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Optimal impulsive control in periodic ecosystem. (English) Zbl 1129.49308
Summary: 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 C. W. Clark [Mathematical bioeconomics. The optimal management of renewable resources. (1976; Zbl 0364.90002)] and M. Fan and K. Wang [Math. Biosci. 152, 165–177 (1998; Zbl 0940.92030)] for a population described by autonomous or nonautonomous logistic model with continuous harvest in renewable resources.

##### MSC:
 49N90 Applications of optimal control and differential games 92D40 Ecology
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##### References:
 [1] Agnew, T.T., Optimal exploitation of a fishery employing a non-linear harvesting function, Ecol. modelling, 6, 47-57, (1979) [2] Angelova, J.; Dishliev, A., Optimization problems for one-impulsive models from population dynamics, Nonlinear anal., 39, 483-497, (2000) · Zbl 0942.34010 [3] Artstein, Z., Chattering limit for a model of harvesting in a rapidly changing environment, Appl. math. optim., 28, 133-147, (1993) · Zbl 0795.90006 [4] D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66, 1993. · Zbl 0815.34001 [5] Bainov, D.D.; Simeonov, P.S., Systems with impulse effect, (1982), Ellis, Horwood Ltd. Chichester · Zbl 0661.34060 [6] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math. comput. modelling, 26, 59-72, (1997) · Zbl 1185.34014 [7] Berkovitz, L.D., Optimal control theory, (1974), Springer New York, Heidelberg, Berlin · Zbl 0295.49001 [8] Clark, C.W., Mathematical bioeconomics: the optimal management of renewable resources, (1976), Wiley New York · Zbl 0364.90002 [9] Y. Cohen, Application of optimal impulse control to optimal foraging problems, in: Applications of Control Theory in Ecology, Lecture Notes in Biomathematics, vol. 73, Springer, Berlin, 1987, pp. 39-56. [10] Fan, M.; Wang, K., Optimal harvesting policy for single population with periodic coefficients, Math. biosci., 152, 165-177, (1998) · Zbl 0940.92030 [11] Goh, B.S., Management and analysis of biological populations, (1980), Elsevier Scientific Publishing Company Amsterdam · Zbl 0453.92015 [12] Hirstova, S.C.; Bainov, D.D., Existence of periodic solutions of nonlinear systems of differential equations with impulsive effect, J. math. anal. appl., 125, 192-202, (1987) [13] L.S. Jennings, K.L. Teo, C.J. Goh, MISER3.2 Optimal Control Software: Theory and User Manual, Department of Mathematics, the University of Western Australia, Australia, 1997 $$\langle$$http://www.cado.uwa.edu.au/miser/⟩. [14] John, T.L., Variational calculus and optimal control, (1996), Springer New York [15] Lakmeche, A.; Arino, O., Bifurcation of non-trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynamics continuous, discrete impulsive systems, 7, 165-287, (2000) · Zbl 1011.34031 [16] Laksmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore [17] Liu, X., Impulsive stabilization and applications to population growth models, Rocky mountain J. math., 25, 381-395, (1995) · Zbl 0832.34039 [18] Liu, Y.; Teo, K.L.; Jennings, L.S.; Wang, S., On a class of optimal control problems with state jumps, J. optim. theory appl., 98, 65-82, (1998) · Zbl 0908.49023 [19] Panetta, J.C., A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment, Bull. math. biol., 58, 425-447, (1996) · Zbl 0859.92014 [20] Tang, S.Y.; Cheke, R.A., State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. math. biol., 50, 257-292, (2005) · Zbl 1080.92067 [21] Tang, S.Y.; Chen, L.S., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. math. biol., 44, 185-199, (2002) · Zbl 0990.92033 [22] Tang, S.Y.; Chen, L.S., Multiple attractors in stage-structured population models with birth pulses, Bull. math. biol., 65, 479-495, (2003) · Zbl 1334.92371
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