Optimal impulsive control in periodic ecosystem

In this paper, the impulsive exploitation of single species modelled by periodic Logistic equation is considered. First, it is shown that the generally periodic Kolmogorov system with impulsive harvest has a unique positive solution which is globally asymptotically stable for the positive solution. Further, choosing the maximum annual biomass yield as the management objective, we investigate the optimal harvesting policies for periodic logistic equation with impulsive harvest. When the optimal harvesting effort maximizes the annual biomass yield, the corresponding optimal population level, and the maximum annual biomass yield are obtained. Their explicit expressions are obtained in terms of the intrinsic growth rate, the carrying capacity, and the impulsive moments. In particular, it is proved that the maximum biomass yield is in fact the maximum sustainable yield (MSY). The results extend and generalize the classical results of Clark [Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976] and Fan [Optimal harvesting policy for single population with periodic coefficients, Math. Biosci. 152 (1998) 165-177] for a population described by autonomous or nonautonomous logistic model with continuous harvest in renewable resources.