ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Effect of External Wastage and Illegal Harvesting on the Fishery Model of the Halda River Ecosystem in Bangladesh

Journal of Applied Nonlinear Dynamics 11(1) (2022) 33--56 | DOI:10.5890/JAND.2022.03.003

Md. Nazmul Hasan$^1$ , Md. Sharif Uddin$^1$, Md. Haider Ali Biswas$^2$

$^{1}$ Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh

$^2$ Mathematics Discipline, Khulna University, Khulna, Bangladesh

Abstract

The Halda, a 98-kilometre long major tributary of Karnaphuli River in the Chattogram Hill Tracts, is the only source of naturally fertilized eggs of carp fishes in South Asia and a great contributor to Bangladesh fisheries sector. Waste from large factories, Hathazari Peaking Power Plant and a housing estate are polluting the water body of Halda river to such an extent that the indigenous sweet-water brood fishes are facing death and the quantity of their release of carp spawn is decreasing. The present paper examines a predator-prey fishery system by taking into account the toxin waste which can lead to polluted system. Both fish species obey the logistic population growth with their respective environmental carrying capacities. The equilibria existed in the model are investigated together with the local and global stability. Bifurcation diagrams are studied to examine the dynamical behaviors of the system. Bionomic equilibria, optimal harvesting policy and Optimal Control Theory are applied to reduce the external toxic substance. Finally, a numerical simulation of the model has been discussed to illustrate the effect of toxicity and their control upon both the predator and the prey species.

References

1.  [1] Hallam, T.G. and Clark, C.E. (1983), Effect of toxicants on populations: A qualitive approach I. Equilibrium environmental exposure, Ecological Modeling., 18, 291-304.
2.  [2] Hallam, T.G. and De Luna, J.T. (1984), Effect of toxicants on populations: A qualitive approach I. Equilibrium environmental and food chain pathways, J. Theor. Bio. Comput. Biol. Med., 109 411-429.
3.  [3] Freedman, H.I. and Shukla, J.B. (1991), Models for the effect of toxicant in single-species and predator-prey systems, J. Math. Biol., 30, 15-30.
4.  [4] Chattopadhyay, J. (1996), Effect of toxic substances on a two-species competitive system, Ecological Modelling., 84, 287-289.
5.  [5] Huang, Q., Wang, H., and Lewis, M, A. (2015), The impect of environmental toxins on predator-prey dynamics, J. Theor. Biol., 378, 12-20.
6.  [6] Huang, Q., Parshotam, L., Wang, H., and Lewis, M.A. (2013), A model for the impact of contaminants on fish population dynamics, J. Theor. Biol., 334, 71-79.
7.  [7] Scott, G.R. and Sloman, K.A. (2004), Effect of environmental polluntant on complex fish behaviour: integrating behavioural and physiological indicators of toxicity, Aquatic Toxicology, 68 369-392.
8.  [8] Kar, T.K. and Chaudhuri, K.S.J. (2003), On non-selective harvesting of two competing fish species in the presence of toxicity, Ecological Modelling., 161, 125-137.
9.  [9] Das, T., Mukherjee, R.N., and chaudhuri, K.S. (2009), Harvesting of a predator-prey fishery in the presence of toxicity, Applied Mathematical Modelling, 33, 2282-2292.
10.  [10] Crawford, J.D. (1991), Introduction to bifurcation theory, Rev. Mod. Phys., 63(4), 991-1037.
11.  [11] Katsukawa, T. (2004), A numerical investigation of the optimal control rule for decision-making in fisheries management, fish sci., 70, 123-131.
12.  [12] Kotani, K., Kakinaka, M., and Matsuda, H. (2008), Optimal escapement levels on renewable resource management under process uncertainty: Some implications of convex unit harvest cost, Evn. Econ. Pol.Sc., 9, 107-118.
13.  [13] Hasan, N., Biswas, H.A., and Uddin, S. (2019), An Ecological model for sustainable wildlife management of ecosystem based on optimal control theory, Commun. Math. Biol. Neurosci., 17.
14.  [14] Islam, M.S., Akbar, A., Akhtar, A., Kibria, M.M., and Bhuyan, M.S. (2017), Water Quality Assessment Along With Pollution Sources of the Halda River, Journal of the Asiatic Society of Bangladesh, Science, 61-70.
15.  [15] Clark, C.W. (1976), Mathematical Bioeconomics: The optimal management of renewable resources, Wiley. Princeton. Univ. Press, New York.
16.  [16] Biswas, M.H.A. (2011), A Necessary conditions for optimal control problems with and without state constraints: a comparative study, WSEAS Transactions on Systems and Control, 6(6), 217-228.
17.  [17] Biswas, M.H.A. and de Pinho, M.D.R. (2011), A Nonsmooth Maximum Principle for Optimal Control Problems with State and Mixed Constraints-Convex Case, Discrete and Continuous Dynamical Systems, (Special), 174-183.
18.  [18] Biswas, M.H.A. and de Pinho, M.D.R. (2012), A variant of nonsmooth maximum principle for state constrained problems, IEEE 51st IEEE Conference on Decision and Control (CDC), 7685-7690.
19.  [19] Biswas, M.H.A. and Haque, M.M. (2016), Nonlinear Dynamical systems in Modeling and Control of Infectious Disease, Book chapter of differential and difference equations with applications, 164, 149-158.
20.  [20] Lenhart, S. and Workman, J. (2007), Optimal Control Applied to Biological Models, Boca Raton, Chapmal Hall/CRC.
21.  [21] Boyce, W.E., DiPrima, R.C., and Meade, D.B. (2017), Elementary differential equations, 11 th Edition, John Wiley, New York.
22.  [22] Lukes, D.L. (1982), Differential Equations: Classical to Controlled, Academic Press, New York.
23.  [23] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelize, R.V., and Mishchenko, E.F. (1967), The Mathematical Theory of Optimal Processes, Wiley, New York.
24.  [24] Fleming, W. and Rishel, R. (2005), Deterministic and Stochastic Optimal Control, Springer-Verlag, New York.
25.  [25] Ang, T.K., Safuan, H.M., and Kavikumar, J. (2018), Dynamical behavior of prey-predator fishery model wih th harvesting affected by toxic substances, MATHEMATIKA., 34(1), 143-151.
26.  [26] Hasan, N., Biswas, H.A., and Uddin, S. (2019), An Ecological Model for Sustainable Forest Management of Eco-system Based on Optimal Control Theory, Journal of Nepal Mathematical Society (JNMS), 2(1).
27.  [27] Hale, J. (1989), Ordinary differential equation, Klieger Publising Company, Malabar.
28.  [28] Mortoza, S.G., Panja, P., and Mondal, S.K. (2018), Dynamics of apredator prey model with stage-structure on both species and anti-predator behavior, Informatics in Medicine Unlocked, 10, 50-57.
29.  [29] Keong, A.T., Safuan, H.M., and Jacob, K. (2018), Dynamical behaviors of prey-predator fishery model with harvesting affected by toxic substances, MATEMATIKA: Malaysian Journal of Industrial and Applied Mathematics, 34(1), 143-151.
30.  [30] Satar, H.A. and Naji, R.K.(2019), Stability and Bifurcation of a Prey-PredatorScavenger Model in the Existence of Toxicant and Harvesting. International Journal of Mathematics and Mathematical Sciences, 1573516.
31.  [31] Rni, R. and Gakkhar, S. (2019), The impact of provision of additional food to predator in predator-prey model with combined harvesting gin the presence of toxicity, Journal of Applied Mathematics and Computing, 60(1-2), 673-701.
32.  [32] Ang, T.K., Safuan, H.M., and Kavikumar, J. (2018), The impacts of harvesting activities on prey-predator fishery model in the presence of toxin, Journal of Science and Technology, 10(2), 128-135.
33.  [33] Majeed, A.A. (2018), The dynamics of an ecological model involving stage structures in both populations with toxin, Journal of Advanced Research in Dynamical and Control System, 10 (13), 1120-1131.
34.  [34] Matia, S.N. and Alam, S. (2013), Prey-predator dynamics under herd behavior of prey, Univ. J. Appl. Math, 1(4), 251-257.
35.  [35] Cheng, K.S. (1981), Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12(4), 541-548.
36.  [36] Hsu, S.B. (1977), On global stability of a predator-prey system, Math. Biosci., 39, 1-10.
37.  [37] Hsu, S.B. and Huang, T.W. (1995), Global stability for a class of predator-prey system, SIAM J. Appl. Math., 55, 763-783.
38.  [38] Sugie, J., Kohno, R., and Miyazaki, R. (1997), On a predator-prey system of Holling type, Proc. Am. Math. Soc., 125, 2041-2050.