Impact of Harvesting on a Bioeconomic Predator–Prey Fishery Model Subject to Environmental Toxicant

  • Tau Keong Ang
  • Hamizah M. SafuanEmail author
  • Harvinder S. Sidhu
  • Zlatko Jovanoski
  • Isaac N. Towers


The present paper studies a predator–prey fishery model which incorporates the independent harvesting strategies and nonlinear impact of an anthropogenic toxicant. Both fish populations are harvested with different harvesting efforts, and the cases for the presence and non-presence of harvesting effort are discussed. The prey fish population is assumed to be infected by the toxicant directly which causes indirect infection to predator fish population through the feeding process. Each equilibrium of the proposed system is examined by analyzing the respective local stability properties. Dynamical behavior and bifurcations are studied with the assistance of threshold conditions influencing the persistence and extinction of both predator and prey. Bionomic equilibrium solutions for three possible cases are investigated with certain restrictions. Optimal harvesting policy is explored by utilizing the Pontryagin’s Maximum Principle to optimize the profit while maintaining the sustainability of the marine ecosystem. Bifurcation analysis showed that the harvesting parameters are the key elements causing fishery extinction. Numerical simulations of bionomic and optimal equilibrium solutions showed that the presence of toxicant has a detrimental effect on the fish populations.


Predator–prey Harvesting Toxicant Biotechnical productivity Bionomic equilibrium Optimal harvesting policy 



Authors thank the anonymous reviewers for their critical comments and helpful suggestions to improve the quality and presentation of this work. The present research is supported by Research Management Centre (RMC) University Tun Hussein Onn Malaysia (Postgraduate Research Grant Code: U992) and Incentive Grant Scheme For Publication (U677).


  1. Arrow KJ, Kurz M (1970) Public investment, the rate of return and optimal fiscal policy. John Hopfkins, BaltimoreGoogle Scholar
  2. Barnes B, Sidhu HS (2013) The impact of marine closed areas on fishing yield under a variety of management strategies and stock depletion levels. Ecol Model 269:113–125CrossRefGoogle Scholar
  3. Barnett S (1971) A new formulation of the theorems of Hurwitz, Routh and Sturm. IMA J Appl Math 8:240–250MathSciNetCrossRefzbMATHGoogle Scholar
  4. Berryman AA (1992) The origins and evolution of predator–prey theory. Ecology 73:1530–1535CrossRefGoogle Scholar
  5. Botsford LW, Micheli F, Hastings A (2003) Principles for the design of marine reserves. Ecol Appl 13:S25–S31CrossRefGoogle Scholar
  6. Breen M, Graham N, Pol M, He P, Reid D, Suuronen P (2016) Selective fishing and balanced harvesting. Fish Res 184:2–8CrossRefGoogle Scholar
  7. Chakraborty K, Das S, Kar TK (2011) Optimal control of effort of a stage structured prey–predator fishery model with harvesting. Nonlinear Anal Real Wolrd Appl 12:3452–3467MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chakraborty S, Pal S, Bairagi N (2012) Predator–prey interaction with harvesting: mathematical study with biological ramifications. Appl Math Model 36:4044–4059MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dai G, Tang M (1998) Coexistence region and global dynamics of a harvested predator–prey system. SIAM J Appl Math 58:193–210MathSciNetCrossRefzbMATHGoogle Scholar
  10. Das T, Mukherjee RN, Chaudhuri KS (2009) Harvesting of a prey–predator fishery in the presence of toxicity. Appl Math Model 33:2282–2292MathSciNetCrossRefzbMATHGoogle Scholar
  11. Ermentrout B (2010) XPPAUT 6:00Google Scholar
  12. FAO (2016) The State of World Fisheries and Aquaculture 2016 (SOFIA): contributing to food security and nutrition for all, 200. Food and Agriculture Organization, RomeGoogle Scholar
  13. Gaines SD, White C, Carr MH, Palumbi SR (2010) Designing marine reserve networks for both conservation and fisheries management. Proc Natl Acad Sci USA 107:18286–18293CrossRefGoogle Scholar
  14. Ganguli C, Kar TK, Mondal PK (2017) Optimal harvesting of a prey–predator model with variable carrying capacity. Int J Biomath 10:1750069MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hu D, Cao H (2017) Stability and bifurcation analysis in a predator–prey system with Michaelis–Menten type predator harvesting. Nonlinear Anal Real World Appl 33:58–82MathSciNetCrossRefzbMATHGoogle Scholar
  16. Huang FC, Xiao DM (2004) Analyses of bifurcations and stability in a predator–prey system with Holling Type-IV functional response. Acta Mathematicae Applicatae Sinica, English Series 20:167–178MathSciNetCrossRefzbMATHGoogle Scholar
  17. Huang Q, Parshotam L, Wang H, Bampfylde C, Lewis MA (2013) A model for the impact of contaminants on fish population dynamics. J Theor Biol 334:71–79MathSciNetCrossRefzbMATHGoogle Scholar
  18. Huang J, Ruan S, Song J (2014) Bifurcations in a predator–prey system of Leslie type with generalized Holling Type-III functional response. J Differ Equ 257:1721–1752MathSciNetCrossRefzbMATHGoogle Scholar
  19. Huang Q, Wang H, Lewis MA (2015) The impact of environmental toxins on predator–prey dynamics. J Theor Biol 378:12–30MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kar TK (2003) Selective harvesting in a prey–predator fishery with time delay. Math Comput Model 38:449–458MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kar TK (2005) Stability and optimal harvesting of a prey–predator model with stage structure for predator. Appl Math 32:279–291MathSciNetzbMATHGoogle Scholar
  22. Kar TK, Chaudhuri KS (2003) On non-selective harvesting of two competing fish species in the presence of toxicity. Ecol Model 161:125–137CrossRefGoogle Scholar
  23. Maplesoft (2008) Maplesoft v. 16, Maplesoft, Waterloo, Ontario, CanadaGoogle Scholar
  24. MathWorks (2014) MATLAB v. R2015a, The MathWorks, USAGoogle Scholar
  25. Neubert MG (2003) Marine reserves and optimal harvesting. Ecol Lett 6:843–849CrossRefGoogle Scholar
  26. Peng G, Jiang Y, Li C (2009) Bifurcations of a Holling Type-II predator–prey system with constant rate harvesting. Int J Bifurc Chaos 19:2499–2514MathSciNetCrossRefzbMATHGoogle Scholar
  27. Shah MA (2013) Optimal control theory and fishery model. J Dev Agric Econ 5:476–481CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  • Tau Keong Ang
    • 1
  • Hamizah M. Safuan
    • 1
    Email author
  • Harvinder S. Sidhu
    • 2
  • Zlatko Jovanoski
    • 2
  • Isaac N. Towers
    • 2
  1. 1.Department of Mathematics and Statistics, Faculty of Applied Sciences and TechnologyUniversity Tun Hussein Onn MalaysiaBatu PahatMalaysia
  2. 2.Applied and Industrial Mathematics Research Group, School of Physical, Environmental and Mathematical SciencesUNSW CanberraCanberraAustralia

Personalised recommendations