Optimality and sustainability of delayed impulsive harvesting

Abstract

We consider a logistic differential equation subject to impulsive delayed harvesting, where the deduction information is a function of the population size at the time of one of the previous impulses. A close connection to the dynamics of high-order difference equations is used to conclude that while the inclusion of a delay in the impulsive condition does not impact the optimality of the yield, sustainability may be highly affected and is generally delay-dependent. Maximal and other types of yields are explored, and sharp stability tests are obtained for the model, as well as explicit sufficient conditions. It is also shown that persistence of the solution is not guaranteed for all positive initial conditions, and extinction in finite time is possible, as is illustrated in the simulations.