Mathematical analysis of a model for a plant-herbivore system

The apple twig borer (Amphicerus bicaudatus) is an insect pest of the grape vine, causing considerable damage to the grape vine in early spring. A simple difference equation model is formulated and analysed for this plant-herbivore system based on two control strategies, cane removal and pesticide application. The system has two equilibria, one where the pest is present and one where the pest is absent. Regions are found in parameter space for global stability of the equilibria and in the absence of global stability it is shown that there exist periodic or quasiperiodic solutions.