Stability and Bifurcation of a 3D Eco Epidemiological Predator Prey Model with Pesticide

Aqiila Ollyana Savitri, Dian Savitri

Abstract


Eco--epidemiological predator--prey models provide an important mathematical framework for understanding the interaction between disease transmission, predation, and human intervention in ecological systems. This study investigates a three--dimensional deterministic model incorporating saturated disease incidence, Holling type II predation, and pesticide application. Analytical techniques are employed to determine the existence and local stability of biologically feasible equilibrium points, while numerical simulations using a fourth--order Runge--Kutta method illustrate the dynamical behavior of the system under different parameter regimes. The analysis reveals the possibility of disease--free, predator--free, and interior coexistence equilibria, as well as bistability depending on parameter values and initial conditions. Bifurcation analysis identifies critical thresholds in disease transmission and predator conversion efficiency that govern transitions between predator persistence and extinction. These findings provide theoretical insights for integrated pest management strategies by emphasizing the balance between chemical control and ecological stability.

Keywords


bifurcation; eco-epidemiological model; pesticide effect; predator–prey system; stability

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References


[1] R. Babu and I. A. Wani, “Dynamics of interaction of one prey and two competing predators with population heterogeneity,” Indonesian Journal of Mathematics and Applications, vol. 2, no. 2, pp. 78–88, 2024, doi: 10.21776/ub.ijma.2024.002.02.2.

[2] D. N. Prabhumani, M. Shanmugam, S. P. Manickasundaram, and N. G. Thangaraj, “Dynamics of an eco-epidemiological model with infected prey in fractional order,” Mathematical Modelling and Control, vol. 5, no. 3, pp. 292–304, 2025, doi: 10.3934/mmc.2025020.

[3] T. Gaber, R. Herdiana, and Widowati, “Dynamical analysis of an eco-epidemiological model experiencing the crowding effect of infected prey,” Communications in Mathematical Biology and Neuroscience, vol. 2024, no. 3, 2024, doi: 10.28919/cmbn/8353.

[4] K. Akshaya, S. Muthukumar, M. Siva Pradeep, T. Nandha Gopal, and N. P. Deepak, “Analysis of diseased one prey two predator model with Holling type II functional response,” Telematique, vol. 24, no. 1, pp. 150–161, 2025.

[5] I. Domínguez-Alemán, I. Domínguez-Alemán, J. C. Hernández-Gómez, and F. J. Ariza-Hernández, “A predator-prey fractional model with disease in the prey species,” Mathematical Biosciences and Engineering, vol. 21, no. 3, pp. 3713–3741, 2024, doi: 10.3934/mbe.2024164.

[6] W. W. Mohammed et al., “Stochastic predator–prey interactions with disease dynamics: Fixed point and numerical investigations,” Boundary Value Problems, 2025, doi: 10.1186/s13661-025-02159-8.

[7] R. Das and S. K. Sasmal, “Fear induced coexistence in eco-epidemiological systems with infected prey,” Mathematical Biosciences and Engineering, vol. 22, no. 11, pp. 2897–2922, 2025, doi: 10.3934/mbe.2025107.

[8] K. Sarkar and S. Khajanchi, “An eco-epidemiological model with the impact of fear,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 32, p. 083126, 2022, doi: 10.1063/5.0099584.

[9] M. S. Rahman, S. Pramanik, and E. Venturino, “An ecoepidemic model with healthy prey herding and infected prey drifting away,” Nonlinear Analysis: Modelling and Control, vol. 28, no. 2, pp. 326–364, 2023, doi: 10.15388/namc.2023.28.31549.

[10] Y. Enatsu, J. Roy, and M. Banerjee, “Hunting cooperation in a prey–predator model with maturation delay,” Journal of Biological Dynamics, vol. 18, no. 1, p. 2332279, 2024, doi: 10.1080/17513758.2024.2332279.

[11] A. H. Naser and D. K. Bahlool, “Modeling disease dynamics in a prey–predator system with competition, fear, and cooperative hunting,” Computation, vol. 13, no. 11, p. 254, 2025, doi: 10.3390/computation13110254.

[12] B. Kumar, R. K. Sinha, and S. Hussain, “Comparative analysis of eco-epidemic model with disease in prey and gestation delay in predator: Implications for population dynamics,” Advances in Continuous and Discrete Models, 2025, doi: 10.1186/s13662-025-03863-6.

[13] M. K. Dewi, D. Savitri, and Abadi, “Dynamical behavior in the competitive model incorporating the fear effect of prey due to allelopathy with shared biotic resources,” BAREKENG: Journal of Mathematics and Its Applications, vol. 18, no. 4, pp. 2663–2674, 2024, doi: 10.30598/barekengvol18iss4pp2663-2674.

[14] K. G. Mekonen, K. P. Rao, and A. F. Bezabih, “Mathematical modeling of infectious disease and prey-predator interaction with optimal control,” International Journal of Mathematics and Mathematical Sciences, vol. 2024, Art. no. 5444627, 2024, doi: 10.1155/2024/5444627.

[15] H. Qiu, T. Zhang, and R. Hou, “Optimal control of an infected prey–predator model with fear effect,” Nonlinear Analysis: Modelling and Control, vol. 29, no. 4, pp. 670–692, 2024, doi: 10.15388/namc.2024.29.35236.

[16] W. Qin, Y. Xia, and Y. Yang, “An eco-epidemic model for assessing the application of integrated pest management strategies,” Mathematical Biosciences and Engineering, vol. 20, no. 9, pp. 16506–16527, 2023, doi: 10.3934/mbe.2023736.

[17] T. Herlambang, D. Rahmalia, N. Aini, and D. F. Karya, “Predator-prey dynamical model with infected prey by virus and its optimal control using pesticide,” in AIP Conference Proceedings, vol. 2577, 2022.

[18] K. Mu’tamar, J. Naiborhu, R. Saragih, and D. Handayani, “Nonlinear tracking control for prey stabilization in predator-prey model using backstepping,” BAREKENG: Journal of Mathematics and Its Applications, vol. 19, no. 3, pp. 1825–1840, 2025, doi: 10.30598/barekengvol19iss3pp1825-1840.

[19] R. Du, X. Liang, Y. Na, and F. Xu, “Optimal control of an eco-epidemiological reaction-diffusion model,” Mathematics, vol. 13, no. 13, p. 2069, 2025, doi: 10.3390/math13132069.

[20] L. Li and J.-E. Zhang, “Input-to-state stability of stochastic nonlinear system with delayed impulses,” Mathematical Biosciences and Engineering, vol. 21, no. 2, pp. 2233–2253, 2024, doi: 10.3934/mbe.2024098.

[21] J. G. Clapp, J. L. Malmberg, and J. D. Holbrook, “Examining pathogen avoidance in predator-prey and scavenging systems,” Frontiers in Ecology and Evolution, vol. 12, Art. no. 1481290, 2024, doi: 10.3389/fevo.2024.1481290.

[22] R. Sivakumar and S. Vijaya, “Eco-epidemiology of prey and competitive predator species in the SEI model,” The Scientific Temper, vol. 15, no. spl-2, pp. 243–252, 2024, doi: 10.58414/SCIENTIFICTEMPER.2024.15.spl-2.37.

[23] A. A. Raezah, J. Chowdhury, and F. A. Basir, “Global stability of the interior equilibrium and the stability of Hopf bifurcating limit cycle in a model for crop pest control,” AIMS Mathematics, vol. 9, no. 9, pp. 24229–24246, 2024, doi: 10.3934/math.20241179.

[24] N. P. Deepak, S. Muthukumar, M. Siva Pradeep, and T. Nandha Gopal, “Stability analysis of fractional order Holling type II prey predator model with diseased prey,” in AIP Conference Proceedings, vol. 3122, p. 040003, 2024, doi: 10.1063/5.0216020.

[25] D. N. Prabhumani, M. Shanmugam, S. P. Manickasundaram, and N. G. Thangaraj, “Dynamical analysis of a fractional order prey-predator model in Crowley–Martin functional response with prey harvesting,” Engineering Proceedings, vol. 56, p. 300, 2023, doi: 10.3390/ASEC2023-15975.




DOI: https://doi.org/10.18860/cauchy.v11i1.40676

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