Dynamic properties of a class of forced positive higher-order scalar difference equations: persistency, stability and convergence

Persistency, stability and convergence properties are considered for a class of nonlinear, forced, positive, scalar higher-order difference equations. Sufficient conditions for these properties to hold are derived, broadly in terms of the interplay of the linear and nonlinear components of the difference equations. The convergence results presented include asymptotic response properties when the system is subject to (asymptotically) almost periodic forcing. The equations under consideration arise in a number of ecological and biological contexts, with the Allen-Clark population model appearing as a special case. We illustrate our results by several examples from population dynamics.