Research Article
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Year 2023, , 65 - 75, 07.08.2023
https://doi.org/10.33187/jmsm.1177403

Abstract

References

  • [1] C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976.
  • [2] M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of predator and prey, Nat. Resour. Model., 3 (1988), 63-90.
  • [3] M. Mesterton-Gibbons, A technique for finding optimal two species harvesting policies, Ecol. Model., 92 (1996), 235-244.
  • [4] O. Flaaten, On the bioeconomics of predator-prey fishery, Fish. Res., 37 (1998), 179-191.
  • [5] T. G. Hallam, C. W. Clark, Non-autonomous logistic equations as models of populations in a deteriorating environment, J. Theoret. Biol, 93 (1982), 303-313.
  • [6] T. K. Kar, K. S. Chaudhuri, On non-selective harvesting of two competing fish species in the presence of toxicity, Ecol. Model., 161 (2003), 125-137.
  • [7] J. Chattopadhyay, Effect of toxic substanceson a two species competitive system, Ecol. Model., 84 (1996), 287-289.
  • [8] T. G. Hallam, J. T. De Luna, Effects of toxicants on populations: a qualitative approach 3. Environmental and food chain pathways, J. Theoret. Biol., 199 (1984), 411-429.
  • [9] H. I. Freedman, J. B. Shukla, Models for the effect of toxicant in a single species and predator-prey systems, J. Math. Biol., 30 (1900), 15-30.
  • [10] D. Mukherjee, Persistence and global stability of a population in a polluted environment with delay, J. Biol. Syst., 10(3) (2002), 1-8.
  • [11] J. M. Smith, Mathematical Models in Biology, Cambridge University Press. Cambridge, 1968.
  • [12] M. Haque, S. Sarwardi, Effect of toxicity on a harvested fishery model, Model. Earth Syst. Environ., 2 (2016), 122.
  • [13] D. Hu, H. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Anal. Real World Appl., 33 (2017), 58-82.
  • [14] H. N. Agiza, E. M. Elabbasy, EI-Metwally, A. A. Elasdany, Chaotic dynamics of a discrete predator-prey model with Holling type II, Nonlinear Anal. Real World Appl., 10 (2009), 116-129.
  • [15] Q. Din, Complexity and chaos control in a discrete time prey-predator model, Commun. Nonlinear Sci. Numer. Simul., 49 (2017), 113-134.
  • [16] M. E. Elettreby, T. Nabil, A. Khawagi, Stability and bifurcation analysis of a discrete predator-prey model with mixed Holling interaction, Computer Modeling in Engineering and Sciences., 122 (2020), 907-921.
  • [17] A. Hening, Coexistence, extinction, and optimal harvesting in discrete-time stochastic population models, J. Nonlinear Sci., 31 (2021), 1-50.
  • [18] W. Ding, S. Lenhart, H. Behncke, Discrete time optimal harvesting of fish populations with age structure, Lett. Biomath., 1 (2014), 193-207.
  • [19] G. Y. Chen, Z. D. Teng, On the stability in a discrete two species competition system. J. Appl. Math. Comput., 38 (2012), 25-39.
  • [20] S. Elaydi, Discrete Chaos with Applications in Science and Engineering, Chapman and Hall/CRC, Boca Raton, Fla, USA, 2000.
  • [21] X. S. Luo, G. Chen, B. H. Wang, J. Q. Fang, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos Solitons & Fractals, 18 (2003), 775-783.
  • [22] L. Wang, M. Wang, Ordinary Difference Equations, Xin Jiang University Press, Urmuqi, 1989.

Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity

Year 2023, , 65 - 75, 07.08.2023
https://doi.org/10.33187/jmsm.1177403

Abstract

This paper analyses a discrete-time Michaelis-Menten type harvested fishery model in the presence of toxicity. Boundary and interior (positive) fixed points are examined. Using an iteration scheme and the comparison principle of difference equations, we determined the sufficient condition for global stability of the interior fixed point. It is shown that the sufficient criterion for Neimark-Sacker bifurcation and flip bifurcation can be established. It is observed that the system behaves in a chaotic manner when a specific set of system parameters is selected, which are controlled by a hybrid control method. Examples are cited to illustrate our conclusions.

References

  • [1] C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976.
  • [2] M. Mesterton-Gibbons, On the optimal policy for the combined harvesting of predator and prey, Nat. Resour. Model., 3 (1988), 63-90.
  • [3] M. Mesterton-Gibbons, A technique for finding optimal two species harvesting policies, Ecol. Model., 92 (1996), 235-244.
  • [4] O. Flaaten, On the bioeconomics of predator-prey fishery, Fish. Res., 37 (1998), 179-191.
  • [5] T. G. Hallam, C. W. Clark, Non-autonomous logistic equations as models of populations in a deteriorating environment, J. Theoret. Biol, 93 (1982), 303-313.
  • [6] T. K. Kar, K. S. Chaudhuri, On non-selective harvesting of two competing fish species in the presence of toxicity, Ecol. Model., 161 (2003), 125-137.
  • [7] J. Chattopadhyay, Effect of toxic substanceson a two species competitive system, Ecol. Model., 84 (1996), 287-289.
  • [8] T. G. Hallam, J. T. De Luna, Effects of toxicants on populations: a qualitative approach 3. Environmental and food chain pathways, J. Theoret. Biol., 199 (1984), 411-429.
  • [9] H. I. Freedman, J. B. Shukla, Models for the effect of toxicant in a single species and predator-prey systems, J. Math. Biol., 30 (1900), 15-30.
  • [10] D. Mukherjee, Persistence and global stability of a population in a polluted environment with delay, J. Biol. Syst., 10(3) (2002), 1-8.
  • [11] J. M. Smith, Mathematical Models in Biology, Cambridge University Press. Cambridge, 1968.
  • [12] M. Haque, S. Sarwardi, Effect of toxicity on a harvested fishery model, Model. Earth Syst. Environ., 2 (2016), 122.
  • [13] D. Hu, H. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Anal. Real World Appl., 33 (2017), 58-82.
  • [14] H. N. Agiza, E. M. Elabbasy, EI-Metwally, A. A. Elasdany, Chaotic dynamics of a discrete predator-prey model with Holling type II, Nonlinear Anal. Real World Appl., 10 (2009), 116-129.
  • [15] Q. Din, Complexity and chaos control in a discrete time prey-predator model, Commun. Nonlinear Sci. Numer. Simul., 49 (2017), 113-134.
  • [16] M. E. Elettreby, T. Nabil, A. Khawagi, Stability and bifurcation analysis of a discrete predator-prey model with mixed Holling interaction, Computer Modeling in Engineering and Sciences., 122 (2020), 907-921.
  • [17] A. Hening, Coexistence, extinction, and optimal harvesting in discrete-time stochastic population models, J. Nonlinear Sci., 31 (2021), 1-50.
  • [18] W. Ding, S. Lenhart, H. Behncke, Discrete time optimal harvesting of fish populations with age structure, Lett. Biomath., 1 (2014), 193-207.
  • [19] G. Y. Chen, Z. D. Teng, On the stability in a discrete two species competition system. J. Appl. Math. Comput., 38 (2012), 25-39.
  • [20] S. Elaydi, Discrete Chaos with Applications in Science and Engineering, Chapman and Hall/CRC, Boca Raton, Fla, USA, 2000.
  • [21] X. S. Luo, G. Chen, B. H. Wang, J. Q. Fang, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos Solitons & Fractals, 18 (2003), 775-783.
  • [22] L. Wang, M. Wang, Ordinary Difference Equations, Xin Jiang University Press, Urmuqi, 1989.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences, Applied Mathematics (Other)
Journal Section Articles
Authors

Debasis Mukherjee 0000-0002-5149-5985

Publication Date August 7, 2023
Submission Date September 19, 2022
Acceptance Date March 17, 2023
Published in Issue Year 2023

Cite

APA Mukherjee, D. (2023). Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity. Journal of Mathematical Sciences and Modelling, 6(2), 65-75. https://doi.org/10.33187/jmsm.1177403
AMA Mukherjee D. Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity. Journal of Mathematical Sciences and Modelling. August 2023;6(2):65-75. doi:10.33187/jmsm.1177403
Chicago Mukherjee, Debasis. “Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity”. Journal of Mathematical Sciences and Modelling 6, no. 2 (August 2023): 65-75. https://doi.org/10.33187/jmsm.1177403.
EndNote Mukherjee D (August 1, 2023) Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity. Journal of Mathematical Sciences and Modelling 6 2 65–75.
IEEE D. Mukherjee, “Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity”, Journal of Mathematical Sciences and Modelling, vol. 6, no. 2, pp. 65–75, 2023, doi: 10.33187/jmsm.1177403.
ISNAD Mukherjee, Debasis. “Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity”. Journal of Mathematical Sciences and Modelling 6/2 (August 2023), 65-75. https://doi.org/10.33187/jmsm.1177403.
JAMA Mukherjee D. Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity. Journal of Mathematical Sciences and Modelling. 2023;6:65–75.
MLA Mukherjee, Debasis. “Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity”. Journal of Mathematical Sciences and Modelling, vol. 6, no. 2, 2023, pp. 65-75, doi:10.33187/jmsm.1177403.
Vancouver Mukherjee D. Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity. Journal of Mathematical Sciences and Modelling. 2023;6(2):65-7.

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