Temperature-mediated stability of the interaction between spider mites and predatory mites in orchards

The nonlinear behavior of the Holling-Tanner predatory-prey differential equation system, employed by R.M. May to illustrate the apparent robustness of Kolmogorov’s Theorem when applied to such exploitation systems, is re-examined by means of the numerical bifurcation code AUTO 86 with model parameters chosen appropriately for a temperature-dependent mite interaction on fruit trees. The most significant result of this analysis is that there exists a temperature range wherein multiple stable states can occur, in direct violation of May’s interpretation of this system’s satisfaction of Kolmogorov’s Theorem: namely, that linear stability predictions have global consequences. In particular these stable states consist of a focus (spiral point) and a limit cycle separated from each other in the phase plane by an unstable limit cycle, all of which are associated with the single community equilibrium point of the system. The ecological implications of such metastability, hysteresis, and threshold behavior for the occurrence of outbreaks, the persistence of oscillations, the resiliency of the system, and the biological control of mite populations are discussed.