For a Depensatory Fishery System Hybrid Modeling and Optimal Control of Harvest Policies

Year 2025, Volume: 8 Issue: 1, 1 - 6

Nurgul Gökgöz , Oğuzhan Çifdalöz

https://doi.org/10.33187/jmsm.1595004

Abstract

Recent decades have brought an increasing concern for the sustainability of renewable resources, such as agricultural land, freshwater, forests, and fisheries. Management and control of them have been conducted through some institutions and governments, which mainly focus on efficiently managing those resources since they are affected by social and ecological uncertainties like climate change, difficulty in the application strategies, or uncertainties and noise in the data collection. Control engineering procedures represent a flexible and reasonable way to investigate and solve the difficulties, uncertainties, and noise listed above by formulating the problem mathematically.

In this work, we investigate fisheries and revenue optimization by using a hybrid model. The harvest of the fishery is done during some seasons of the year, which suggests that the model should include both discrete and continuous dynamics. To investigate the bio-economic system, the problem is formulated by two-hybrid dynamical fishery models. Those formulations are used to investigate optimal control and the stability of the sustainability of the system. In this respect, we investigate the optimal effort for the maximization of the revenue where the continuation of sustainability is preserved. Moreover, which parameters should be taken into account to check the stability in this case are determined. Whenever the system is unstable, the optimal effort for the sustainability of the system is determined.

Keywords

Bio-economic model, Fishery, Hybrid dynamical systems, Optimal control, Stability

References

• [1] D. S. Powlson, P. J. Gregory, W. R. Whalley, J. N. Quinton, D. W. Hopkins, A. P. Whitmore, P. R. Hirsch, K. W. T. Goulding, Soil management in relation to sustainable agriculture and ecosystem services, Food Policy, 36 (2011), 72–87.
• [2] P. Jules, Agricultural sustainability: Concepts, principles and evidence, Philos. Trans. R. Soc. B, 363 (2007), 447–465.
• [3] N. J. Cunniffe, R. C. Cobb, R. K. Meentemeyer, D. M. Rizzo, C. A. Gilligan, Modeling when, where, and how to manage a forest epidemic, motivated by sudden oak death in California, P. Natl. Acad. Sci. USA, 113(20), (2016), 5640–5645.
• [4] E. F. Moran, E. Ostrom, Seeing the Forest and the Trees: Human-Environment Interactions in Forest Ecosystems, The MIT Press, Massachusetts, 2005.
• [5] P. H. Gleick, Transitions to freshwater sustainability, P. Natl. Acad. Sci. USA, 115(36) (2018), 8863–8871.
• [6] R. A. Pauloo, A. Escriva-Bou, H. Dahlke, A. Fencl, H. Guillon, G. E. Fogg, Domestic well vulnerability to drought duration and unsustainable groundwater management in California’s central valley, Environ. Res. Lett., 15(4) (2020), 044010.
• [7] C. W. Clark, Fisheries bioeconomics: why is it so widely misunderstood?, Popul. Ecol., 48(2) (2006), 95–98.
• [8] G. Sethi, C. Costello, A. Fisher, M. Hanemann, Fishery management under multiple uncertainty, J. Environ. Econ. Manag., 50(2), (2005), 300–318.
• [9] C. Mullon, P. Freon, P. Cury, The dynamics of collapse in world fisheries, Fish Fish., 6(2) (2005), 111–120.
• [10] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2012.
• [11] C. W. Clark, Mathematical Bioeconomics, John Wiley & Sons, New York, 1990.
• [12] G. I. Bischi, F. Lamantia, D. Radi, A prey–predator fishery model with endogenous switching of harvesting strategy, Appl. Math. Comput., 219(20), (2013), 10123–10142.
• [13] E. L. Councill, Incorporating stochasticity in the study of exploited fish population dynamics: Implications for the study of post-recruitment harvest strategies, Math. Biosci., 273 (2016), 16–22.
• [14] A. S. Crepin, J. Norberg, K.-G. Maler, Coupled economic-ecological systems with slow and fast dynamics - modelling and analysis method, Ecol. Econ., 70, (2011), 1448–1458.
• [15] Y. Pei, M. Chen, X. Liang, C. Li, Model-based on fishery management systems with selective harvest policies, Math. Comput. Simulat., 156 (2019), 377–395.
• [16] O. Çifdalöz, Sustainable management of a renewable fishery resource with depensation dynamics from a control systems perspective, Gazi Univ. J. Sci., 35(3) (2022), 936–955.
• [17] J. M. Anderies, A. A. Rodriguez, M. A. Janssen, O. Cifdaloz, Panaceas, uncertainty, and the robust control framework in sustainability science, P. Natl. Acad. Sci. USA, 104(39) (2007), 1294–1298.
• [18] D. Das, T. K. Kar, Feedback control and its impact on generalist predator-prey system with prey harvesting, Nonlinear Anal-Model., 24(5) (2019), 718–732.
• [19] D. De Giovanni, F. Lamantia, Dynamic harvesting under imperfect catch control, J. Optim. Theory Appl., 176(1) (2018), 252–267.
• [20] J. M. Anderies, M. A. Janssen, Robustness of social-ecological systems: implications for public policy, Policy Stud. J., 41(3) (2013), 513–536.
• [21] M. L. Schoon, M. E. Cox, Understanding disturbances and responses in social-ecological systems, Soc. Nat. Resour., 25(2) (2012), 141–155.
• [22] M. S. Branicky, Stability of switched and hybrid systems, Proceedings of 33rd IEEE Conference on Decision and Control, 4 (1994), 3498–3503.
• [23] M. S. Branicky, S. K. Mitter, Algorithms for optimal hybrid control, Proceedings of the 34th Conference on Decision & Control, 3 (1995), 2661–2666.
• [24] R. Goebel, R. G. Sanfelice, A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability, and Robustness, Princeton University Press, Princeton, 2009.
• [25] D. Liberzon, Switching in Systems and Control, Birkhauser, 2003.
• [26] Q. Lin, R. Loxton, K. L. Teo, Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dyn. Contin. Discrete Impuls. Syst. B: Appl. Algorithms, 18 (2011), 59–76.
• [27] Q. Lin, R. Loxton, K. L. Teo, Optimal control of nonlinear switched systems: Computational methods and applications, J. Oper. Res. Soc. China, 1 (2013), 275–311.
• [28] G. I. Bischi, F. Lamantia, F. Tramontana, Sliding and oscillations in fisheries with on–off harvesting and different switching times, Commun. Nonlinear Sci. Numer. Simul., 19(1) (2014), 216–229.
• [29] D. Radi, F. Lamantia, T. Tichy, Hybrid dynamics of multi-species resource exploitation, Decis. Econ. Finance, 44 (2021), 559–577.
• [30] L. Huang, J. Wang, Global dynamics of piecewise smooth systems with switches depending on both discrete times and status, SIAM J. Appl. Dyn. Syst., 23(3) (2024), 2533–2556.
• [31] N. Gökgöz Küçüksaksakallı, Stability and Optimal Control of Hybrid Models of Fishery, Msc. Thesis, Cankaya University, Ankara, 2022.
• [32] V. L. Smith, On models of commercial fishing, J. Pol. Econ., 77, (1969), 181–198.
• [33] N. Gökgöz, H. Öktem, Modeling of tumor-immune system interaction with stochastic hybrid systems with memory: A piecewise linear approach, Adv. Theory Nonlinear Anal. Appl., 5(1) (2021), 25–38.
• [34] F. Yılmaz, H. Öz Bakan, G. W. Weber, Strong-order conditions of runge-kutta method for stochastic optimal control problems, Appl. Numer. Math., 157 (2020), 470–489.
• [35] E. Kropat, G. W. Weber, S. Z. Alparslan-Gök, A. Özmen, Inverse problems in complex multi-modal regulatory networks based on uncertain clustered data, in: A. A. Pinto, D. Zilberman (Editors), Modeling, Dynamics, Optimization and Bioeconomics I, Springer International Publishing, 2014, pp. 437–451.
• [36] A. Çevik, G.-W. Weber, B. M. Eyüboğlu, K. Karlı-Oğuz, Voxel-mars: A method for early detection of Alzheimer’s disease by classification of structural brain MRI, Ann. Oper. Res., 258 (2017), 31–57.
• [37] T. Ewertowski, B. Ç. Güldoğuş, S. Kuter, S. Akyüz, G. W. Weber, J. Sadlowska-Wrzesinska, E. Racek, The use of machine learning techniques for assessing the potential of organizational resilience, Cent. Eur. J. Oper. Res., 32 (2024), 685–710.