Stability and Hopf bifurcation of an intraguild prey-predator fishery model with two delays and Michaelis-Menten type predator harvest

Min Hou; Tonghua Zhang; Sanling Yuan; Min Hou; Tonghua Zhang; Sanling Yuan

doi:10.3934/mbe.2024251

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 4: 5687-5711. doi: 10.3934/mbe.2024251

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Stability and Hopf bifurcation of an intraguild prey-predator fishery model with two delays and Michaelis-Menten type predator harvest

1.
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2.
Department of Mathematics, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia

Academic Editor: Xiangsheng Wang

Received: 19 January 2024 Revised: 03 April 2024 Accepted: 08 April 2024 Published: 22 April 2024

In this paper, we have proposed and investigated an intraguild predator-prey system incorporating two delays and a harvesting mechanism based on the Michaelis-Menten principle, and it was assumed that the two species compete for a shared resource. Firstly, we examined the properties of the relevant characteristic equations to derive sufficient conditions for the asymptotical stability of equilibria in the delayed model and the existence of Hopf bifurcation. Using the normal form method and the central manifold theorem, we analyzed the stability and direction of periodic solutions arising from Hopf bifurcations. Our theoretical findings were subsequently validated through numerical simulations. Furthermore, we explored the impact of harvesting on the quantity of biological resources and examined the critical values associated with the two delays.
- predator-prey system,
- Michaelis-Menten type harvesting,
- delay,
- Hopf bifurcation,
- stability
Citation: Min Hou, Tonghua Zhang, Sanling Yuan. Stability and Hopf bifurcation of an intraguild prey-predator fishery model with two delays and Michaelis-Menten type predator harvest[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5687-5711. doi: 10.3934/mbe.2024251

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Abstract

In this paper, we have proposed and investigated an intraguild predator-prey system incorporating two delays and a harvesting mechanism based on the Michaelis-Menten principle, and it was assumed that the two species compete for a shared resource. Firstly, we examined the properties of the relevant characteristic equations to derive sufficient conditions for the asymptotical stability of equilibria in the delayed model and the existence of Hopf bifurcation. Using the normal form method and the central manifold theorem, we analyzed the stability and direction of periodic solutions arising from Hopf bifurcations. Our theoretical findings were subsequently validated through numerical simulations. Furthermore, we explored the impact of harvesting on the quantity of biological resources and examined the critical values associated with the two delays.

References

[1]	A. J. Lotka, Elements of Physical Biology, Williams & Wilkins, 1925. https://doi.org/10.1038/116461b0
[2]	V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES. J. Mar. Sci., 3 (1928), 3–51. https://doi.org/10.1093/icesjms/3.1.3 doi: 10.1093/icesjms/3.1.3
[3]	N. Bacaër, A Short History of Mathematical Population Dynamics, Springer, 618 (2011). https://doi.org/10.1007/978-0-85729-115-8
[4]	J. A. Rosenheim, H. K. Kaya, L. E. Ehler, B. A. Jaffee, Intraguild predation among biological-control agents: theory and evidence, Biol. Control, 5 (1995), 303–335. https://doi.org/10.1006/bcon.1995.1038 doi: 10.1006/bcon.1995.1038
[5]	J. M. Fedriani, T. K. Fuller, R. M. Sauvajot, E. C. York, Competition and intraguild predation among three sympatric carnivores, Oecologia, 125 (2000), 258–270. https://doi.org/10.1007/s004420000448 doi: 10.1007/s004420000448
[6]	E. T. Borer, C. J. Briggs, W. W. Murdoch, S. L. Swarbrick, Testing intraguild predation theory in a field system: does numerical dominance shift along a gradient of productivity?, Ecol. Lett., 6 (2003), 929–935. https://doi.org/10.1046/j.1461-0248.2003.00515.x doi: 10.1046/j.1461-0248.2003.00515.x
[7]	G. A. Polis, C. A. Myers, R. D. Holt, The ecology and evolution of intraguild predation: potential competitors that eat each other, Annu. Rev. Ecol. S., 20 (1989), 297–330. https://doi.org/10.1146/annurev.es.20.110189.001501 doi: 10.1146/annurev.es.20.110189.001501
[8]	G. A. Polis, R. D. Holt, Intraguild predation: the dynamics of complex trophic interactions, Trends Ecol. Evol., 7 (1992), 151–154. https://doi.org/10.1016/0169-5347(92)90208-S doi: 10.1016/0169-5347(92)90208-S
[9]	D. R. Hart, Intraguild predation, invertebrate predators, and trophic cascades in lake food webs, J. Theor. Biol., 218 (2002), 111–128. https://doi.org/10.1006/jtbi.2002.3053 doi: 10.1006/jtbi.2002.3053
[10]	H. V. Moeller, M. G. Neubert, M. D. Johnson, Intraguild predation enables coexistence of competing phytoplankton in a well-mixed water column, Ecology, 100 (2019), e02874. https://doi.org/10.1002/ecy.2874 doi: 10.1002/ecy.2874
[11]	H. M. Safuan, H. S. Sidhu, Z. Jovanoski, I. N. Towers, Impacts of biotic resource enrichment on a predator-prey population, Bull. Math. Biol., 75 (2013), 1798–1812. https://doi.org/10.1007/s11538-013-9869-7 doi: 10.1007/s11538-013-9869-7
[12]	K. A. Fordjour, R. D. Parshad, M. A. Beauregard, Dynamics of a predator-prey model with generalized Holling type functional response and mutual interference, Math. Biosci., 326 (2020), 108407. https://doi.org/10.1016/j.mbs.2020.108407 doi: 10.1016/j.mbs.2020.108407
[13]	M. H. Mohd, Diversity in interaction strength promotes rich dynamical behaviours in a three-species ecological system, Appl. Math. Comput., 353 (2019), 243–253. https://doi.org/10.1016/j.amc.2019.02.007 doi: 10.1016/j.amc.2019.02.007
[14]	X. Meng, N. Qin, H. Huo, Dynamics of a food chain model with two infected predators. International Journal of Bifurcation and Chaos, Int. J. Bifurcation Chaos, 31 (2021). https://doi.org/10.1142/S021812742150019X doi: 10.1142/S021812742150019X
[15]	S. Korpinen, E. Bonsdorff, Eutrophication and Hypoxia: Impacts of Nutrient and Organic Enrichment, Cambridge University Press, (2015), 202–243. https://doi.org/10.1017/CBO9781139794763.008
[16]	X. Chen, X. Wang, Qualitative analysis and control for predator-prey delays system, Chaos, Solitons Fractals, 123 (2019), 361–372. https://doi.org/10.1016/j.chaos.2019.04.023 doi: 10.1016/j.chaos.2019.04.023
[17]	Y. Lv, Y. Pei, Y. Wang, Bifurcations and simulations of two predator-prey models with nonlinear harvesting, Chaos, Solitons Fractals, 120 (2019), 158–170. https://doi.org/10.1016/j.chaos.2018.12.038 doi: 10.1016/j.chaos.2018.12.038
[18]	S. Chakravarty, L. N. Guin, S. Ghosh, Mathematical modelling of intraguild predation and its dynamics of resource harvesting, Int. J. Nonlinear Anal., 13 (2022), 837–861. https://doi.org/10.22075/ijnaa.2022.26067.3215 doi: 10.22075/ijnaa.2022.26067.3215
[19]	X. Wang, Y. Wang, Novel dynamics of a predator-prey system with harvesting of the predator guided by its population, Appl. Math. Model., 42 (2017), 636–654. https://doi.org/10.1016/j.apm.2016.10.006 doi: 10.1016/j.apm.2016.10.006
[20]	T. Yu, S. Yuan, Dynamic analysis of a stage-structured forest population model with non-smooth continuous threshold harvesting, Appl. Math. Model., 120 (2023), 1–24. https://doi.org/10.1016/j.apm.2023.03.026 doi: 10.1016/j.apm.2023.03.026
[21]	K. Chaudhuri, A bioeconomic model of harvesting a multispecies fishery, Ecol. Model., 32 (1986), 267–279. https://doi.org/10.1016/0304-3800(86)90091-8 doi: 10.1016/0304-3800(86)90091-8
[22]	T. Das, R. N. Mukherjee, K. S. Chaudhuri, Harvesting of a prey-predator fishery in the presence of toxicity, Appl. Math. Model., 33 (2009), 2282–2292. https://doi.org/10.1016/j.apm.2008.06.008 doi: 10.1016/j.apm.2008.06.008
[23]	T. K. Ang, H. M. Safuan, Harvesting in a toxicated intraguild predator-prey fishery model with variable carrying capacity, Chaos, Solitons Fractals, 126 (2019), 158–168. https://doi.org/10.1016/j.chaos.2019.06.004 doi: 10.1016/j.chaos.2019.06.004
[24]	C. W. Clark, Aggregation and fishery dynamics: a theoretical study of schooling and the purse seine tuna fisheries, Fish B-Noaa, 77 (1979), 317–337.
[25]	R. P. Gupta, P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278–295. https://doi.org/10.1016/j.jmaa.2012.08.057 doi: 10.1016/j.jmaa.2012.08.057
[26]	D. Hu, H. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Anal. Real World Appl., 33 (2017), 58–82. https://doi.org/10.1016/j.nonrwa.2016.05.010 doi: 10.1016/j.nonrwa.2016.05.010
[27]	T. K. Ang, H. M. Safuan, Dynamical behaviors and optimal harvesting of an intraguild prey-predator fishery model with Michaelis-Menten type predator harvesting, Biosystems, 202 (2021), 104357. https://doi.org/10.1016/j.biosystems.2021.104357 doi: 10.1016/j.biosystems.2021.104357
[28]	U. S. B. U. Sharif, M. H. Mohd, Combined influences of environmental enrichment and harvesting mediate rich dynamics in an intraguild predation fishery system, Ecol. Modell., 474 (2022), 110140. https://doi.org/10.1016/j.ecolmodel.2022.110140 doi: 10.1016/j.ecolmodel.2022.110140
[29]	Y. Shao, Global stability of a delayed predator-prey system with fear and Holling-type Ⅱ functional response in deterministic and stochastic environments, Math. Comput. Simul., 200 (2022), 65–77. https://doi.org/10.1016/j.matcom.2022.04.013 doi: 10.1016/j.matcom.2022.04.013
[30]	X. Wang, M. Peng, X. Liu, Stability and Hopf bifurcation analysis of a ratio-dependent predator-prey model with two time delays and Holling type Ⅲ functional response, Appl. Math. Comput., 268 (2015), 496–508. https://doi.org/10.1016/j.amc.2015.06.108 doi: 10.1016/j.amc.2015.06.108
[31]	A. Kumar, B. Dubey, Modeling the effect of fear in a prey-predator system with prey refuge and gestation delay, Int. J. Bifurcation Chaos, 29 (2019), 1950195. https://doi.org/10.1142/S0218127419501955 doi: 10.1142/S0218127419501955
[32]	K. Li, J. Wei, Stability and Hopf bifurcation analysis of a prey-predator system with two delays, Chaos, Solitons Fractals, 42 (2009), 2606–2613. https://doi.org/10.1016/j.chaos.2009.04.001 doi: 10.1016/j.chaos.2009.04.001
[33]	B. Dubey, A. Kumar, A. P. Maiti, Global stability and Hopf-bifurcation of prey-predator system with two discrete delays including habitat complexity and prey refuge, Commun. Nonlinear Sci., 67 (2019), 528–554. https://doi.org/10.1016/j.cnsns.2018.07.019 doi: 10.1016/j.cnsns.2018.07.019
[34]	S. Li, S. Yuan, Z. Jin, H. Wang, Bifurcation analysis in a diffusive predator-prey model with spatial memory of prey, Allee effect and maturation delay of predator, J. Differ. Equations, 357 (2023), 32–63. https://doi.org/10.1016/j.jde.2023.02.009 doi: 10.1016/j.jde.2023.02.009
[35]	T. K. Kar, U. K. Pahari, Modelling and analysis of a prey-predator system with stage-structure and harvesting, Nonlinear Anal. Real World Appl., 8 (2007), 601–609. https://doi.org/10.1016/j.nonrwa.2006.01.004 doi: 10.1016/j.nonrwa.2006.01.004
[36]	C. Xu, S. Yuan, Stability and Hopf bifurcation in a delayed predator-prey system with herd behavior, Abstr. Appl. Anal., 2014 (2014), 568943. https://doi.org/10.1155/2014/568943 doi: 10.1155/2014/568943
[37]	R. Shi, J. Yu, Hopf bifurcation analysis of two zooplankton-phytoplankton model with two delays, Chaos, Solitons Fractals, 100, (2017), 62–73. https://doi.org/10.1016/j.chaos.2017.04.044 doi: 10.1016/j.chaos.2017.04.044
[38]	Y. Kuang, Delay Differential Equations: with Applications in Population Dynamics, Academic Press, 1993.
[39]	S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Math. Med. Biol.: J. IMA, 10 (2003), 863–874. https://doi.org/10.1093/imammb/18.1.41 doi: 10.1093/imammb/18.1.41
[40]	B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981. https://doi.org/10.1137/1024123
[41]	X. Lin, H. Wang, Stability analysis of delay differential equations with two discrete delays, Can. Appl. Math. Q., 20 (2012), 519–533.
[42]	Q. An, E. Beretta, Y. Kuang, C. Wang, H. Wang, Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters, J. Differ. Equations, 266 (2019), 7073–7100. https://doi.org/10.1016/j.jde.2018.11.025 doi: 10.1016/j.jde.2018.11.025

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