The generalized Beverton–Holt equation and the control of populations

Abstract

This paper is devoted to the investigation of the positivity, stability and control of the solutions of a generalized Beverton–Holt equation arising in population dynamics which is potentially subject to bounded discontinuities at sampling instants due to the harvesting (i.e. fishing/hunting) quota and eventual independent consumption. Other generalizations are that the intrinsic growth rate and the environment carrying capacity gains are allowed to be time-varying sequences. The interpretation of the appearance of discontinuities in the solution is the presence of impulsive terms in the corresponding continuous-time differential equation. The parallel interpretation in ecology is that there are two different recruitment levels at each current sampling time due, firstly, to the evolution of the population driven by its intrinsic growth rate and the environment carrying capacity and, subsequently, a second level arises due to harvesting and independent consumption from the existing spawning stock of the population. The “left” recruitment level occurs immediately before each current sampling time while the “right” one occurs just after related to the sampling period size. By this reason, the mathematical formulation presented distinguishes between “left” and “right” sides of the sampling times. The control actions on the population stock and recruitment might be performed in a direct fashion either through the environment carrying capacity in close habitats gains or the harvesting quota in open air environments. Both actions may be combined in some open air habitats subject to some artificial control.