Optimal Harvesting of Structured Populations*t

A general harvesting model is presented that allows the dynamics of the population to be divided into two distinct phases, viz, the harvesting-season dynamics, modeled by an n-dimensional system of ordinary nonlinear differential equations, and the spawning season, modeled by n difference equations. A maximum principle for this type of system is presented. The concept of “maximum sustainable yield” for periodic forms of such systems is introduced and discussed. The model is then simplified to exhibit linear-bilinear dynamics and in this form is shown to be a natural extension of the Beverton-Holt model in fisheries management. A method for deriving maximum-sustainable-yield solutions is presented, using this formulation and its corresponding maximum principle. By considering the solution in the limit as the control constraint set [0, b] becomes unbounded above, the concept of the “ultimate” sustainable yield is introduced. Finally, models including only scalar harvesting are introduced as being of practical value. The question is explored as to how maximum-sustainable-yield solutions are to be constructed from the maximum principle.