Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle

  • Received: 01 September 2019 Revised: 01 December 2019
  • Primary: 35K57, 37C65; Secondary: 92D25

  • This paper deals with the initial value problem of a predator-prey system with dispersal and delay, which does not admit the classical comparison principle. When the initial value has nonempty compact support, the initial value problem formulates that two species synchronously invade the same habitat in population dynamics. By constructing proper auxiliary equations and functions, we confirm the faster invasion speed of two species, which equals to the minimal wave speed of traveling wave solutions in earlier works.

    Citation: Shuxia Pan. Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle[J]. Electronic Research Archive, 2019, 27: 89-99. doi: 10.3934/era.2019011

    Related Papers:

  • This paper deals with the initial value problem of a predator-prey system with dispersal and delay, which does not admit the classical comparison principle. When the initial value has nonempty compact support, the initial value problem formulates that two species synchronously invade the same habitat in population dynamics. By constructing proper auxiliary equations and functions, we confirm the faster invasion speed of two species, which equals to the minimal wave speed of traveling wave solutions in earlier works.



    加载中


    [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Partial Differential Equations and Related Topics, J.A. Goldstein Eds., Lecture Notes in Mathematics, Vol. 446. Springer, Berlin, German, (1975), 5–49.

    0427837

    [2] Bao X., Li W.-T. (2020) Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats. Nonlinear Anal. Real World Appl. 51: 102975, 26 pp.
    [3] Bao X., Li W.-T., Shen W., Wang Z.-C. (2018) Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems. J. Differential Equations 265: 3048-3091.
    [4] P. W. Bates, On some nonlocal evolution equations arising in materials science, In: Nonlinear Dynamics and Evolution Equations (Ed. by H. Brunner, X.Q. Zhao, X. Zou), Fields Inst. Commun., 48 (2006), 13–52, AMS, Providence.

    2223347

    [5] Coville J., Dupaigne L. (2007) On a non-local equation arising in population dynamics. Proc. Roy. Soc. Edinburgh Sect. A 137: 725-755.
    [6] Ding W., Liang X. (2015) Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media. SIAM J. Math. Anal. 47: 855-896.
    [7] Ducrot A. (2013) Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system,. J. Math. Pures Appl. 100: 1-15.
    [8] Ducrot A. (2016) Spatial propagation for a two component reaction-diffusion system arising in population dynamics. J. Differ. Equations 260: 8316-8357.
    [9] A. Ducrot, J. S. Guo, G. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), Art. 146, 25 pp.

    10.1007/s00033-019-1188-x

    3999344

    [10] Dunbar S. R. (1983) Travelling wave solutions of diffusive Lotka-Volterra equations. J. Math. Biol. 17: 11-32.
    [11] Fagan W. F., Bishop J. G. (2000) Trophic interactions during primary succession: Herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155: 238-251.
    [12] Fang J., Zhao X. Q. (2014) Traveling waves for monotone semiflows with weak compactness. SIAM J. Math. Anal. 46: 3678-3704.
    [13] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In: Trends in Nonlinear Analysis (Ed. by M. Kirkilionis, S. Kr$\ddot{o}$mker, R. Rannacher, F. Tomi), 153–191, Springer: Berlin, 2003.

    1999098

    [14] L. Hopf, Introduction to Differential Equations of Physics, Dover: New York, 1948.

    0025035

    [15] Jin Y., Zhao X. Q. (2009) Spatial dynamics of a periodic population model with dispersal. Nonlinearity 22: 1167-1189.
    [16] X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019), ID: 269.

    10.3390/math7030269

    [17] X. Li, S. Pan and H. B. Shi, Minimal wave speed in a dispersal predator-prey system with delays, Boundary Value Problems, 2018 (2018), Paper No. 49, 26 pp.

    10.1186/s13661-018-0966-2

    3782680

    [18] Liang X., Zhao X. Q. (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm. Pure Appl. Math. 60: 1-40.
    [19] Lin G. (2011) Spreading speeds of a Lotka-Volterra predator-prey system: the role of the predator. Nonlinear Analysis 74: 2448-2461.
    [20] Lin G. (2012) Asymptotic spreading fastened by inter-specific coupled nonlinearities: A cooperative system. Physica D 241: 705-710.
    [21] Lin G., Li W. T. (2012) Asymptotic spreading of competition diffusion systems: The role of interspecific competitions. Eur. J. Appl. Math. 23: 669-689.
    [22] Lin G., Li W. T., Ruan S. (2011) Spreading speeds and traveling waves of a competitive recursion. J. Math. Biol. 62: 165-201.
    [23] Lin G., Pan S., Yan X. P. (2019) Spreading speeds of epidemic models with nonlocal delays. Mathe. Biosci. Eng. 16: 7562-7588.
    [24] Lin G., Ruan S. (2014) Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays. J. Dyn. Differ. Equ. 26: 583-605.
    [25] Liu X. L., Pan S. (2019) Spreading speed in a nonmonotone equation with dispersal and delay. Mathematics 7: 291.
    [26] Lui R. (1989) Biological growth and spread modeled by systems of recursions. Ⅰ. Mathematical theory. Math. Biosci. 93: 269-295.
    [27] Martin R. H., Smith H. L. (1990) Abstract functional differential equations and reaction-diffusion systems,. Trans. Amer. Math. Soc. 321: 1-44.
    [28] J. D. Murray, Mathematical Biology, II. Spatial Models and Biomedical Applications., Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag: New York, 2003.

    1952568

    [29] Owen M. R., Lewis M. A. (2001) How predation can slow, stop or reverse a prey invasion. Bull. Math. Biol. 63: 655-684.
    [30] Pan S. (2013) Asymptotic spreading in a Lotka-Volterra predator-prey system. J. Math. Anal. Appl. 407: 230-236.
    [31] S. Pan, Convergence and traveling wave solutions for a predator-prey system with distributed delays, Mediterr. J. Math., 14 (2017), Art. 103, 15 pp.

    10.1007/s00009-017-0905-y

    3633363

    [32] Pan S. (2017) Invasion speed of a predator-prey system. Appl. Math. Lett. 74: 46-51.
    [33] Pan S., Lin G., Wang J. (2019) Propagation thresholds of competitive integrodifference systems. J. Difference Equ. Appl. 25: 1680-1705.
    [34] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press: Oxford, UK, 1997.
    [35] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS: Providence, RI, USA, 1995.

    1319817

    [36] Wang M., Zhang Y. (2018) Dynamics for a diffusive prey-predator model with different free boundaries. J. Differential Equations 264: 3527-3558.
    [37] Weinberger H. F. (1982) Long-time behavior of a class of biological model. SIAM J. Math. Anal. 13: 353-396.
    [38] Weinberger H. F., Lewis M. A., Li B. (2002) Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45: 183-218.
    [39] Weng P., Zhao X. Q. (2006) Spreading speed and traveling waves for a multi-type SIS epidemic model. J. Differential Equations 229: 270-296.
    [40] Ye Q., Li Z., Wang M., Wu Y. (2011) Introduction to Reaction Diffusion Equations.Science Press.
    [41] Yu Z., Yuan R. (2011) Travelling wave solutions in non-local convolution diffusive competitive-cooperative systems. IMA J. Appl. Math. 76: 493-513.
    [42] Zhang G., Li W. T., Lin G. (2009) Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure. Math. Comput. Model. 49: 1021-1029.
    [43] X. Q. Zhao, Spatial dynamics of some evolution systems in biology, In Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, Y. Du, H. Ishii, W.Y. Lin, Eds.; World Scientific: Singapore, 2009,332–363.

    10.1142/9789812834744_0015

    2532932

  • Reader Comments
  • © 2019 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(1547) PDF downloads(256) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog