Home Stability analysis of a delayed predator–prey model with nonlinear harvesting efforts using imprecise biological parameters
Article
Licensed
Unlicensed Requires Authentication

Stability analysis of a delayed predator–prey model with nonlinear harvesting efforts using imprecise biological parameters

  • Amit K. Pal EMAIL logo
Published/Copyright: July 13, 2021

Abstract

In this paper, the dynamical behaviors of a delayed predator–prey model (PPM) with nonlinear harvesting efforts by using imprecise biological parameters are studied. A method is proposed to handle these imprecise parameters by using a parametric form of interval numbers. The proposed PPM is presented with Crowley–Martin type of predation and Michaelis–Menten type prey harvesting. The existence of various equilibrium points and the stability of the system at these equilibrium points are investigated. Analytical study reveals that the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate the main analytical findings.


Corresponding author: Amit K. Pal, Department of Mathematics, S. A. Jaipuria College, Kolkata 700005, India, E-mail:

Acknowledgments

The author is grateful to the learned reviewers and the Editor for their careful reading, valuable comments and helpful suggestions, which have helped him to improve the presentations of this work significantly.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] A. J. Lotka, Elements Of Physical Biology, Baltimore, Williams & Wilkins, 1925.Search in Google Scholar

[2] V. Volterra, Leconssen la Theorie Mathematique de la Leitte Pou Lavie, Paris, Gauthier-Villars, 1931.Search in Google Scholar

[3] K. S. Chaudhury and S. Saha Ray, “On the combined harvesting of prey-predator system,” J. Biol. Syst., vol. 4, no. 3, pp. 373–389, 1996.10.1142/S0218339096000259Search in Google Scholar

[4] S. Djilali and S. Bentout, “Pattern formations of a delayed diffusive predator-prey model with predator harvesting and prey social behavior,” Math. Methods Appl. Sci., vol. 44, no. 11, pp. 9128–9142, 2021. https://doi.org/10.1002/mma.7340.Search in Google Scholar

[5] M. E. M. Hacini, D. Hammoudi, S. Djilali, and S. Bentout, “Optimal harvesting and stability of a predator-prey model for fish populations with schooling behavior,” Theor. Biosci., vol. 140, no. 2, pp. 225–239, 2021. https://doi.org/10.1007/s12064-021-00347-5.Search in Google Scholar

[6] P. Lenzini and J. Rebaza, “Nonconstant predator harvesting on ratio-dependent predator-prey models,” Appl. Math. Sci., vol. 4, no. 16, pp. 791–803, 2010.Search in Google Scholar

[7] D. Manna, A. Maiti, and G. P. Samanta, “Analysis of a predator-prey model for exploited fish populations with schooling behaviour,” Appl. Math. Comput., vol. 317, pp. 35–48, 2018. https://doi.org/10.1016/j.amc.2017.08.052.Search in Google Scholar

[8] N. Zhang, F. Chen, Q. Su, and T. Wu, “Dynamic behaviors of a harvesting Leslie-Gower predator-prey model,” Discrete Dynam Nat. Soc., vol. 2011, p. 473949, 2011. https://doi.org/10.1155/2011/473949.Search in Google Scholar

[9] E. Berreta and Y. Kuang, “Global analysis in some delayed ratio-dependent predator -prey system,” Nonlinear Anal., vol. 32, pp. 381–408, 1998.10.1016/S0362-546X(97)00491-4Search in Google Scholar

[10] L. Haiyin and Y. Takeuchi, “Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response,” J. Math. Anal. Appl., vol. 374, pp. 644–654, 2011.10.1016/j.jmaa.2010.08.029Search in Google Scholar

[11] S. B. Hsu, T. B. Hwang, and Y. Kuang, “Global analysis of the Micaelis-Menten type ratio-dependent predator-prey system,” J. Math. Biol., vol. 42, pp. 489–506, 2001. https://doi.org/10.1007/s002850100079.Search in Google Scholar PubMed

[12] A. Oaten and W. Murdoch, “Functional response and stability in predator-prey system,” Am. Nat., vol. 109, pp. 289–298, 1975. https://doi.org/10.1086/282998.Search in Google Scholar

[13] D. Xiao and S. Ruan, “Global dynamics of a ratio-dependent predator-prey system,” J. Math. Biol., vol. 43, pp. 268–290, 2001. https://doi.org/10.1007/s002850100097.Search in Google Scholar PubMed

[14] R. K. Upadhyay and R. K. Naji, “Dynamics of three species food chain model with Crowley-Martin type functional response,” Chaos, Solit. Fractals, vol. 42, pp. 1337–1346, 2009. https://doi.org/10.1016/j.chaos.2009.03.020.Search in Google Scholar

[15] P. H. Crowley and E. K. Martin, “Functional response and interference within and between year classes of a dragently population,” J. N. Amer. Benth. Soc., vol. 8, pp. 211–221, 1989. https://doi.org/10.2307/1467324.Search in Google Scholar

[16] Q. Dong, W. Ma, and M. Sun, “The asymptotic behavior of a Chemostat model with Crowley-Martin type functional response and time delays,” J. Math. Chem., vol. 51, pp. 1231–1248, 2013. https://doi.org/10.1007/s10910-012-0138-z.Search in Google Scholar

[17] A. Mondal, A. K. Pal, and G. P. Samanta, “Stability and bifurcation analysis of a delayed three species food chain model with Crowley-Martin response function,” Appl. Appl. Math.: Int. J., vol. 13, no. 2, pp. 709–749, 2018.Search in Google Scholar

[18] G. T. Sklaski and J. F. Gillian, “Functional response with predator interference: variable alternative to Holling type II model,” Ecology, vol. 82, pp. 3083–3092, 2001.10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2Search in Google Scholar

[19] R. K. Upadhyay, S. N. Raw, and V. Rai, “Dynamic complexities in a tri-tropic food chain model with Holling type II and Crowley- Martin functional response,” Nonlinear Anal. Model Contr., vol. 15, pp. 361–375, 2010. https://doi.org/10.15388/na.15.3.14331.Search in Google Scholar

[20] C. W. Clark, Mathematical Bioeconomics : The Optimal Management Of Renewable Resources, Newyork, Wiley, 1976.Search in Google Scholar

[21] C. W. Clark, Bioeconomic Modeling and Fisheries Management, New York, Wiley, 1985.Search in Google Scholar

[22] D. S. Shiffman and D. Hammerschlag, “Preferred conservation policies of shark researchers,” Conserv. Biol., vol. 30, pp. 805–815, 2016. https://doi.org/10.1111/cobi.12668.Search in Google Scholar

[23] J. Wang, H. Cheng, H. Liu, et al.., “Periodic solution and control optimization of a prey-predator model with two types of harvesting,” Adv. Differ. Equ., vol. 41, 2018. https://doi.org/10.1186/s13662-018-1499-9.Search in Google Scholar

[24] R. Yuan, W. Jiang, and Y. Wang, “Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delay and prey harvesting,” J. Math. Anal. Appl., vol. 422, no. 2, pp. 1072–1090, 2015. https://doi.org/10.1016/j.jmaa.2014.09.037.Search in Google Scholar

[25] T. Das, R. N. Mukherjee, and K. S. Chaudhury, “Bioeconomic harvesting of a prey-predator fishery,” J. Biol. Dynam., vol. 3, no. 5, pp. 447–462, 2009. https://doi.org/10.1080/17513750802560346.Search in Google Scholar

[26] D. P. Hu and H. J. Cao, “Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting,” Nonlinear Anal. R. World Appl., vol. 33, pp. 58–82, 2017. https://doi.org/10.1016/j.nonrwa.2016.05.010.Search in Google Scholar

[27] R. P. Gupta and P. Chandra, “Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting,” J. Math. Anal. Appl., vol. 398, pp. 278–295, 2003.10.1016/j.jmaa.2012.08.057Search in Google Scholar

[28] M. MacDonald, Biological Delay System: Linear Stability Theory, Cambridge, Cambridge University Press, 1989.Search in Google Scholar

[29] C. Celik, “The stability and Hopf bifurcation for a predator-prey system with time delay,” Chaos, Solitons Fractals, vol. 37, no. 1, pp. 87–99, 2008. https://doi.org/10.1016/j.chaos.2007.10.045.Search in Google Scholar

[30] Y. Chen, J. Yu, and C. Sun, “Stability and Hopf bifurcation analysis in a three-level food chain system with delay,” Chaos, Solit. Fractals, vol. 31, no. 3, pp. 683–694, 2007. https://doi.org/10.1016/j.chaos.2005.10.020.Search in Google Scholar

[31] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, New York, Academic Press, 1993.Search in Google Scholar

[32] A. Maiti, A. K. Pal, and G. P. Samanta, “Effect of time-delay on a food chain model,” Appl. Math. Comput., vol. 200, pp. 189–203, 2008. https://doi.org/10.1016/j.amc.2007.11.011.Search in Google Scholar

[33] R. Xua, Q. Gan, and Z. Ma, “Stability and bifurcation analysis on a ratio-dependent predator-prey model with time delay,” J. Comput. Appl. Math., vol. 230, pp. 187–203, 2009. https://doi.org/10.1016/j.cam.2008.11.009.Search in Google Scholar

[34] R. C. Bassanezi, L. C. Barros, and A. Tonelli, “Attractors and asymptotic stability for fuzzy dynamical systems,” Fuzzy Set Syst., vol. 113, pp. 473–483, 2000. https://doi.org/10.1016/s0165-0114(98)00142-0.Search in Google Scholar

[35] M. Peixoto, L. C. Barros, and R. C. Bassanezi, “Predator-prey fuzzy model,” Ecol. Model., vol. 214, pp. 39–44, 2008. https://doi.org/10.1016/j.ecolmodel.2008.01.009.Search in Google Scholar

[36] M. Guo, X. Xu, and R. Li, “Impulsive functional differential inclusions and fuzzy populations models,” Fuzzy Set Syst., vol. 138, pp. 601–615, 2003. https://doi.org/10.1016/s0165-0114(02)00522-5.Search in Google Scholar

[37] A. K. Pal, P. Dolai, and G. P. Samanta, “Dynamics of a delayed competitive system Affected by toxic substances with imprecise biological parameters,” Filomat, vol. 31, no. 16, pp. 5271–5293, 2017, https://doi.org/10.2298/FIL1716271P.Search in Google Scholar

[38] Jana, D., Dolai, P., Pal, A. K., Samanta, G. P., “On the stability and Hopf-bifurcation of a multi-delayed competitive population system affected by toxic substances with imprecise biological parameters,” Model. Earth. Syst. Environ., vol. 2, p. 110, 2016. https://doi.org/10.1007/s40808-016-0156-0.Search in Google Scholar

[39] D. Pal, G. S. Mahapatra, and G. P. Samanta, “A proportional harvesting dynamical model with fuzzy intrinsic growth rate and harvesting quantity,” Pac. Asian J. Math., vol. 6, pp. 199–213, 2012.Search in Google Scholar

[40] D. Pal, G. S. Mahapatra, and G. P. Samanta, “Optimal harvesting of prey-predator system with interval biological parameters: a bioeconomic model,” Math. Biosci., vol. 241, pp. 181–187, 2013. https://doi.org/10.1016/j.mbs.2012.11.007.Search in Google Scholar PubMed

[41] S. Bentout, S. Djilali, and S. Kumar, “Mathematical analysis of the influence of prey escaping from prey herd on three species fractional predator-prey interaction model,” Phys. Stat. Mech. Appl., vol. 572, p. 125840, 2021. https://doi.org/10.1016/j.physa.2021.125840.Search in Google Scholar

[42] S. Djilali and S. Bentout, “Spatiotemporal patterns in a diffusive predator-prey model with prey social behavior,” Acta Appl. Math., vol. 169, pp. 125–143, 2020. https://doi.org/10.1007/s10440-019-00291-z.Search in Google Scholar

[43] S. Djilali, S. Bentout, B. Ghanbari, and S. Kumar, “Spatial patterns in a vegetation model with internal competition and feedback regulation,” Eur. Phys. J. Plus, vol. 136, p. 256, 2021. https://doi.org/10.1140/epjp/s13360-021-01251-z.Search in Google Scholar

[44] J. K. Hale, Theory Of Functional Differential Equations, Heidelberg, Springer, 1977.10.1007/978-1-4612-9892-2Search in Google Scholar

[45] H. Freedman and V. S. H. Rao, “The trade-off between mutual interference and time lags in predator-prey systems,” Bull. Math. Biol., vol. 45, pp. 991–1004, 1983. https://doi.org/10.1007/bf02458826.Search in Google Scholar

Received: 2021-05-14
Revised: 2021-06-24
Accepted: 2021-06-25
Published Online: 2021-07-13
Published in Print: 2021-10-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 2.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/zna-2021-0131/pdf?lang=en
Scroll to top button