# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020332
##### References:
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Nonlinear Sci., 23 (2013), 751-775.  doi: 10.1007/s00332-013-9167-4.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] B. Yang, Y. Cai, K. Wang and W. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] I. Barbǎlat, Système d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511662829.  Google Scholar [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.  Google Scholar [12] B. $\emptyset$ksendal, Stochastic Differential Equations, Springer-Verlag, 1985. doi: 10.1007/978-3-662-13050-6.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] G. B. Kallianpur, Stochastic differential equations and diffusion processes, Technometrics, 25 (1983), 208. doi: 10.1080/00401706.1983.10487861.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
Numerical simulations for sample paths
Numerical simulations for time average
The effects of intermittent control in one-dimensional situation
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