# American Institute of Mathematical Sciences

September  2021, 26(9): 5047-5066. doi: 10.3934/dcdsb.2020332

## Stochastic modelling and analysis of harvesting model: Application to "summer fishing moratorium" by intermittent control

 1 Department of Mathematics, Harbin Institute of Technology(Weihai), Weihai 264209, China 2 Department of Basic Course, Xingtai Polytechnic College, Xingtai 054000, China

* Corresponding author: Xiaoling Zou

Received  November 2019 Revised  June 2020 Published  September 2021 Early access  November 2020

As we all know, "summer fishing moratorium" is an internationally recognized management measure of fishery, which can protect stock of fish and promote the balance of marine ecology. In this paper, "intermittent control" is used to simulate this management strategy, which is the first attempt in theoretical analysis and the intermittence fits perfectly the moratorium. As an application, a stochastic two-prey one-predator Lotka-Volterra model with intermittent capture is considered. Modeling ideas and analytical skills in this paper can also be used to other stochastic models. In order to deal with intermittent capture in stochastic model, a new time-averaged objective function is proposed. Besides, the corresponding optimal harvesting strategies are obtained by using the equivalent method (equivalency between time-average and expectation). Theoretical results show that intermittent capture can affect the optimal harvesting effort, but it cannot change the corresponding optimal time-averaged yield, which are accord with observations. Finally, the results are illustrated by practical examples of marine fisheries and numerical simulations.

Citation: Xiaoling Zou, Yuting Zheng. Stochastic modelling and analysis of harvesting model: Application to "summer fishing moratorium" by intermittent control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5047-5066. doi: 10.3934/dcdsb.2020332
##### References:
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Teng, Exponential stabilization and synchronization of neural networks with time-varying delays via periodically intermittent control, Nonlinearity, 23 (2010), 2369-2391.  doi: 10.1088/0951-7715/23/10/002. [17] Z. Y. Huang, A comparison theorem for solutions of stochastic differential equations and its applications, Proc. Amer. Math. Soc., 91 (1984), 611-617.  doi: 10.1090/S0002-9939-1984-0746100-9. [18] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations, 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017. [19] B. Johnson, R. Narayanakumar, P. S. Swathilekshmi, R. Geetha and C. Ramachandran, Economic performance of motorised and non-mechanised fishing methods during and after-ban period in ramanathapuram district of tamil nadu, Indian J. Fish., 64 (2017), 160-165.  doi: 10.21077/ijf.2017.64.special-issue.76248-22. [20] G. B. Kallianpur, Stochastic differential equations and diffusion processes, Technometrics, 25 (1983), 208. doi: 10.1080/00401706.1983.10487861. [21] W. Li and K. Wang, Optimal harvesting policy for stochastic logistic population model, Appl. Math. Comput., 218 (2011), 157-162.  doi: 10.1016/j.amc.2011.05.079. [22] M. Liu and K. Wang, Dynamics of a two-prey one-predator system in random environments, J. Nonlinear Sci., 23 (2013), 751-775.  doi: 10.1007/s00332-013-9167-4. [23] O. Ovaskainen and B. Meerson, Stochastic models of population extinction, Trends Ecol. Evol., 25 (2010), 643-652.  doi: 10.1016/j.tree.2010.07.009. [24] N.-T. Shih, Y.-H. Cai and I.-H. Ni, A concept to protect fisheries recruits by seasonal closure during spawning periods for commercial fishes off taiwan and the east china sea, J. Appl. Ichthyol., 25 (2009), 676-685.  doi: 10.1111/j.1439-0426.2009.01328.x. [25] L. Wang, D. Jiang and G. S. K. Wolkowicz, Global asymptotic behavior of a multi-species stochastic chemostat model with discrete delays, J. Dyn. Differ. Equ., 32 (2020), 849-872.  doi: 10.1007/s10884-019-09741-6. [26] W. Xia and J. Cao, Pinning synchronization of delayed dynamical networks via periodically intermittent control, Chaos, 19 (2009), 013120, 8pp. doi: 10.1063/1.3071933. [27] B. Yang, Y. Cai, K. Wang and W. Wang, Optimal harvesting policy of logistic population model in a randomly fluctuating environment, Phys. A, 526 (2019), 120817, 17pp. doi: 10.1016/j.physa.2019.04.053. [28] Y. Ye, Assessing effects of closed seasons in tropical and subtropical penaeid shrimp fisheries using a length-based yield-per-recruit model, ICES J. Mar. Sci., 55 (1998), 1112-1124.  doi: 10.1006/jmsc.1998.0415. [29] C. Zhang, W. Li and K. Wang, Graph-theoretic method on exponential synchronization of stochastic coupled networks with Markovian switching, Nonlinear Anal-Hybri., 15 (2015), 37-51.  doi: 10.1016/j.nahs.2014.07.003. [30] G. Zhang and Y. Shen, Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control, Neural Networks, 55 (2014), 1-10.  doi: 10.1016/j.neunet.2014.03.009. [31] X. Zou and K. Wang, Optimal harvesting for a stochastic lotka-volterra predator-prey system with jumps and nonselective harvesting hypothesis, Optim. Control Appl. Methods., 37 (2016), 641-662.  doi: 10.1002/oca.2185. [32] X. Zou and K. Wang, Optimal harvesting for a stochastic n-dimensional competitive lotka-volterra model with jumps, Discrete Cont. Dyn-B, 20 (2015), 683-701.  doi: 10.3934/dcdsb.2015.20.683. [33] X. Zou, Y. Zheng, L. Zhang and J. Lv, Survivability and stochastic bifurcations for a stochastic Holling type II predator-prey model, Commun. Nonlinear Sci Numer. Simulat., 83 (2020), 105136, 20 pp. doi: 10.1016/j.cnsns.2019.105136.

show all references

##### References:
 [1] L. J. S. Allen and A. M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biocsi., 163 (2000), 1-33.  doi: 10.1016/S0025-5564(99)00047-4. [2] L. H. R. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Math. Biosci., 152 (1998), 63-85.  doi: 10.1016/S0025-5564(98)10018-4. [3] L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, Journal of Mathematical Biology, 37 (1998), 155-177.  doi: 10.1007/s002850050124. [4] I. Barbǎlat, Système d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. [5] J. Batsleer, A. D. Rijnsdorp, K. G. Hamon, H. M. J. van Overzee and J. J. Poos, Mixed fisheries management: Is the ban on discarding likely to promote more selective and fuel efficient fishing in the dutch flatfish fishery?, Fish Res., 174 (2016), 118-128.  doi: 10.1016/j.fishres.2015.09.006. [6] J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465.  doi: 10.1126/science.197.4302.463. [7] A. Bottaro, Y. Yasutake, T. Nomura, M. Casadio and P. Morasso, Bounded stability of the quiet standing posture: An intermittent control model, Hum. Movement Sci., 27 (2008), 473-495.  doi: 10.1016/j.humov.2007.11.005. [8] C. W. Clark, Mathematical Bioeconomics. The Optimal Management of Renewable Resources, , Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. [9] J. H. Connell, On the prevalence and relative importance of interspecific competition: Evidence from field experiments, Am. Nat., 122 (1983), 661-696.  doi: 10.1086/284165. [10] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511662829. [11] J.-M. Ecoutin, M. Simier, J.-J. Albaret, R. Laë, J. Raffray, O. Sadio and L. T. de Morais, Ecological field experiment of short-term effects of fishing ban on fish assemblages in a tropical estuarine mpa, Ocean Coastal Manage., 100 (2014), 74-85.  doi: 10.1016/j.ocecoaman.2014.08.009. [12] B. $\emptyset$ksendal, Stochastic Differential Equations, Springer-Verlag, 1985. doi: 10.1007/978-3-662-13050-6. [13] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation sis epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X. [14] Y. Guo, W. Zhao and X. Ding, Input-to-state stability for stochastic multi-group models with multi-dispersal and time-varying delay, Appl. Math. Comput., 343 (2019), 114-127.  doi: 10.1016/j.amc.2018.07.058. [15] S. Hong and N. Hong, H$^{\infty}$ switching synchronization for multiple time-delay chaotic systems subject to controller failure and its application to aperiodically intermittent control, Nonlinear Dyn., 92 (2018), 869-883. [16] C. Hu, J. Yu, H. Jiang and Z. Teng, Exponential stabilization and synchronization of neural networks with time-varying delays via periodically intermittent control, Nonlinearity, 23 (2010), 2369-2391.  doi: 10.1088/0951-7715/23/10/002. [17] Z. Y. Huang, A comparison theorem for solutions of stochastic differential equations and its applications, Proc. Amer. Math. Soc., 91 (1984), 611-617.  doi: 10.1090/S0002-9939-1984-0746100-9. [18] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations, 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017. [19] B. Johnson, R. Narayanakumar, P. S. Swathilekshmi, R. Geetha and C. Ramachandran, Economic performance of motorised and non-mechanised fishing methods during and after-ban period in ramanathapuram district of tamil nadu, Indian J. Fish., 64 (2017), 160-165.  doi: 10.21077/ijf.2017.64.special-issue.76248-22. [20] G. B. Kallianpur, Stochastic differential equations and diffusion processes, Technometrics, 25 (1983), 208. doi: 10.1080/00401706.1983.10487861. [21] W. Li and K. Wang, Optimal harvesting policy for stochastic logistic population model, Appl. Math. Comput., 218 (2011), 157-162.  doi: 10.1016/j.amc.2011.05.079. [22] M. Liu and K. Wang, Dynamics of a two-prey one-predator system in random environments, J. Nonlinear Sci., 23 (2013), 751-775.  doi: 10.1007/s00332-013-9167-4. [23] O. Ovaskainen and B. Meerson, Stochastic models of population extinction, Trends Ecol. Evol., 25 (2010), 643-652.  doi: 10.1016/j.tree.2010.07.009. [24] N.-T. Shih, Y.-H. Cai and I.-H. Ni, A concept to protect fisheries recruits by seasonal closure during spawning periods for commercial fishes off taiwan and the east china sea, J. Appl. Ichthyol., 25 (2009), 676-685.  doi: 10.1111/j.1439-0426.2009.01328.x. [25] L. Wang, D. Jiang and G. S. K. Wolkowicz, Global asymptotic behavior of a multi-species stochastic chemostat model with discrete delays, J. Dyn. Differ. Equ., 32 (2020), 849-872.  doi: 10.1007/s10884-019-09741-6. [26] W. Xia and J. Cao, Pinning synchronization of delayed dynamical networks via periodically intermittent control, Chaos, 19 (2009), 013120, 8pp. doi: 10.1063/1.3071933. [27] B. Yang, Y. Cai, K. Wang and W. Wang, Optimal harvesting policy of logistic population model in a randomly fluctuating environment, Phys. A, 526 (2019), 120817, 17pp. doi: 10.1016/j.physa.2019.04.053. [28] Y. Ye, Assessing effects of closed seasons in tropical and subtropical penaeid shrimp fisheries using a length-based yield-per-recruit model, ICES J. Mar. Sci., 55 (1998), 1112-1124.  doi: 10.1006/jmsc.1998.0415. [29] C. Zhang, W. Li and K. Wang, Graph-theoretic method on exponential synchronization of stochastic coupled networks with Markovian switching, Nonlinear Anal-Hybri., 15 (2015), 37-51.  doi: 10.1016/j.nahs.2014.07.003. [30] G. Zhang and Y. Shen, Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control, Neural Networks, 55 (2014), 1-10.  doi: 10.1016/j.neunet.2014.03.009. [31] X. Zou and K. Wang, Optimal harvesting for a stochastic lotka-volterra predator-prey system with jumps and nonselective harvesting hypothesis, Optim. Control Appl. Methods., 37 (2016), 641-662.  doi: 10.1002/oca.2185. [32] X. Zou and K. Wang, Optimal harvesting for a stochastic n-dimensional competitive lotka-volterra model with jumps, Discrete Cont. Dyn-B, 20 (2015), 683-701.  doi: 10.3934/dcdsb.2015.20.683. [33] X. Zou, Y. Zheng, L. Zhang and J. Lv, Survivability and stochastic bifurcations for a stochastic Holling type II predator-prey model, Commun. Nonlinear Sci Numer. Simulat., 83 (2020), 105136, 20 pp. doi: 10.1016/j.cnsns.2019.105136.
Numerical simulations for sample paths
Numerical simulations for time average
The effects of intermittent control in one-dimensional situation
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