Controlling one-dimensional unimodal population maps by harvesting at a constant rate

Abstract

Controlling chaos and periodic oscillations in dynamical systems is a well known problem. It has been studied by several types of algorithms, some of which were only demonstrated numerically with specific examples. We study here a class of one-dimensional discrete maps of the form ${x}_{n+1}{=f(x}_{n}),$ where $f(x)$ is a unimodal function satisfying a few natural conditions as a model for the dynamics of a single species population. For managing the population we seek to suppress any possible chaotic or periodic behavior that may emerge. The paper proposes a simple, rigorously proved algorithm for controlling unimodal maps, implemented by harvesting the population at a constant rate. This forces the orbits of the map to converge to a limit that we can compute a priori. The result holds for any initial conditions within an interval that we specify. Our control algorithm is easy to implement, requires no updated information on the population and no changes in the parameters of the system, which, in general, are fixed by the properties of the population. It can therefore be useful for exploiting a population while maintaining it at a fixed density.