Control issues for the Beverton–Holt equation in ecology by locally monitoring the environment carrying capacity: Non-adaptive and adaptive cases

Abstract

This paper proposes a control algorithm to govern the solution of the Beverton–Holt equation (BHE) under the potentially presence of additive disturbances. The BHE to be controlled is defined by certain intrinsic growth rate and environment carrying capacity sequences, the last one being susceptible of local modifications around nominal values. In fact, the control action provides the carrying capacity which makes that the solution of the current BHE tracks a reference sequence given by another BHE defined by appropriate intrinsic growth rate and environment carrying capacity sequences. In this context, the fact that the inverse of the BHE is a discrete time-varying linear system is taken into account where the inverse of the carrying capacity sequence plays the role of control sequence. The current and the reference BHEs have to be close enough to each other in order that local modifications of the carrying capacity be able to meet the tracking objective. A feedback control law is designed to achieve such an objective with a zero tracking-error in the ideal case of known intrinsic growth rate sequence and no presence of disturbances. An adaptive control law, with the associated parameter estimation algorithm, is considered when the intrinsic growth rate is fully or partially unknown and disturbances are present. Such a control strategy guarantees a bounded tracking-error with the error converging asymptotically to zero in case that additive disturbances also converge to zero. Some results obtained from a simulation example illustrate the effectiveness of this control strategy.