Stabilization with target oriented control for higher order difference equations

TL;DR

This paper introduces a target-oriented control method for higher order difference equations, enabling stabilization of fixed points in ecological and biological models, including pest control with delayed density dependence.

Contribution

It proposes a novel control approach that stabilizes fixed points in higher order difference equations, applicable to ecological models and pest management.

Findings

Method stabilizes fixed points in a wide class of models

Applicable to ecological systems with harvesting and recruitment

Effective for pest control with delayed density dependence

Abstract

For a physical or biological model whose dynamics is described by a higher order difference equation $u_{n+1}=f(u_n,u_{n-1}, \dots, u_{n-k+1})$, we propose a version of a target oriented control $u_{n+1}=cT+(1-c)f(u_n,u_{n-1}, \dots, u_{n-k+1})$, with $T\ge 0$, $c\in [0,1)$. In ecological systems, the method incorporates harvesting and recruitment and for a wide class of $f$, allows to stabilize (locally or globally) a fixed point of $f$. If a point which is not a fixed point of $f$ has to be stabilized, the target oriented control is an appropriate method for achieving this goal. As a particular case, we consider pest control applied to pest populations with delayed density-dependence. This corresponds to a proportional feedback method, which includes harvesting only, for higher order equations.