Stabilization with target oriented control for higher order difference equations

•We extend target oriented control to higher-order difference equations.•The introduced control can stabilize any prescribed point.•Sufficient conditions for local and global stabilization are presented.•Estimates for the stabilizing control intensities are given and tested numerically.•The results are applied to Pielou and Ricker delayed population models.

For a physical or biological model whose dynamics is described by a higher order difference equation un+1=f(un,un−1,…,un−k+1)un+1=f(un,un−1,…,un−k+1), we propose a version of a target oriented control un+1=cT+(1−c)f(un,un−1,…,un−k+1)un+1=cT+(1−c)f(un,un−1,…,un−k+1), with T≥0T≥0, c∈[0,1)c∈[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.