Loading [Contrib]/a11y/accessibility-menu.js

The global stability of coexisting equilibria for three models of mutualism

  • Received: 01 January 2015 Accepted: 29 June 2018 Published: 01 October 2015
  • MSC : Primary: 92D25, 92D40; Secondary: 34D20, 34D23, 93D30.

  • We analyze the dynamics of three models of mutualism, establishing the global stability of coexisting equilibria by means of Lyapunov's second method. This further establishes the usefulness of certain Lyapunov functionals of an abstract nature introduced in an earlier paper. As a consequence, it is seen that the use of higher order self-limiting terms cures the shortcomings of Lotka-Volterra mutualisms, preventing unbounded growth and promoting global stability.

    Citation: Paul Georgescu, Hong Zhang, Daniel Maxin. The global stability of coexisting equilibria for three models of mutualism[J]. Mathematical Biosciences and Engineering, 2016, 13(1): 101-118. doi: 10.3934/mbe.2016.13.101

    Related Papers:

    [1] N. H. AlShamrani, A. M. Elaiw . Stability of an adaptive immunity viral infection model with multi-stages of infected cells and two routes of infection. Mathematical Biosciences and Engineering, 2020, 17(1): 575-605. doi: 10.3934/mbe.2020030
    [2] Xinran Zhou, Long Zhang, Tao Zheng, Hong-li Li, Zhidong Teng . Global stability for a class of HIV virus-to-cell dynamical model with Beddington-DeAngelis functional response and distributed time delay. Mathematical Biosciences and Engineering, 2020, 17(5): 4527-4543. doi: 10.3934/mbe.2020250
    [3] Toshikazu Kuniya, Tarik Mohammed Touaoula . Global stability for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearity. Mathematical Biosciences and Engineering, 2020, 17(6): 7332-7352. doi: 10.3934/mbe.2020375
    [4] Andrei Korobeinikov, William T. Lee . Global asymptotic properties for a Leslie-Gower food chain model. Mathematical Biosciences and Engineering, 2009, 6(3): 585-590. doi: 10.3934/mbe.2009.6.585
    [5] Jinliang Wang, Jingmei Pang, Toshikazu Kuniya . A note on global stability for malaria infections model with latencies. Mathematical Biosciences and Engineering, 2014, 11(4): 995-1001. doi: 10.3934/mbe.2014.11.995
    [6] Yu Ji . Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences and Engineering, 2015, 12(3): 525-536. doi: 10.3934/mbe.2015.12.525
    [7] Miled El Hajji, Frédéric Mazenc, Jérôme Harmand . A mathematical study of a syntrophic relationship of a model of anaerobic digestion process. Mathematical Biosciences and Engineering, 2010, 7(3): 641-656. doi: 10.3934/mbe.2010.7.641
    [8] Jinliang Wang, Ran Zhang, Toshikazu Kuniya . A note on dynamics of an age-of-infection cholera model. Mathematical Biosciences and Engineering, 2016, 13(1): 227-247. doi: 10.3934/mbe.2016.13.227
    [9] Pengyan Liu, Hong-Xu Li . Global behavior of a multi-group SEIR epidemic model with age structure and spatial diffusion. Mathematical Biosciences and Engineering, 2020, 17(6): 7248-7273. doi: 10.3934/mbe.2020372
    [10] Tewfik Sari, Miled El Hajji, Jérôme Harmand . The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat. Mathematical Biosciences and Engineering, 2012, 9(3): 627-645. doi: 10.3934/mbe.2012.9.627
  • We analyze the dynamics of three models of mutualism, establishing the global stability of coexisting equilibria by means of Lyapunov's second method. This further establishes the usefulness of certain Lyapunov functionals of an abstract nature introduced in an earlier paper. As a consequence, it is seen that the use of higher order self-limiting terms cures the shortcomings of Lotka-Volterra mutualisms, preventing unbounded growth and promoting global stability.


    [1] Comput. Math. Appl., 67 (2014), 2127-2143.
    [2] in Evolutionary Conservation Biology (eds. R. Ferrière, U. Dieckmann and D. Couvet), Cambridge University Press, (2004), 305-326.
    [3] Princeton University Press, Princeton, 2010.
    [4] SIAM J. Appl. Math., 67 (2006), 337-353.
    [5] Nonlinear Anal.: Real World Appl., 11 (2010), 3653-3665.
    [6] Appl. Math. Comput., 219 (2013), 8496-8507.
    [7] Appl. Math. Comput., 226 (2014), 754-764.
    [8] Am. Nat., 113 (1979), 261-275.
    [9] Bull. Math. Biol., 68 (2006), 1851-1872.
    [10] J. Math. Biol., 8 (1979), 159-171.
    [11] in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, J. Wu and Y. Zhou), Series in Contemporary Applied Mathematics (CAM), Higher Education Press, 11 (2009), 216-236.
    [12] in Population Dynamics, Vol 3 of Encyclopedia of Ecology (eds. S.E. Jorgensen and B.D. Fath), Elsevier, (2008), 2485-2491.
    [13] Ecology, 91 (2010), 1286-1295.
    [14] Math. Med. Biol., 21 (2004), 75-83.
    [15] Bull. Math. Biol., 68 (2006), 615-626.
    [16] Math. Med. Biol., 26 (2009), 309-321.
    [17] in Theoretical Ecology: Principles and Application (ed. R. M. May), Saunders, (1976), 78-104.
    [18] Math. Biosci. Eng., 6 (2009), 603-610.
    [19] Math. Biosci. Eng., 10 (2013), 369-378.
    [20] Ecology, 88 (2007), 3004-3011.
    [21] Forest Science, 19 (1973), 2-22.
    [22] J. Exp. Bot., 10 (1959), 290-300.
    [23] J. Theor. Biol., 74 (1978), 549-558.
    [24] Appl. Math. Comput., 219 (2012), 2493-2497.
    [25] Abstraction & Application, 9 (2013), 50-61.
    [26] Biomatemática, 23 (2013), 139-146.
    [27] Am. Nat., 124 (1984), 843-862.
  • This article has been cited by:

    1. Saikat Batabyal, Debaldev Jana, Jingjing Lyu, Rana D. Parshad, Explosive predator and mutualistic preys: A comparative study, 2020, 541, 03784371, 123348, 10.1016/j.physa.2019.123348
    2. Rusliza Ahmad, Global stability of two-species mutualism model with proportional harvesting, 2017, 4, 2313626X, 74, 10.21833/ijaas.2017.07.011
    3. Gabriel Dimitriu, Răzvan Ştefănescu, Ionel M. Navon, Comparative numerical analysis using reduced-order modeling strategies for nonlinear large-scale systems, 2017, 310, 03770427, 32, 10.1016/j.cam.2016.07.002
    4. Paul Georgescu, Daniel Maxin, Hong Zhang, Global stability results for models of commensalism, 2017, 10, 1793-5245, 1750037, 10.1142/S1793524517500371
    5. Paul Georgescu, Daniel Maxin, Laurentiu Sega, Hong Zhang, 2019, 40, 9780444641526, 85, 10.1016/bs.host.2018.09.001
    6. D. Maxin, P. Georgescu, L. Sega, L. Berec, Global stability of the coexistence equilibrium for a general class of models of facultative mutualism, 2017, 11, 1751-3758, 339, 10.1080/17513758.2017.1343871
    7. Ke Qi, Zhijun Liu, Lianwen Wang, Qinglong Wang, Survival and stationary distribution of a stochastic facultative mutualism model with distributed delays and strong kernels, 2021, 18, 1551-0018, 3160, 10.3934/mbe.2021157
    8. Paul Georgescu, Hong Zhang, Facultative Mutualisms and θ-Logistic Growth: How Larger Exponents Promote Global Stability of Co-Existence Equilibria, 2023, 11, 2227-7390, 4373, 10.3390/math11204373
    9. Paul Georgescu, Hong Zhang, Global stability of coexistence equilibria for n-species models of facultative mutualism, 2024, 00225193, 111961, 10.1016/j.jtbi.2024.111961
  • Reader Comments
  • © 2016 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(3033) PDF downloads(672) Cited by(9)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog