The Dynamics of a Predator-Prey Model Involving Disease Spread In Prey and Predator Cannibalism

Nurul Imamah Ah, Wuryansari Muharini Kusumawinahyu, Agus Suryanto, Trisilowati Trisilowati

Abstract


In this article, dynamics of predator prey model with infection spread in prey and cannibalism in predator is analyzed. The model has three populations, namely susceptible prey, infected prey, and predator. It is assumed that there is no migration in both prey and predator populations. The dynamical analysis shows that the model has six equilibria, namely the trivial equilibrium point, the prey extinction point, the disease free and predator extinction equilibrium point, the disease-free equilibrium point, the predator extinction equilibrium point, and the coexistence equilibrium point. The first equilibrium is unstable, and the other equilibria conditionally local asymptotically stable. The positivity and boundedness of the solution are also shown. The analytical result is supported by numerical simulation. It is shown that in such a high cannibalization the coexistence equilibrium is locally asymptotically stable.

Keywords


Cannibalism; Predator Prey Model; Disease; Dynamical Analysis

Full Text:

PDF

References


A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992. DOI:10.2307/1940005

H. S. Panigoro and E. Rahmi, “Computational dynamics of a Lotka-Volterra Model with additive Allee effect based on Atangana-Baleanu fractional derivative,” Jambura Journal of Biomathematics (JJBM), vol. 2, no. 2, pp. 96–103, 2021. DOI:10.34312/jjbm.v2i2.11886

D. Mukherjee, “Fear induced dynamics on Leslie-Gower predator-prey system with Holling-type IV functional response,” Jambura Journal of Biomathematics (JJBM), vol. 3, no. 2, pp. 49–57, 2022. DOI:10.34312/jjbm.v3i2.14348

W. O. Kermack and A. Mckendrick, “Contributions to the mathematical theory of epidemicsI,” Bulletin of Mathematical Biology, vol. 53, no. 1-2, pp. 33–55, 1991. DOI:10.1016/S0092-8240(05)80040-0

Y. P. Zhang, M. J. Ma, P. Zuo, and X. Liang, “Analysis of a eco-epidemiological model with disease in the predator,” Applied Mechanics and Materials, vol. 536-537, pp. 861–864, 2014. DOI:10.4028/www.scientific.net/AMM.536-537.861

Y. Xiao and L. Chen, “Modeling and analysis of a predator-prey model with disease in the prey,” Mathematical Biosciences, vol. 171, no. 1, pp. 59–82, 2001. DOI:10.1016/S0025-5564(01)00049-9

D. Greenhalgh and M. Haque, “A predator-prey model with disease in the prey species only,” Mathematical Methods in the Applied Sciences, vol. 30, no. 8, pp. 911–929, 2007. DOI:10.1002/mma.815

L. Han, Z. Ma, and H. W. Hethcote, “Four predator prey models with infectious diseases,” Mathematical and Computer Modelling, vol. 34, no. 7-8, pp. 849–858, 2001. DOI:10.1016/S0895-7177(01)00104-2

H. W. Hethcote, W. Wang, L. Han, and Z. Ma, “A predator - Prey model with infected prey,” Theoretical Population Biology, vol. 66, no. 3, pp. 259–268, 2004. DOI:10.1016/j.tpb.2004.06.010

M. S. Rahman and S. Chakravarty, “A predator-prey model with disease in prey,” Nonlinear Analysis: Modelling and Control, vol. 18, no. 2, pp. 191–209, 2013. DOI:10.15388/NA.18.2.14022

X. Zhou, J. Cui, X. Shi, and X. Song, “A modified Leslie-Gower predator-prey model with prey infection,” Journal of Applied Mathematics and Computing, vol. 33, no. 2010, pp. 471–487, 2009. DOI:10.1007/s12190-009-0298-6

A. Lahay, M. R. F. Payu, S. L. Mahmud, H. S. Panigoro, and P. Zakaria, “Dynamics of a predator-prey model incorporating infectious disease and quarantine on prey,” Jambura Journal of Biomathematics (JJBM), vol. 3, no. 2, pp. 75–81, 2022. DOI:10.34312/jjbm.v3i2.17162

R. Ibrahim, L. Yahya, E. Rahmi, and R. Resmawan, “Analisis dinamik model predator-prey tipe Gause dengan wabah penyakit pada prey,” Jambura Journal of Biomathematics (JJBM), vol. 2, no. 1, pp. 20–28, 2021. DOI:10.34312/jjbm.v2i1.10363

J. Chattopadhyay and O. Arino, “A predator-prey model with disease in the prey,” Nonlinear Analysis: Theory, Methods & Applications, vol. 36, no. 6, pp. 747–766, 1999. DOI:10.1016/S0362-546X(98)00126-6

S. Biswas, S. Samanta, and J. Chattopadhyay, “A cannibalistic eco-epidemiological model with disease in predator population,” Journal of Applied Mathematics and Computing, vol. 57, no. 2018, pp. 161–197, 2017. DOI:0.1007/s12190-017-1100-9

S. Maisaroh, R. Resmawan, and E. Rahmi, “Analisis kestabilan model predator-prey dengan infeksi penyakit pada prey dan pemanenan proporsional pada predator,” Jambura Journal of Biomathematics (JJBM), vol. 1, no. 1, pp. 8–15, 2020. DOI:10.34312/jjbm.v1i1.5948

Y. Kang, M. Rodriguez-Rodriguez, and S. Evilsizor, “Ecological and evolutionary dynamics of two-stage models of social insects with egg cannibalism,” Journal of Mathematical Analysis and Applications, vol. 430, no. 1, pp. 324–353, 2015. DOI:10.1016/j.jmaa.2015.04.079

H. Deng, F. Chen, Z. Zhu, and Z. Li, “Dynamic behaviors of LotkaVolterra predatorprey model incorporating predator cannibalism,” Advances in Difference Equations, vol. 2019, no. 2019, pp. 1–17, 2019. DOI:10.1186/s13662-019-2289-8

S. Biswas, S. Samanta, and J. Chattopadhyay, “A model based theoretical study on cannibalistic preypredator system with disease in both populations,” Differential Equations and Dynamical Systems, vol. 23, no. 2015, pp. 327–370, 2014. DOI:10.1007/s12591-014-0211-0

M. Rayungsari, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, “Dynamics analysis of a predatorprey fractional-order model incorporating predator cannibalism and refuge,” Frontiers in Applied Mathematics and Statistics, vol. 9, pp. 1–12, 2023. DOI:10.3389/fams.2023.1122330

F. Zhang, Y. Chen, and J. Li, “Dynamical analysis of a stage-structured predator-prey model with cannibalism,” Mathematical Biosciences, vol. 307, pp. 33–41, 2019. DOI:10.1016/j.mbs.2018.11.004




DOI: https://doi.org/10.37905/jjbm.v4i2.21495

Copyright (c) 2023 Nurul Imamah Ah, Wuryansari Muharini Kusumawinahyu, Agus Suryanto, Trisilowati Trisilowati

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Jambura Journal of Biomathematics (JJBM) has been indexed by:

                          EDITORIAL OFFICE OF JAMBURA JOURNAL OF BIOMATHEMATICS

 Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Gorontalo
Jl. Prof. Dr. Ing. B. J. Habibie, Moutong, Tilongkabila, Kabupaten Bone Bolango 96554, Gorontalo, Indonesia
 Email: editorial.jjbm@ung.ac.id
 +6281356190818 (Call/SMS/WA)
 Jambura Journal of Biomathematics (JJBM) by Department of Mathematics Universitas Negeri Gorontalo is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.  Powered by Public Knowledge Project OJS.