Search Advanced

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 6064-6084. doi: 10.3934/mbe.2020322
Previous Article Next Article
Research article Special Issues

A consumer-resource competition model with a state-dependent delay and stage-structured consumer species

  • 1.
    Key Laboratory of Eco-environments in Three Gorges Reservoir Region (Ministry of Education), School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
  • 2.
    School of Mathematics and Statistics, Nanning Normal University, Nanning 530023, China
  • Received: 28 June 2020 Accepted: 04 September 2020 Published: 14 September 2020

  • In this paper, a new consumer-resource competition model with a state-dependent maturity delay is developed, which incorporates one resource species and two stage-structured consumer species. The main innovation is that the model directly manifests the relationship between resources and maturity time of consumers through a correction term, $1-\tau'(x(t))x'(t)$. Firstly, the well-posedness of the solution is studied. At the same time, the existence and uniqueness of all equilibria are discussed. Then, the linearized stabilities of the equilibria are achieved. Finally, some sufficient conditions which ensure the global attractivity of the coexistence equilibrium are obtained.

    Citation: Yan Wang, Xianning Liu, Yangjiang Wei. A consumer-resource competition model with a state-dependent delay and stage-structured consumer species[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 6064-6084. doi: 10.3934/mbe.2020322

    Related Papers:

  • In this paper, a new consumer-resource competition model with a state-dependent maturity delay is developed, which incorporates one resource species and two stage-structured consumer species. The main innovation is that the model directly manifests the relationship between resources and maturity time of consumers through a correction term, $1-\tau'(x(t))x'(t)$. Firstly, the well-posedness of the solution is studied. At the same time, the existence and uniqueness of all equilibria are discussed. Then, the linearized stabilities of the equilibria are achieved. Finally, some sufficient conditions which ensure the global attractivity of the coexistence equilibrium are obtained.


    加载中


    [1] J. H. Connell, The influence of interspecific competition and other factors on the distribution of the barnacle chthamalus stellatus, Ecology, 42 (1961), 710-723.
    [2] P. K. Dayton, Competition, disturbance and community organisation; the provision and subsequent utilization of space in a rocky intertidal community, Ecol. Monogr., 41 (1971), 351-389.
    [3] R. S. Miller, Pattern and process in competition, Adv. Ecol. Res., 4 (1967), 1-74.
    [4] F. G. Bader, J. S. Meyer, A. G. Fredrickson, H. M. Tsuchiya, Comments on microbial growth rate, Biotechnol. Bioeng., 17 (1975), 279-283.
    [5] A. M. De Roos, J. A. J. Metz, L. Persson, Ontogenetic symmetry and asymmetry in energetics, J. Math. Biol., 66 (2013), 889-914.
    [6] L. Persson, A. M. de Roos, Symmetry breaking in ecological systems through different energy efficiencies of juveniles and adults, Ecology, 94 (2013), 1487-1498.
    [7] D. Tilman, Resource competition and community structure, Monogr. Popul. Biol., 17 (1982), 1-296.
    [8] K. Gopalsamy, Convergence in a resource-based competition system, Bull. Math. Biol., 48 (1986), 681-699.
    [9] H. D. Landahl, B. D. Hansen, A three stage population model with cannibalism, Bull. Math. Biol., 37 (1975), 11-17.
    [10] P. J. Wangersky, W. J. Cunningham, On time lags in equations of growth, Proc. Natl. Acad. Sci. USA, 42 (1956), 699-702.
    [11] P. J. Wangersky, W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139.
    [12] W. G. Aiello, H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.
    [13] K. Gopalsamy, Time lags and global stability in two-species competition, Bull. Math. Biol., 42 (1980), 729-737.
    [14] R. Gambell, Birds and mammals-antarctic whales, in Antarctica (eds. W. Bonner and D. Walton), Pergamon Press, NewYork, (1985), 223-241.
    [15] W. G. Aiello, H. I. Freedman, J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.
    [16] J. F. M. Al-Omari, S. A. Gourley, Stability and traveling fronts in Lotka-Volterra competition models with stage structure, SIAM J. Appl. Math., 63 (2003), 2063-2086.
    [17] J. F. M. Al-Omari, The effect of state dependent delay and harvesting on a stage-structured predator-prey model, Appl. Math. Comput., 271 (2015), 142-153.
    [18] T. Cassidy, M. Craig, A. R. Humphries, Equivalences between age structured models and state dependent distributed delay differential equations, Math. Biosci. Eng., 16 (2019), 5419-5450.
    [19] F. Chen, D. Sun, J. Shi, Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl., 288 (2003), 136-146.
    [20] Q. Hu, X.-Q. Zhao, Global dynamics of a state-dependent delay model with unimodal feedback, J. Math. Anal. Appl., 399 (2013), 133-146.
    [21] M. Kloosterman, S. A. Campbell, F. J. Poulin, An NPZ model with state-dependent delay due to size-structure in juvenile zooplankton, SIAM J. Appl. Math., 76 (2016), 551-577.
    [22] Y. Lv, Y. Pei, R. Yuan, Modeling and analysis of a predator-prey model with state-dependent delay, Int. J. Biomath., 11 (2018), 1850026(1-22).
    [23] F. M. G. Magpantay, N. Kosovalic, An age-structured population model with state-dependent delay: derivation and numerical integration, SIAM J. Numer. Anal., 52 (2014), 735-756.
    [24] A. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, J. Math. Anal. Appl., 326 (2007), 1031-1045.
    [25] L. Zhang, S. Guo, Slowly oscillating periodic solutions for the Nicholson's blowflies equation with state-dependent delay, Math. Methods Appl. Sci., 145 (2017), 4893-4903.
    [26] Z. Zhou, Z. Yang, Periodic solutions in higher-dimensional Lotka-Volterra neutral competition systems with state-dependent delays, Appl. Math. Comput., 189 (2007), 986-995.
    [27] Y. Lv, R. Yuan, Y. Pei, T. Li, Global stability of a competitive model with state-dependent delay, J. Dyn. Differ. Equ., 29 (2017), 501-521.
    [28] Y. Wang, X. Liu, Y. Wei, Dynamics of a stage-structured single population model with state-dependent delay, Adv. Differ. Equ., 2018 (2018), 364.
    [29] M. V. Barbarossa, K. P. Hadeler, C. Kuttler, State-dependent neutral delay equations from population dynamics, J. Math. Biol., 69 (2014), 1027-1056.
    [30] Y. Lou, X.-Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573-603.
    [31] N. Macdonald, Biological Delay Systems: Linear Stability Theory, Cambridge University Press, New York, 1989.
    [32] J. M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics, Philadephia, 1998.
    [33] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, CRC Press, 1985.
    [34] K. L. Cooke, W. Z. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 1417-1426.
    [35] X. Li, J. Wei, On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays, Chaos Soliton. Fract., 26 (2005), 519-526.
  • Reader Comments
  • © 2020 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
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(3292) PDF downloads(60) Cited by(1)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog