Research article Special Issues

Biological advection and cross-diffusion with parameter regimes

  • Received: 17 October 2018 Accepted: 27 November 2019 Published: 23 December 2019
  • MSC : 35K57, 35Q92

  • Advection and cross-diffusion terms are obtained as dispersal strategies of biological species. The focus of the paper is their connection to a given population dynamics. In particular, meaningful parameter regimes as biological dispersal are obtained. Eventually, we obtain a systematic approach to construct an advection or a cross-diffusion term from a given population dynamics and find meaningful parameter regimes as biological advection and cross-diffusion.

    Citation: Jaywan Chung, Yong-Jung Kim, Ohsang Kwon, Chang-Wook Yoon. Biological advection and cross-diffusion with parameter regimes[J]. AIMS Mathematics, 2019, 4(6): 1721-1744. doi: 10.3934/math.2019.6.1721

    Related Papers:

  • Advection and cross-diffusion terms are obtained as dispersal strategies of biological species. The focus of the paper is their connection to a given population dynamics. In particular, meaningful parameter regimes as biological dispersal are obtained. Eventually, we obtain a systematic approach to construct an advection or a cross-diffusion term from a given population dynamics and find meaningful parameter regimes as biological advection and cross-diffusion.



    加载中


    [1] R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons Ltd., Chichester, 2003.
    [2] J. Cebrian, Energy flows in ecosystems, Science, 349 (2015), 1053-1054.
    [3] E. Cho and Y.-J. Kim, Starvation driven diffusion as a survival strategy of biological organisms, Bull. Math. Biol., 75 (2013), 845-870.
    [4] D. Cohen and S. A. Levin, Dispersal in patchy environments: the effects of temporal and spatial structure, Theoret. Population Biol., 39 (1991), 63-99.
    [5] L. Desvillettes, Y.-J. Kim, A. Trescases, et al. A logarithmic chemotaxis model featuring global existence and aggregation, Nonlinear Analysis: Real World Applications, 50 (2019), 562-582.
    [6] U. Dieckman, B. O'Hara and W. Weisser, The evolutionary ecology of dispersal, Trends Ecol. Evol., 14 (1999), 88-90.
    [7] J. Dockery, V. Hutson, K. Mischaikow, et al. The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol., 37 (1998), 61-83.
    [8] I. A. Hatton, K. S. McCann, J. M. Fryxell, et al. The predator-prey power law: Biomass scaling across terrestrial and aquatic biomes, Science, 349 (2015).
    [9] R. Holt and M. McPeek, Chaotic population dynamics favors the evolution of dispersal, The American Naturalist, 148 (1996), 709-718.
    [10] V. Hutson, K. Mischaikow and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.
    [11] M. Johnson and M. Gaines, Evolution of dispersal: Theoretical models and empirical tests using birds and mammels, Ann. Rev. Ecol. Syst., 21 (1990), 449-480.
    [12] M. Keeling, Spatial models of interacting populations, advanced ecological theory: Principles and applications, J. McGlade, ed. Blackwell Science, Oxford, 1999.
    [13] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
    [14] E. F. Keller, L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
    [15] Y.-J. Kim, O. Kwon and F. Li, Evolution of dispersal toward fitness, Bull. Math. Biol., 75 (2013), 2474-2498.
    [16] Y.-J. Kim, O. Kwon and F. Li, Global asymptotic stability and the ideal free distribution in a starvation driven diffusion, J. Math. Biol., 68 (2014), 1341-1370.
    [17] Y.-J. Kim and H. Seo, Model for heterogeneous diffusion, SIAM Appl. Math., 2019.
    [18] Y.-J. Kim, H. Seo and C. Yoon, Asymmetric dispersal and evolutional selection in two-patch system, Discrete Contin. Dyn. Syst., 2019.
    [19] K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst., 28 (2010), 1051-1067.
    [20] M. McPeek and R. Holt, The evolution of dispersal in spatially and temporally varying environments, The American Naturalist, 140 (1992), 1010-1027.
    [21] J. D. Murray, Non-existence of wave solutions for the class of reaction-diffusion equations given by the Volterra interacting-population equations with diffusion, J. Theoret. Biol., 52 (1975), 459-469.
    [22] T. Nagylaki, Introduction to theoretical population genetics, Biomathematics, vol. 21, SpringerVerlag, Berlin, 1992.
    [23] W.-M. Ni, The mathematics of diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.
    [24] A. Okubo and S. A. Levin, Diffusion and ecological problems: modern perspectives, second ed., Interdisciplinary Applied Mathematics, vol. 14, Springer-Verlag, New York, 2001.
    [25] J. G. Skellam, Some phylosophical aspects of mathematical modelling in empirical science with special reference to ecology, Mathematical Models in Ecology, Blackwell Sci. Publ., London, 1972.
    [26] J. G. Skellam, The formulation and interpretation of mathematical models of diffusionary processes in population biology, The mathematical theory of the dynamics of biological populations, Academic Press, New York, 1973.
    [27] J. M. J. Travis and C. Dytham, Habitat persistence, habitat availability and the evolution of dispersal, Proc. Roy. Soc. London B., 266 (1999), 723-728.
  • 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(4195) PDF downloads(502) Cited by(4)

Article outline

Figures and Tables

Figures(11)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog