A Control Theory point of view on Beverton–Holt equation in population dynamics and some of its generalizations

• Published in: Applied mathematics and computation Vol. 199; no. 2; pp. 464 - 481
• Main Authors: De la Sen, M., Alonso-Quesada, S.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01-06-2008; Elsevier
• Subjects: Algebra; Algebraic geometry; Beverton–Holt equation; Combinatorics; Combinatorics. Ordered structures; Control systems; Controllability; Designs and configurations; Difference equations; Exact sciences and technology; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Ordinary differential equations; Sciences and techniques of general use; Stability; Controllability; Control systems; Stability; Difference equations; Beverton–Holt equation; Transcendental equation; Growth rate; Recurrence relation; Numerical linear algebra; Control system; Discrete system; Difference equation; Attractor; Equation system; Direct method; Beverton-Holt equation; Numerical analysis; Linear system; Matrix inversion; Applied mathematics; Population dynamics; Control theory; Numerical stability
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2007.10.021

This paper is devoted to develop some “ad hoc” Control Theory formalism useful for the famous Beverton–Holt equation arising in population dynamics. In particular, the inverse equation is redefined for a finite set of consecutive samples under the equivalent form of a discrete linear dynamic system whose input sequence is defined by the sequence of carrying capacity gains and the unforced dynamics is directly related to the intrinsic growth rate. For that purpose, the environment carrying capacity gains are allowed to be time-varying and designed for control purposes. The controllability property is also investigated on this dynamic extended system as well as the stability, equilibrium points and attractor oscillating trajectories. The properties of the dynamic system associated with the Beverton–Holt inverse equation allow extrapolate in a simple dual way the above properties to the standard Beverton–Holt equation. Some generalizations are given for the case when there are extra parameters in the equation or when the system is subject to the presence of additive disturbances. In all cases, a reference model being also of Beverton–Holt type is proposed to be followed by the control system.