The dynamics of a simple Laissez-Faire model with two predators

  • Received: 01 May 2008 Accepted: 29 June 2018 Published: 01 December 2008
  • MSC : 34D05, 34D20, 92D25

  • In this paper, we study the dynamics of a laissez-faire predator--prey model with both a specialist and a generalist predator. We analyze the stabilities of equilibria by performing linearized stability analyses. We then reexamine the stability of the equilibrium where the prey and predator coexist by constructing a Lyapunov function. If we hold the generalist predator population constant, treating it as a bifurcation parameter, we show that our model can possess multiple (up to three) limit cycles that surround an equilibrium in the interior of the first quadrant. Our model shows rich dynamics including fold, transcritical, pitchfork, Hopf, cyclic-fold, and Bautin bifurcations as well as heteroclinic connections. If we instead vary the generalist predator population slowly across bifurcations, the model exhibits bursting behavior as it alternates between a repetitive spiking phase and a quiescent phase.

    Citation: Gunog Seo, Mark Kot. The dynamics of a simple Laissez-Faire model with two predators[J]. Mathematical Biosciences and Engineering, 2009, 6(1): 145-172. doi: 10.3934/mbe.2009.6.145

    Related Papers:

    [1] Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi . Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences and Engineering, 2018, 15(4): 883-904. doi: 10.3934/mbe.2018040
    [2] Yun Kang, Sourav Kumar Sasmal, Komi Messan . A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences and Engineering, 2017, 14(4): 843-880. doi: 10.3934/mbe.2017046
    [3] Christian Cortés García, Jasmidt Vera Cuenca . Impact of alternative food on predator diet in a Leslie-Gower model with prey refuge and Holling Ⅱ functional response. Mathematical Biosciences and Engineering, 2023, 20(8): 13681-13703. doi: 10.3934/mbe.2023610
    [4] Xue Xu, Yibo Wang, Yuwen Wang . Local bifurcation of a Ronsenzwing-MacArthur predator prey model with two prey-taxis. Mathematical Biosciences and Engineering, 2019, 16(4): 1786-1797. doi: 10.3934/mbe.2019086
    [5] Saheb Pal, Nikhil Pal, Sudip Samanta, Joydev Chattopadhyay . Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model. Mathematical Biosciences and Engineering, 2019, 16(5): 5146-5179. doi: 10.3934/mbe.2019258
    [6] Mengyun Xing, Mengxin He, Zhong Li . Dynamics of a modified Leslie-Gower predator-prey model with double Allee effects. Mathematical Biosciences and Engineering, 2024, 21(1): 792-831. doi: 10.3934/mbe.2024034
    [7] Xiao Wu, Shuying Lu, Feng Xie . Relaxation oscillations of a piecewise-smooth slow-fast Bazykin's model with Holling type Ⅰ functional response. Mathematical Biosciences and Engineering, 2023, 20(10): 17608-17624. doi: 10.3934/mbe.2023782
    [8] Lazarus Kalvein Beay, Agus Suryanto, Isnani Darti, Trisilowati . Hopf bifurcation and stability analysis of the Rosenzweig-MacArthur predator-prey model with stage-structure in prey. Mathematical Biosciences and Engineering, 2020, 17(4): 4080-4097. doi: 10.3934/mbe.2020226
    [9] Kawkab Al Amri, Qamar J. A Khan, David Greenhalgh . Combined impact of fear and Allee effect in predator-prey interaction models on their growth. Mathematical Biosciences and Engineering, 2024, 21(10): 7211-7252. doi: 10.3934/mbe.2024319
    [10] Peter A. Braza . A dominant predator, a predator, and a prey. Mathematical Biosciences and Engineering, 2008, 5(1): 61-73. doi: 10.3934/mbe.2008.5.61
  • In this paper, we study the dynamics of a laissez-faire predator--prey model with both a specialist and a generalist predator. We analyze the stabilities of equilibria by performing linearized stability analyses. We then reexamine the stability of the equilibrium where the prey and predator coexist by constructing a Lyapunov function. If we hold the generalist predator population constant, treating it as a bifurcation parameter, we show that our model can possess multiple (up to three) limit cycles that surround an equilibrium in the interior of the first quadrant. Our model shows rich dynamics including fold, transcritical, pitchfork, Hopf, cyclic-fold, and Bautin bifurcations as well as heteroclinic connections. If we instead vary the generalist predator population slowly across bifurcations, the model exhibits bursting behavior as it alternates between a repetitive spiking phase and a quiescent phase.


  • This article has been cited by:

    1. Ashrafi M. Niger, Abba B. Gumel, Mathematical analysis of the role of repeated exposure on malaria transmission dynamics, 2008, 16, 0971-3514, 251, 10.1007/s12591-008-0015-1
    2. Gamaliel Blé, Luis Miguel Valenzuela, Manuel Falconi, Coexistence of populations in a Leslie-Gower tritrophic model with Holling-type functional responses, 2024, 10, 24058440, e38207, 10.1016/j.heliyon.2024.e38207
  • Reader Comments
  • © 2009 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2409) PDF downloads(491) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog