Implementation of non-standard finite difference on a predator prey model considering cannibalism on predator and harvesting on prey

Prisalo Luis, Putri Zahra Kamalia, Olumuyiwa James Peter, Dipo Aldila

Abstract


The type of interaction between two different species in the same ecosystem plays an important role in the coexistence between these species. One type of interaction between species is predator-prey interaction. Several important factors are crucial to guarantee the existence of predator and prey in the same ecosystem, such as the carrying capacity of the ecosystem for the survival of prey, the intensity of predation, cannibalism in the predator population, and many other factors. External factors such as human intervention, such as harvesting, increase the complexity of the problem. Here in this article, we discuss a predator-prey model that takes predation and harvesting in prey populations into account. We implement a Non-Standard Finite Difference (NSFD) numerical scheme to solve our model due to it good performance on stability and approximation. Mathematical analysis on the existence and stability of equilibrium points from the discrete model was analyzed in detail. We implement a Nonstandard Finite Difference (NSFD) scheme to ensure numerical stability across various simulation scenarios. It is shown that NSFD has a better numerical stability compared to the standard numerical scheme like Euler or fourth-order Runge-Kutta method. From the sensitivity of autonomous simulation, we have shown that increases of cannibalism in predator populations will reduce predator populations, and as a result, the population of prey will increase due to the lack of number of predators. We also showed that increasing harvesting in prey populations may cause extinction in prey and predator populations. Furthermore, we have shown how periodic harvesting on prey populations may cause a critical condition on the existence of prey populations that takes a longer period to get recovered.

Keywords


Non-standard finite difference; Predator-prey; Periodic harvesting; Population dynamic

Full Text:

PDF

References


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

J. Li et al., “Impact of cannibalism on dynamics of a structured predator–prey system,” Applied Mathematical Modelling, vol. 78, pp. 1–19, 2020. DOI:10.1016/j.apm.2019.09.022

M. S. Islam and M. S. Rahman, “Dynamics of cannibalism in the predator in a prey refuge functional response dependent predator-prey model,” IJEES, vol. 45, 2024.

Z. Hammouch et al., “Dynamics investigation and numerical simulation of fractional-order predator-prey model with holling type II functional response,” Discrete and Continuous Dynamical Systems-S, vol. 18, no. 5, pp. 1230–1266, 2025. DOI:10.3934/dcdss.2024181

P. Sireeshadevi and G. R. Kumar, “Dynamical behavior of an eco-epidemiological model incorporating holling type-II functional response with prey refuge and constant prey harvesting,” Network Biology, vol. 14, no. 3, pp. 228–241, 2024.

Ö. A. Gümüs¸, “Dynamics of a prey-predator system with harvesting effect on prey,” Chaos Theory and Applications, vol. 4, no. 3, pp. 144–151, 2022. DOI:10.51537/chaos.1183113

N. Imamah Ah et al., “The Dynamics of a Predator-Prey Model Involving Disease Spread In Prey and Predator Cannibalism,” Jambura Journal of Biomathematics (JJBM), vol. 4, no. 2, pp. 119–125, 2023. DOI:10.37905/jjbm.v4i2.21495

L. K. Beay et al., “A Stage-structure Leslie-Gower Model with Linear Harvesting and Disease in Predator,” Jambura Journal of Biomathematics, vol. 4, no. 2, pp. 155–163, 2023. DOI:10.37905/jjbm.v4i2.22047

A. Lahay et al., “Dynamics of a predator-prey model incorporating infectious disease and quarantine on prey,” Jambura Journal of Mathematics, vol. 3, no. 2, pp. 75–81, 2022. DOI:10.37905/jjbm.v4i2.22047

A. Sirén and K. Parvinen, “A spatial bioeconomic model of the harvest of wild plants and animals,” Ecological Economics, vol. 116, pp. 201–210, 2015. DOI:10.1016/j.ecolecon.2015.04.015

Rashi et al., “Cooperation and harvesting-induced delays in a predator–prey model with prey fear response: A crossing curves approach,“ Chaos, Solitons & Fractals, vol. 194, p. 116132, 2025. DOI:10.1016/j.chaos.2025.116132

B. E. Kashem and H. F. Al-Husseiny, “The dynamic of two prey–one predator food web model with fear and harvesting,“ Partial Differential Equations in Applied Mathematics, vol. 11, p. 100875, 2024. DOI:10.1016/j.padiff.2024.100875

S. Roy and P. K. Tiwari, “Bistability in a predator–prey model characterized by the crowley–martin functional response: Effects of fear, hunting cooperation, additional foods and nonlinear harvesting,“ Mathematics and Computers in Simulation, vol. 228, pp. 274–297, 2025. DOI:10.1016/j.matcom.2024.09.001

N. Ahmed et al., “Pattern formation and analysis of reaction–diffusion ratio-dependent prey–predator model with harvesting in predator,“ Chaos, Solitons & Fractals, vol. 251, p. 115164, 2024. DOI:10.1016/j.chaos.2024.115164

N. Sarif et al., “Spatio-temporal dynamics in a delayed prey–predator model with nonlinear prey refuge and harvesting,“ Chaos, Solitons & Fractals, vol. 186, p. 115247, 2024. DOI:10.1016/j.chaos.2024.115247

Y. Gao, M. Banerjee, and T. V. Ta, “Dynamics of infectious diseases in predator–prey populations: A stochastic model, sustainability, and invariant measure,“ Mathematics and Computers in Simulation, vol. 227, pp. 103–120, 2025. DOI:10.1016/j.matcom.2024.07.031

M. Wang and S. Yao, “The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey,“ Communications in Nonlinear Science and Numerical Simulation, vol. 132, p. 107902, 2024. DOI:10.1016/j.cnsns.2024.107902

A. Xiang and L. Wang, “Boundedness and stabilization in a predator-prey model with prey-taxis and disease in predator species,“ Journal of Mathematical Analysis and Applications, vol. 522, no. 1, p. 126953, 2023. DOI:10.1016/j.jmaa.2022.126953

Y. Cao et al., “Investigating the spread of a disease on the prey and predator interactions through a nonsingular fractional model,“ Results in Physics, vol. 32, p. 105084, 2022. DOI:10.1016/j.rinp.2021.105084

N. Li and M. Yan, “Bifurcation control of a delayed fractional-order prey-predator model with cannibalism and disease,“ Physica A: Statistical Mechanics and its Applications, vol. 600, p. 127600, 2022. DOI:10.1016/j.physa.2022.127600

S. G. Mortoja, P. Panja, and S. K. Mondal, “Dynamics of a predator-prey model with nonlinear incidence rate, Crowley-martin type functional response and disease in prey population,“ Ecological Genetics and Genomics, vol. 10, p. 100035, 2019. DOI:10.1016/j.egg.2018.100035

W. Mbava, J. Mugisha, and J. Gonsalves, “Prey, predator and super-predator model with disease in the super-predator,“ Applied Mathematics and Computation, vol. 297, pp. 92–114, 2017. DOI:10.1016/j.amc.2016.10.034

G. Barabás, M. J. Michalska-Smith, and S. Allesina, “Self-regulation and the stability of large ecological networks,” Nature ecology & evolution, vol. 1, no. 12, pp. 1870–1875, 2017.

H. Deng et al., “Dynamic behaviors of lotka–volterra predator–prey model incorporating predator cannibalism,” Advances in Difference Equations, vol. 2019, pp. 1–17, 2019. DOI:10.1186/s13662-019-2289-8

M. Rayungsari et al., “A nonstandard numerical scheme for a predator-prey model involving predator cannibalism and refuge,” Communication in Biomathematical Sciences, vol. 6, no. 1, pp. 11–23, 2023. DOI:10.5614/cbms.2023.6.1.2

R. E. Mickens, Nonstandard finite difference models of differential equations. world scientific, 1993. DOI:10.1142/2081




DOI: https://doi.org/10.37905/jjbm.v6i1.30550

Copyright (c) 2025 Prisalo Luis, Putri Zahra Kamalia, Olumuyiwa James Peter, Dipo Aldila

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 (WA Only)
 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.