Stability and Hopf bifurcation in an eco-epidemiological system with the cost of anti-predator behaviors

Chunmei Zhang; Suli Liu; Jianhua Huang; Weiming Wang; Chunmei Zhang; Suli Liu; Jianhua Huang; Weiming Wang

doi:10.3934/mbe.2023354

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 8146-8161. doi: 10.3934/mbe.2023354

Previous Article Next Article

Research article

Stability and Hopf bifurcation in an eco-epidemiological system with the cost of anti-predator behaviors

1.
College of Sciences, National University of Defense Technology, Changsha 410073, China
2.
School of Mathematics, Jilin University, Changchun 130012, China
3.
School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, China

Academic Editor: Yang Kuang

Received: 30 November 2022 Revised: 08 January 2023 Accepted: 15 February 2023 Published: 27 February 2023

The fear effect is a powerful force in prey-predator interaction, eliciting a variety of anti-predator responses which lead to a reduction of prey growth rate. To study the impact of the fear effect on population dynamics of the eco-epidemiological system, we develop a predator-prey interaction model that incorporates infectious disease in predator population as well as the cost of anti-predator behaviors. Detailed mathematical results, including well-posedness of solutions, stability of equilibria and the occurrence of Hopf bifurcation are provided. It turns out that population density diminishes with increasing fear, and the fear effect can either destabilize the stability or induce the occurrence of periodic behavior. The theoretical results here provide a sound foundation for understanding the effect of the anti-predator behaviors on the eco-epidemiological interaction.
- eco-epidemiological system,
- fear effect,
- stability,
- Hopf bifurcation
Citation: Chunmei Zhang, Suli Liu, Jianhua Huang, Weiming Wang. Stability and Hopf bifurcation in an eco-epidemiological system with the cost of anti-predator behaviors[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8146-8161. doi: 10.3934/mbe.2023354

Related Papers:

Abstract

The fear effect is a powerful force in prey-predator interaction, eliciting a variety of anti-predator responses which lead to a reduction of prey growth rate. To study the impact of the fear effect on population dynamics of the eco-epidemiological system, we develop a predator-prey interaction model that incorporates infectious disease in predator population as well as the cost of anti-predator behaviors. Detailed mathematical results, including well-posedness of solutions, stability of equilibria and the occurrence of Hopf bifurcation are provided. It turns out that population density diminishes with increasing fear, and the fear effect can either destabilize the stability or induce the occurrence of periodic behavior. The theoretical results here provide a sound foundation for understanding the effect of the anti-predator behaviors on the eco-epidemiological interaction.

References

[1]	K. P. Das, A mathematical study of a predator-prey dynamics with disease in predator, ISRN Appl. Math., 2011 (2011), 807486. https://doi.org/10.5402/2011/807486 doi: 10.5402/2011/807486
[2]	M. Haque, A predator-prey model with disease in the predator species only, Nonlinear Anal. Real World Appl., 11 (2010), 2224–2236. https://doi.org/10.1016/j.nonrwa.2009.06.012 doi: 10.1016/j.nonrwa.2009.06.012
[3]	D. Greenhalgh, Q. J. A. Khan, J. S. Pettigrew, An eco-epidemiological predator-prey model where predators distinguish between susceptible and infected prey, Math. Meth. Appl. Sci., 40 (2016), 146–166. https://doi.org/10.1002/mma.3974 doi: 10.1002/mma.3974
[4]	A. M. Bate, F. M. Hilker, Complex dynamics in an eco-epidemiological model, Bull. Math. Biol., 75 (2013), 2059–2078.
[5]	Y. Cai, Z. Gui, X. Zhang, H. Shi, W. Wang, Bifurcations and pattern formation in a predator-prey model, Int. J. Bifurcation Chaos, 28 (2018), 1850140. https://doi.org/10.1142/S0218127418501407 doi: 10.1142/S0218127418501407
[6]	E. Venturino, Epidemics in predator-prey models: disease in the predators, Math. Med. Biol., 19 (2002), 185–205. https://doi.org/10.1093/imammb/19.3.185 doi: 10.1093/imammb/19.3.185
[7]	F. Gulland, The Impact of Infectious Diseases on Wild Animal Populations: A Review, Cambridge University Press Cambridge, 1995.
[8]	A. Mondal, A. K. Pal, G. P. Samanta. On the dynamics of evolutionary leslie-gower predator-prey eco-epidemiological model with disease in predator, Ecol. Genet. Genom., 10 (2019), 100034. https://doi.org/10.1016/j.egg.2018.11.002 doi: 10.1016/j.egg.2018.11.002
[9]	I. M. Bulai, F. M. Hilker, Eco-epidemiological interactions with predator interference and infection, Theor. Popul. Biol., 130 (2019), 191–202. https://doi.org/10.1016/j.tpb.2019.07.016 doi: 10.1016/j.tpb.2019.07.016
[10]	E. Venturino. Epidemics in predator-prey models: Disease in the predators, IMA J. Math. Appl. Med., 19 (2002), 185–205. https://doi.org/10.1093/imammb/19.3.185 doi: 10.1093/imammb/19.3.185
[11]	M. Sieber, H. Malchow, F. M. Hilker. Disease-induced modification of prey competition in eco-epidemiological models, Ecol. Complex., 18 (2014), 74–82. https://doi.org/10.1016/j.ecocom.2013.06.002 doi: 10.1016/j.ecocom.2013.06.002
[12]	Y. Xiao, L. Chen. Analysis of a three species eco-epidemiological model, J. Math. Anal. Appl., 258 (2001), 733–754. https://doi.org/10.1006/jmaa.2001.7514 doi: 10.1006/jmaa.2001.7514
[13]	M. A. Aziz-Alaoui, M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett, 16 (2003), 1069–1075. https://doi.org/10.1016/S0893-9659(03)90096-6 doi: 10.1016/S0893-9659(03)90096-6
[14]	A. A. Shaikh, H. Das, An eco-epidemic predator-prey model with Allee effect in prey, Int. J. Bifurcation and Chaos, 30 (2020), 2050194. https://doi.org/10.1142/S0218127420501941 doi: 10.1142/S0218127420501941
[15]	S. Yu, Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Discrete Dyn. Nat. Soc., 2012 (2012), 857–868. https://doi.org/10.1155/2012/208167 doi: 10.1155/2012/208167
[16]	T. Qiao, Y. Cai, S. Fu, W. Wang, Stability and Hopf bifurcation in a predator-prey model with the cost of anti-predator behaviors, Int. J. Bifurcation Chaos, 29 (2019), 1950185. https://doi.org/10.1142/S0218127419501852 doi: 10.1142/S0218127419501852
[17]	S. K. Sasmal, Y. Takeuchi, Dynamics of a predator-prey system with fear and group defense, J. Math. Anal. Appl., 481 (2020), 123471. https://doi.org/10.1016/j.jmaa.2019.123471 doi: 10.1016/j.jmaa.2019.123471
[18]	X. Wang, Y. Tan, Y. Cai, W. Wang. Impact of the fear effect on the stability and bifurcation of a Leslie-Gower predator-prey model, Int. J. Bifurcation and Chaos, 30 (2020), 2050210. https://doi.org/10.1142/S0218127420502107 doi: 10.1142/S0218127420502107
[19]	X. Meng, J. Li, Dynamical behavior of a delayed prey-predator-scavenger system with fear effect and linear harvesting, Int. J. Biomath., 2021 (2021), 2150024. https://doi.org/10.1142/S1793524521500248 doi: 10.1142/S1793524521500248
[20]	J. Wang, Y. Cai, S. Fu, W. Wang, The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge, Chaos, 29 (2019), 083109. https://doi.org/10.1063/1.5111121 doi: 10.1063/1.5111121
[21]	P. Cong, M. Fan, X. Zou, Dynamics of a three-species food chain model with fear effect, Commun. Nonlinear Sci. Numer. Simulat., 2021 (2021), 105809. https://doi.org/10.1016/j.cnsns.2021.105809 doi: 10.1016/j.cnsns.2021.105809
[22]	M. Hossain, N. Pal, S. Samanta, Impact of fear on an eco-epidemiological model, Chaos Solitons Fractals, 134 (2020), 109718. https://doi.org/10.1016/j.chaos.2020.109718 doi: 10.1016/j.chaos.2020.109718
[23]	J. Liu, B. Liu, P. Lv, T. Zhang, An eco-epidemiological model with fear effect and hunting cooperation, Chaos Solitons Fractals, 142 (2021), 110494. https://doi.org/10.1016/j.chaos.2020.110494 doi: 10.1016/j.chaos.2020.110494
[24]	Y. Tan, Y. Cai, R. Yao, M. Hu, W. Wang, Complex dynamics in an eco-epidemiological model with the cost of anti-predator behaviors, Nonlinear Dyn., 107 (2022), 3127–3141. https://doi.org/10.1007/s11071-021-07133-4 doi: 10.1007/s11071-021-07133-4
[25]	X. Wang, X. Zou, Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators, Bull. Math. Biol., 79 (2017), 1325–1359. https://doi.org/10.1007/s11538-017-0287-0 doi: 10.1007/s11538-017-0287-0
[26]	H. Zhang, Y. Cai, S. Fu, W. Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge, Appl. Math. Comput., 356 (2019), 328–337. https://doi.org/10.1016/j.amc.2019.03.034 doi: 10.1016/j.amc.2019.03.034
[27]	X. Wang, L. Zanette, X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179–1204. https://doi.org/10.1007/s00285-016-0989-1 doi: 10.1007/s00285-016-0989-1
[28]	O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratior0in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382. https://doi.org/10.1016/0012-8252(90)90054-Y doi: 10.1016/0012-8252(90)90054-Y

Reader Comments

Your name:*

Email:*
© 2023 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)