DYNAMICS OF THE LESLIE-GOWER PREDATOR-PREY MODEL WITH THE BEDDINGTON-DEANGELIS RESPONSE FUNCTION, INCLUDING THE PRESENCE OF INFECTED PREY AND THE FEAR FACTOR OF SUSCEPTIBLE PREY TOWARD PREDATORS

Authors

  • Miswanto Miswanto Universitas Airlangga
  • Windarto Windarto Universitas Airlangga
  • Eridani Eridani Universitas Airlangga

DOI:

https://doi.org/10.25077/jmua.13.4.373-387.2024

Keywords:

Leslie-Gower, Beddington-DeAngelis Function, Stability Analysis, Numerical Simulation.

Abstract

This article presents a stability analysis of the Leslie-Gower predator-prey model that is extended by taking into account prey infection, the presence of prey fear factors towards predators and the Beddington-DeAngelis response function. The Leslie-Gower model is a classic model that describes the dynamics of predator and prey populations, while the Beddington-DeAngelis response function accommodates the effects of population density and more complex interactions between predators and prey. The infected prey factor describes the prey's resistance to predator attacks becoming weak, while the prey fear factor towards predators affects prey growth.

This study combines the components of infection in the prey population and the prey fear factor into the model to reflect the dynamics of disease and fear factors that can affect model stability. The model studied uses the Beddington-DeAngelis response function which describes the interaction between the prey population and the predator. This study uses two methods, namely analytical methods and numerical simulations. Analytical methods to study the stability analysis of the equilibrium point of the model by exploring the conditions under which the equilibrium point of the model is stable or unstable, focusing on the influence of infection parameters and the Beddington-DeAngelis response function on the stability of the equilibrium point. The results of the analysis show that prey infection and the shape of the response function can significantly affect the stability of the Leslie-Gower predator-prey model. The last section of this article presents numerical simulations that illustrate the stability of the equilibrium point of the model

Author Biographies

Miswanto Miswanto, Universitas Airlangga

Scopus ID = 57022248800

Windarto Windarto, Universitas Airlangga

Scopus ID = 41862696300 

Eridani Eridani, Universitas Airlangga

Scopus id = 35776115700

References

A. Ejaz, Y. Nawaz, M. S. Arif, D. S. Mashat, and K. Abodayeh, “Stability Analysis of Predator-Prey System with Consuming Resource and Disease in Predator Species,†CMES - Computer Modeling in Engineering and Sciences, vol. 131, no. 1, 2022, doi: 10.32604/cmes.2022.019440.

C. Arancibia–Ibarra and J. Flores, “Dynamics of a Leslie–Gower predator–prey model with Holling type II functional response, Allee effect and a generalist predator,†Math Comput Simul, vol. 188, pp. 1–22, Oct. 2021, doi: 10.1016/j.matcom.2021.03.035.

D. Melese Bahir, S. Gakkhar, and D. Melese, “Special Issue 1 Journal of International Academy of Physical Sciences pp. 1-6 *Paper presented in CONIAPS XIII at UPES, Dehradun during,†2011. [Online]. Available: https://www.researchgate.net/publication/266729856.

G. Ranjith Kumar, K. Ramesh, A. Khan, K. Lakshminarayan, and T. Abdeljawad, “Dynamical study of fractional order Leslie-Gower model of predator-prey with fear, Allee effect, and inter-species rivalry,†Results in Control and Optimization, vol. 14, Mar. 2024, doi: 10.1016/j.rico.2024.100403.

J. Cao, L. Ma, and P. Hao, “BIFURCATION ANALYSIS IN A MODIFIED LESLIE-GOWER PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE,†Journal of Applied Analysis and Computation, vol. 13, no. 5, pp. 3026–3053, 2023, doi: 10.11948/20230183.

T. ; Moulipriya; Das and R. N. Mukherjee, “Bifurcation and Stability of Prey-Predator Model with Beddington-DeAngelis Functional Response,†Applications and Applied Mathematics: An International Journal (AAM), vol. 12, no. 1, pp. 350–366, 2017.

A. Ejaz, Y. Nawaz, M. S. Arif, D. S. Mashat, and K. Abodayeh, “Stability Analysis of Predator-Prey System with Consuming Resource and Disease in Predator Species,†CMES - Computer Modeling in Engineering and Sciences, vol. 131, no. 1, 2022, doi: 10.32604/cmes.2022.019440.

C. Arancibia–Ibarra and J. Flores, “Dynamics of a Leslie–Gower predator–prey model with Holling type II functional response, Allee effect and a generalist predator,†Math Comput Simul, vol. 188, pp. 1–22, Oct. 2021, doi: 10.1016/j.matcom.2021.03.035.

D. Melese Bahir, S. Gakkhar, and D. Melese, “Special Issue 1 Journal of International Academy of Physical Sciences pp. 1-6 *Paper presented in CONIAPS XIII at UPES, Dehradun during,†2011. [Online]. Available: https://www.researchgate.net/publication/266729856

G. Ranjith Kumar, K. Ramesh, A. Khan, K. Lakshminarayan, and T. Abdeljawad, “Dynamical study of fractional order Leslie-Gower model of predator-prey with fear, Allee effect, and inter-species rivalry,†Results in Control and Optimization, vol. 14, Mar. 2024, doi: 10.1016/j.rico.2024.100403.

J. Cao, L. Ma, and P. Hao, “BIFURCATION ANALYSIS IN A MODIFIED LESLIE-GOWER PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE,†Journal of Applied Analysis and Computation, vol. 13, no. 5, pp. 3026–3053, 2023, doi: 10.11948/20230183.

T. ; Moulipriya; Das and R. N. Mukherjee, “Bifurcation and Stability of Prey-Predator Model with Beddington-DeAngelis Functional Response,†Applications and Applied Mathematics: An International Journal (AAM), vol. 12, no. 1, pp. 350–366, 2017.

Y. Shao and W. Kong, “A Predator–Prey Model with Beddington–DeAngelis Functional Response and Multiple Delays in Deterministic and Stochastic Environments,†Mathematics, vol. 10, no. 18, Sep. 2022, doi: 10.3390/math10183378.

J. Liu, “Dynamical analysis of a delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis functional response,†2014. [Online]. Available: https://www.advancesindifferenceequations.com/content/2014/1/314

P. J. Pal and P. K. Mandal, “Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and strong Allee effect,†Math Comput Simul, vol. 97, pp. 123–146, 2014, doi: 10.1016/j.matcom.2013.08.007.

S. Sharma and G. P. Samanta, “A Leslie-Gower predator-prey model with disease in prey incorporating a prey refuge,†Chaos Solitons Fractals, vol. 70, no. 1, pp. 69–84, 2015, doi: 10.1016/j.chaos.2014.11.010.

M. Haque, “A predator-prey model with disease in the predator species only,†Nonlinear Anal Real World Appl, vol. 11, no. 4, pp. 2224–2236, Aug. 2010, doi: 10.1016/j.nonrwa.2009.06.012.

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, no. 1, pp. 278–295, Feb. 2013, doi: 10.1016/j.jmaa.2012.08.057.

M. Chen, Y. Takeuchi, and J. F. Zhang, “Dynamic complexity of a modified Leslie–Gower predator–prey system with fear effect,†Commun Nonlinear Sci Numer Simul, vol. 119, May 2023, doi: 10.1016/j.cnsns.2023.107109.

S. Pal, N. Pal, S. Samanta, and J. Chattopadhyay, “Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model,†Mathematical Biosciences and Engineering, vol. 16, no. 5, pp. 5146–5179, 2019, doi: 10.3934/mbe.2019258.

S. Yao, J. Yang, and S. Yuan, “Bifurcation analysis in a modified Leslie-Gower predator-prey model with fear effect and multiple delays,†Mathematical Biosciences and Engineering, vol. 21, no. 4, pp. 5658–5685, 2024, doi: 10.3934/mbe.2024249.

S. Yu, “Global stability of a modified Leslie-Gower model with Beddington-DeAngelis functional response,†2014. [Online]. Available: https://www.advancesindifferenceequations.com/content/2014/1/84

M. He and Z. Li, “Global dynamics of a Leslie–Gower predator–prey model with square root response function,†Appl Math Lett, vol. 140, Jun. 2023, doi: 10.1016/j.aml.2022.108561.

Y. Cai, C. Zhao, W. Wang, and J. Wang, “Dynamics of a Leslie-Gower predator-prey model with additive Allee effect,†Appl Math Model, vol. 39, no. 7, pp. 2092–2106, 2015, doi: 10.1016/j.apm.2014.09.038.

Y. Wu and X. Ai, “Analysis of a stochastic Leslie-Gower predator-prey system with Beddington-DeAngelis and Ornstein-Uhlenbeck process,†Electronic Research Archive, vol. 32, no. 1, pp. 370–385, 2023, doi: 10.3934/ERA.2024018.

Y. Y. Mi, C. Song, and Z. C. Wang, “Global boundedness and dynamics of a diffusive predator–prey model with modified Leslie–Gower functional response and density-dependent motion,†Commun Nonlinear Sci Numer Simul, vol. 119, May 2023, doi: 10.1016/j.cnsns.2023.107115.

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31-10-2024

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Vol. 13 No. 4 (2024)

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