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.