [80] from predator and prey components developed by Leslie(Biometrica 35:213–245, 1948) [67] and Holling (Mem. Entomol. Soc. Can. 45:1–60, 1965) [49], respectively, is examined by means of the numerical bifurcation code AUTO with model parameters chosen appropriately for a temperature dependent mite interaction on fruit tree leaves. In particular the proper temperature-rate relationship for arthropods is developed by the knowledge of the results of singular perturbation theory applied to ordinary differential equations which is introduced as a pastoral interlude. The concepts of linear stability theory, phase-plane analysis, and limit cycle behavior are also introduced as pastoral interludes. The predictions of this model are then compared with general ecological field results and particular laboratory experimental data. The problems extend singular perturbation type analyses to the investigation of polynomials and the specific temperature-rate relationship for the predaceous mite and linear stability analyses to competing species models and predator–prey models employing a Holling-type I functional response as opposed to the type II response included in the May mite model.