ABSTRACT
This chapter analyzes simple ordinary differential equations-based models for a predator–prey mite interaction on apple tree foliage. That model is a specific Kolmogorov-type exploitation ordinary differential equation system assembled by May, from predator and prey components developed by Leslie and Holling, respectively. It is a composite model, in that its parameters are chosen appropriately for a temperature-dependent mite interaction on fruit tree leaves by curve-fitting to the relevant data. In particular, the proper temperature-rate relationship for arthropods is deduced by the knowledge of the results of singular perturbation theory applied to ordinary differential equations. Its linear behavior is examined by a standard linear stability analysis of the equilibrium solution to this model. Its nonlinear behavior is examined by means of Kolmogorov's theorem for dynamical systems and by the numerical bifurcation code AUTO, these results involving limit cycle behavior. The predictions of this model are then compared with general ecological field results and particular laboratory experimental data. In addition, two related models are developed from that model: The first, a generalization of it involving predator dependence for its functional and numerical responses; and the second, a Bazykin-type system derived from it by employing a truncation of a Taylor series expansion in its predator equation. The global behavior of these two systems are determined by AUTO and ecological interpretations of those results discussed.
