A two-agent model applied to the biological control of the sugarcane borer (Diatraea saccharalis) by the egg parasitoid Trichogramma galloi and the larvae parasitoid Cotesia flavipes.

The paper is aimed at a methodological development in biological pest control. The considered one pest two-agent system is modelled as a verticum-type system. Originally, linear verticum-type systems were introduced by one of the authors for modelling certain industrial systems. These systems are hierarchically composed of linear subsystems such that a part of the state variables of each subsystem affect the dynamics of the next subsystem. Recently, verticum-type system models have been applied to population ecology as well, which required the extension of the concept a verticum-type system to the nonlinear case. In the present paper the general concepts and technics of nonlinear verticum-type control systems are used to obtain biological control strategies in a two-agent system. For the illustration of this verticum-type control, these tools of mathematical systems theory are applied to a dynamic model of interactions between the egg and larvae populations of the sugarcane borer (Diatraea saccharalis) and its parasitoids: the egg parasitoid Trichogramma galloi and the larvae parasitoid Cotesia flavipes. In this application a key role is played by the concept of controllability, which means that it is possible to steer the system to an equilibrium in given time. In addition to a usual linearization, the basic idea is a decomposition of the control of the whole system into the control of the subsystems, making use of the verticum structure of the population system. The main aim of this study is to show several advantages of the verticum (or decomposition) approach over the classical control theoretical model (without decomposition). For example, in the case of verticum control the pest larval density decreases below the critical threshold value much quicker than without decomposition. Furthermore, it is also shown that the verticum approach may be better even in terms of cost effectiveness. The presented optimal control methodology also turned out to be an efficient tool for the "in silico" analysis of the cost-effectiveness of different biocontrol strategies, e.g. by answering the question how far it is cost-effective to speed up the reduction of the pest larvae density, or along which trajectory this reduction should be carried out.