A study of a prey-predator model with disease in predator including gestation delay, treatment and linear harvesting of predator species

Nada A. Almuallem; Hasanur Mollah; Sahabuddin Sarwardi; Nada A. Almuallem; Hasanur Mollah; Sahabuddin Sarwardi

doi:10.3934/math.2025660

AIMS Mathematics

2025, Volume 10, Issue 6: 14657-14698. doi: 10.3934/math.2025660

Previous Article Next Article

Research article

A study of a prey-predator model with disease in predator including gestation delay, treatment and linear harvesting of predator species

1.
Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah - 21589, Saudi Arabia
2.
Department of Mathematics and Statistics, Aliah University, IIA/27, New Town, Kolkata -700 160, India

Received: 17 March 2025 Revised: 21 May 2025 Accepted: 26 May 2025 Published: 26 June 2025
MSC : 92D25, 92D40, 34D20, 37G15

In this work, we investigated a predator-prey model based on ecodemiology in which the predator population was infected with a transmissible illness, and the time delay equaled the predator's gestation period. Predators can be divided into two separate groups: The ones that are infected and the ones that are susceptible to infection. Assume that the infected predator's sickness is thought to be treatable. The explanations for the solutions' positive invariance and the reason there are equilibria are stated in the suggested system. We showed the boundedness of the system solutions. Additionally, the local stability of every possible equilibrium point was investigated with respect to delayed as well as non-delayed systems. Furthermore, time delay has been demonstrated to be essential for controlling the dynamics of the system, and the periodic solutions were known to exist through Hopf bifurcation in connection to gestation delay. Also, our finding showed that treatment for infected predators have an influence on the dynamics of the system. Lastly, a computational simulation is used to verify the conceptual inquiry's conclusion.
- prey-predator,
- linear harvesting,
- gestation time delay,
- disease,
- treatment,
- numerical simulations
Citation: Nada A. Almuallem, Hasanur Mollah, Sahabuddin Sarwardi. A study of a prey-predator model with disease in predator including gestation delay, treatment and linear harvesting of predator species[J]. AIMS Mathematics, 2025, 10(6): 14657-14698. doi: 10.3934/math.2025660

Related Papers:

Abstract

In this work, we investigated a predator-prey model based on ecodemiology in which the predator population was infected with a transmissible illness, and the time delay equaled the predator's gestation period. Predators can be divided into two separate groups: The ones that are infected and the ones that are susceptible to infection. Assume that the infected predator's sickness is thought to be treatable. The explanations for the solutions' positive invariance and the reason there are equilibria are stated in the suggested system. We showed the boundedness of the system solutions. Additionally, the local stability of every possible equilibrium point was investigated with respect to delayed as well as non-delayed systems. Furthermore, time delay has been demonstrated to be essential for controlling the dynamics of the system, and the periodic solutions were known to exist through Hopf bifurcation in connection to gestation delay. Also, our finding showed that treatment for infected predators have an influence on the dynamics of the system. Lastly, a computational simulation is used to verify the conceptual inquiry's conclusion.

References

[1]	A. J. Lotka, Elements of physical biology, Williams and Wilkins, Baltimore, 1925.
[2]	V. Volterra, Variations and fluctuations of a number of individuals in animl species living together, Animal Ecology, Mcgraw Hill, 1926.
[3]	N. Bairagi, P. K. Roy, J. Chattopadhyay, Role of infection on the stability of a predator-prey system with several response functions–a comparative study, J. Theor. Biol., 2007. https://doi.org/10.1016/j.jtbi.2007.05.005 doi: 10.1016/j.jtbi.2007.05.005
[4]	S. S. Maity, P. K. Tiwari, Z. Shuai, S. Pal, Role of space in an eco-epidemic predator-prey system with the effect of fear and selective predation, J. Biol. Syst., World Scientific Publishing Co, 2023.
[5]	D. Barman, S. Roy, P. K. Tiwari, S. Alam, Two-fold impacts of fear in a seasonally forced predator-prey system with Cosner functional response, J. Biol. Syst., World Scientific Publishing Co, 2023. https://doi.org/10.1142/S0218339023500183
[6]	R. Kumbhakar, S. Pal, N. Pal, P. K. Tiwari, Bistability and tristability in a predator-prey model with strong Allee effect in prey, J. Biol. Syst., PB - World Scientific Publishing Co, 2023. https://doi.org/10.1142/S0218339023500110
[7]	S. Sarwardi, M. Haque, S. Hossain, Analysis of Bogdanov–Takens bifurcations in a spatiotemporal harvested-predator and prey system with Beddington–DeAngelis-type response function, Nonlinear Dynam., 2020. https://doi.org/10.1007/s11071-020-05549-y
[8]	E. Venturino, Epidemics in predator-prey models: Disease in the predators, IMA J. Math. Appl. Med. Biol., 2002. https://doi.org/10.1093/imammb/19.3.185 doi: 10.1093/imammb/19.3.185
[9]	Z. Guo, W. Li, L. Cheng, Z. Li, An eco-epidemiological model with epidemic and response function in the predator, J. Lanzhou Univ. Nat. Sci., 2009.
[10]	S. Sarwardi, M. Haque, E. Venturino, Global stability and persistence in LG-Holling type Ⅱ diseased predator ecosystems, J. Biol. Phys., 2011. https://doi.org/10.1007/s10867-010-9201-9 doi: 10.1007/s10867-010-9201-9
[11]	A. A. Shaikh, H. Das, N. Ali, Study of LG-Holling type Ⅲ predator-prey model with disease in predator, J. Appl. Math. Comput., 2017. https://doi.org/10.1007/s12190-017-1142-z doi: 10.1007/s12190-017-1142-z
[12]	E. Francomano, M. F. Hilker, M. Paliaga, E. Venturino, Diseased social predators, Bull. Math. Biol., 2017. https://doi.org/10.1007/s11538-017-0325-y doi: 10.1007/s11538-017-0325-y
[13]	M. Haque, A predator-prey model with disease in the predator species only, Nonlinear Anal. Real., 2010. https://doi.org/10.1016/j.nonrwa.2009.06.012 doi: 10.1016/j.nonrwa.2009.06.012
[14]	W. Lihong, F. Zhang, C. Jin, Analysis of an eco-epidemiological model with disease in the prey and predator, Int. J. Math. Res., 2017.
[15]	S. Kant, V. Kumar, Stability analysis of predator-prey system with migrating prey and disease infection in both species, Appl. Math. Model., 2016. https://doi.org/10.1016/j.apm.2016.10.003 doi: 10.1016/j.apm.2016.10.003
[16]	L. Han, Z. Ma, H. W. Hethcote, Four predator prey models with infectious diseases, Math. Comput. Model., 34 (2001), 849–858. https://doi.org/10.1016/S0895-7177(01)00104-2 doi: 10.1016/S0895-7177(01)00104-2
[17]	Y. Hsieh, C. Hsiao, Predator–prey model with disease infection in both populations, Math. Med. Biol., 25 (2008), 247–266. https://doi.org/10.1093/imammb/dqn017 doi: 10.1093/imammb/dqn017
[18]	A. Hugo, S. Massawe, O. Makinde, An eco-epidemiological mathematical model with treatment and disease infection in both prey and predator population, J. Ecol. Nat. Environ., 4 (2012), 266–279. https://doi.org/10.5897/JENE12.013 doi: 10.5897/JENE12.013
[19]	B. Mondal, A. Sarkar, N. Sk, Treatment of infected predators under the influence of fear-induced refuge, Sci. Rep., 13 (2023), 16623. https://doi.org/10.1038/s41598-023-43021-0 doi: 10.1038/s41598-023-43021-0
[20]	M. G. Cojocaru, T. Migot, A. Jaber. Controlling infection in predator-prey systems with transmission dynamics, Infect. Dis. Model., 5 (2020), 1–11. https://doi.org/10.1016/j.idm.2019.12.002 doi: 10.1016/j.idm.2019.12.002
[21]	J. Tripathi, S. Tyagi, S. Abbas, Global analysis of a delayed density dependent predator-prey model with Crowley-Martin functional response, Commun. Nonlinear Sci., 2015.
[22]	L. Wang, R. Xu, G. Feng, Modelling and analysis of an eco-epidemiological model with time delay and stage structure, J. Appl. Math. Comput., 2015. https://doi.org/10.1007/s12190-014-0865-3 doi: 10.1007/s12190-014-0865-3
[23]	A. M. Elaiw, N. A. Almuallem, Global properties of delayed-HIV dynamics models with differential drug efficacy in cocirculating target cells, Appl. Math. Comput., 2015.
[24]	A. M. Elaiw, N. A. Almuallem, Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Mat. Meth. Appl. Sci., 2015.
[25]	Q. Liu, M. Sun, T. Li, Analysis of an SIRS epidemic model with time delay on heterogeneous network, Adv. Differ. Eqn., 2017. https://doi.org/10.1186/s13662-017-1367-z doi: 10.1186/s13662-017-1367-z
[26]	P. Sen, S. Samanta, M. Y. Khan, S. Mandal, P. K. Tiwari, A seasonally forced eco-epidemic model with disease in predator and incubation delay, J. Biol. Syst., World Scientific Publishing Co, 2023.
[27]	R. Xu, S. Zhang, Modelling and analysis of a delayed predator–prey model with disease in the predator, Appl. Math. Comput., 2013.
[28]	X. Zhou, J. Cui, X. Shi, X. Song, A modified Leslie- Gower predator-prey model with prey infection, Appl. Math. Comput., 2010.
[29]	G. P. Hu, X. L. Li, Stability and hopf bifurcation for a delayed predator-prey model with disease in the prey, Chaos Soliton. Fract., 2012.
[30]	Z. Zhang, H. Yang, Hopf bifurcation control in a delayed predator-prey system with prey infection and modified Leslie-Gower scheme, Abst. Appl. Anal., 2013. https://doi.org/10.1155/2013/704320 doi: 10.1155/2013/704320
[31]	J. Zhang, W. Li, X. P. Yan, Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system, Appl. Math. Comput., 2008. https://doi.org/10.1016/j.amc.2007.09.045 doi: 10.1016/j.amc.2007.09.045
[32]	N. Bairagi, D. Jana, On the stability and Hopf bifurcation of a delay-induced predator-prey system with habitat complexity, Appl. Math. Model., 2011. https://doi.org/10.1016/j.apm.2011.01.025 doi: 10.1016/j.apm.2011.01.025
[33]	L. Deng, X. Wang, M. Peng, Hopf bifurcation analysis for a ratio-dependent predator-prey system with two delays and stage structure for the predator, Appl. Math. Comp., 2014. https://doi.org/10.1016/j.amc.2014.01.025 doi: 10.1016/j.amc.2014.01.025
[34]	X. Zhou, J. Cui, Stability and Hopf bifurcation analysis of an eco-epidemiological model with delay, J. Franklin Inst., 2010.
[35]	Y. Kuang, Delay differential equation with application in population dynamics, 1993.
[36]	G. Birkhoff, G. C. Rota, Ordinary differential equations, Ginn Boston., 1982.
[37]	J. K. Hale, S. M. V. Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-4342-7
[38]	B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and applications of Hopf bifurcation, Cambridge University Press Cambridge, New York, 1981.
[39]	Y. Song, J. Wei, Bifurcation analysis for Chen's system with delayed feedback and its application to control of chaos, Chaos Soliton. Fract., 2004.
[40]	A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 1985. https://doi.org/10.1016/0167-2789(85)90011-9

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)