Abstract
This note is concerned with the optimal harvesting of a stochastic logistic model with time delay. The classical optimal harvesting question of this type of model is difficult because it is very difficult to obtain the explicit solution of the corresponding delay Fokker–Planck equation. The main aim of this note was to find a new approach to overcome this problem. In this note, using the ergodic method, sufficient and necessary criteria for the existence of optimal harvesting policy of our model are obtained. At the same time, the optimal harvesting effort and the maximum of harvesting yield are given. This method provides a new approach to study the optimal harvesting problem of stochastic population models, which can be also applied to investigate stochastic multi-species models.
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References
Alvarez, L.H.R., Shepp, L.A.: Optimal harvesting of stochastically fluctuating populations. J. Math. Biol. 37, 155–177 (1998)
Bahar, A., Mao, X.: Stochastic delay population dynamics. Int. J. Pure Appl. Math. 11, 377–400 (2004)
Bao, J., Hou, Z., Yuan, C.: Stability in distribution of neutral stochastic differential delay equations with Markovian switching. Stat. Probab. Lett. 79, 1663–1673 (2009)
Bao, J., Yuan, C.: Comparison theorem for stochastic differential delay equations with jumps. Acta Appl. Math. 116, 119–132 (2011)
Barbalat, I.: Systems dequations differentielles d’osci d’oscillations nonlineaires. Revue Roumaine de Mathematiques Pures et Appliquees 4, 267–270 (1959)
Beddington, J.R., May, R.M.: Harvesting natural populations in a randomly fluctuating environment. Science 197, 463–465 (1977)
Braumann, C.A.: Variable effort harvesting models in random environments: generalization to density-dependent noise intensities. Math. Biosci. 177 & 178, 229–245 (2002).
Bruti-Liberati, N., Platen, E.: Monte Carlo simulation for stochastic differential equations. Encycl. Quant. Financ. (2010). doi:10.1002/9780470061602.eqf13001
Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht (1992)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991)
Lande, R., Engen, S., Saeher, B.E.: Optimal harvesting of fluctuating populations with a risk of extinction. Am. Nat. 145, 728–745 (1995)
Li, W., Wang, K.: Optimal harvesting policy for general stochastic Logistic population model. J. Math. Anal. Appl. 368, 420–428 (2010)
Liu, M., Bai, C.: Optimal harvesting policy for a stochastic predator-prey model. Appl. Math. Lett. 34, 22–26 (2014)
Liu, M., Wang, K.: A remark on stochastic predator-prey system with time delays. Appl. Math. Lett. 26, 318–323 (2013)
Ludwig, D., Varah, J.M.: Optimal harvesting of a randomly fluctuating resource II: numerical methods and results. SIAM J. Appl. Math. 37, 185–205 (1979)
Lungu, E.M., Øksendal, B.: Optimal harvesting from a population in a stochastic crowded environment. Math. Biosci. 145, 47–75 (1997)
Maekey, M.C., Neehaeva, I.G.: Noise and stability in differential delay equations. J. Dynam. Diff. Equ. 6, 395–426 (1994)
Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)
Prato, D., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)
Ruan, S.: Delay differential equations in single species dynamics. In: Arino, O., et al. (eds.) Delay Differential Equations and Applications, pp. 477–517. Springer, New York (2006)
Song, Q.S., Stockbridge, R., Zhu, C.: On optimal harvesting problems in random environments. SIAM J. Control Optim. 49, 859–889 (2011)
Zou, X., Wang, K.: Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps. Nonlinear Anal. Hybrid Syst. 13, 32–44 (2014)
Acknowledgments
The authors thank the editor and reviewers for those valuable comments and suggestions. The authors also thank the Natural Science Foundation of PR China (Nos. 11301207, 11171081, 11271364, 11301112, 11401136), Natural Science Foundation of Jiangsu Province (Nos. BK2011407 and BK20130411), Natural Science Research Project of Ordinary Universities in Jiangsu Province (No. 13KJB110002), Qing Lan Project of Jiangsu Province (2014).
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Communicated by Philip K. Maini.
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Liu, M., Bai, C. Optimal Harvesting of a Stochastic Logistic Model with Time Delay. J Nonlinear Sci 25, 277–289 (2015). https://doi.org/10.1007/s00332-014-9229-2
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DOI: https://doi.org/10.1007/s00332-014-9229-2