Abstract
For stochastic population models with periodic coefficients, traditional approaches used to study the optimal harvesting problems of autonomous stochastic models are invalid due to the inhomogeneity of the models. The aim of this report is to search for a workable way. By using the stochastic periodic solution of the model as a bridge, we obtain the explicit forms of the optimal harvesting effort and the maximum sustained yield for a periodic stochastic Gompertz model. This approach, which can also be used to research multi-species models, offers a possible way to explore the optimal harvesting of periodic stochastic population 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)
Beddington, J.R., May, R.M.: Harvesting natural populations in a randomly fluctuating environment. Science 197, 463–465 (1977)
Braumann, C.A.: Itô versus Stratonovich calculus in random population growth. Math. Biosci. 206, 81–107 (2007)
Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, NewYork (1976)
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)
Dong, L., Chen, L., Sun, L.: Optimal harvesting policies for periodic Gompertz systems. Nonlinear Anal. Real World Appl. 8, 572–578 (2007)
Duan, J.: An Introduction to Stochastic Dynamics. Science Press, Beijing (2015)
Evans, S.N., Ralph, P.L., Schreiber, S.J., Sen, A.: Stochastic population growth in spatially heterogeneous environments. J. Math. Biol. 66, 423–476 (2013)
Fan, M., Wang, K.: Optimal harvesting policy for single population with periodic coefficients. Math. Biosci. 152, 165–177 (1998)
Feng, C., Zhao, H.: Random periodic processes, periodic measures and ergodicity. J. Differ. Equ. 269, 7382–7428 (2020)
Hening, A., Nguyen, D.H., Ungureanu, S.C., Wong, T.: Asymptotic harvesting of populations in random environments. J. Math. Biol. 78, 293–329 (2019)
Hening, A., Tran, K., Phan, T., Yin, G.: Harvesting of interacting stochastic populations. J. Math. Biol. 79, 533–570 (2019)
Hu, G., Li, Y.: Asymptotic behaviors of stochastic periodic differential equation with Markovian switching. Appl. Math. Comput. 264, 403–416 (2015)
Jenkins, D., Watson, A., Miller, G.R.: Predation and red grouse populations. J. Appl. Ecol. 1, 183–195 (1964)
Jiang, D., Shi, N.: A note on nonautonomous logistic equation with random perturbation. J. Math. Anal. Appl. 303, 164–172 (2005)
Khasminskii, R.: Stochastic Stability of Differential Equations, 2nd edn. Springer, Berlin (2012)
Lande, R., Engen, S., Saether, B.: Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press, London (2003)
Li, W., Wang, K.: Optimal harvesting policy for general stochastic logistic population model. J. Math. Anal. Appl. 368, 420–428 (2010)
Liu, M.: Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays. IMA J. Appl. Math. 80, 1431–1453 (2015)
Liu, M., Bai, C.: Analysis of a stochastic tri-trophic food-chain model with harvesting. J. Math. Biol. 73, 597–625 (2016)
Liu, M., Bai, C.: Optimal harvesting of a stochastic mutualism model with regime-switching. Appl. Math. Comput. 373, 125040 (2020)
Liu, M., He, X., Yu, J.: Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays. Nonlinear Anal. Hybrid Syst. 28, 87–104 (2018)
Murton, R.K., Westwood, N.J., Isaacson, A.J.: A study of wood-pigeon shooting: the exploitation of a natural animal population. J. Appl. Ecol. 11, 61–81 (1974)
Qiu, H., Deng, W.: Optimal harvesting of a stochastic delay logistic model with Lévy jumps. J. Phys. A 49, 405601 (2016)
Qiu, H., Deng, W.: Optimal harvesting of a stochastic delay competitive Lotka-Volterra model with Lévy jumps. Appl. Math. Comput. 317, 210–222 (2018)
Tran, K., Yin, G.: Numerical methods for optimal harvesting strategies in random environments under partial observations. Automatica 70, 74–85 (2016)
Troutman, J.L.: Variational Calculus and Optimal Control: Optimization with Elementary Convexity. Springer, New York (1996)
Turelli, M.: Random environments and stochastic calculus. Theor. Popul. Biol. 12, 140–178 (1977)
Xu, C., Boyce, M.S., Daley, D.J.: Harvesting in seasonal environments. J. Math. Biol. 50, 663–682 (2005)
Zeng, Z.: Asymptotically periodic solution and optimal harvesting policy for Gompertz system. Nonlinear Anal. Real World Appl. 12, 1401–1409 (2011)
Zou, X., Zheng, Y.: Stochastic modelling and analysis of harvesting model: application to “summer fishing moratorium” by intermittent control. Discrete Contin. Dyn. Syst. Ser. B 26, 5047–5066 (2021)
Zou, X., Li, W., Wang, K.: Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process. Appl. Math. Lett. 26, 170–174 (2013)
Zu, L., Jiang, D., O’Regan, D., Ge, B.: Periodic solution for a non-autonomous Lotka-Volterra predator-prey model with random perturbation. J. Math. Anal. Appl. 430, 428–437 (2015)
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The author thanks the anonymous referees for their careful reading and valuable comments.
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The author thanks the National Natural Science Foundation of P.R. China (No. 11771174).
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Communicated by Philip K. Maini.
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Liu, M. Optimal Harvesting of Stochastic Population Models with Periodic Coefficients. J Nonlinear Sci 32, 23 (2022). https://doi.org/10.1007/s00332-021-09758-6
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DOI: https://doi.org/10.1007/s00332-021-09758-6