Advertisement

Compactly Supported Solutions of Reaction–Diffusion Models of Biological Spread

  • Maureen P. Edwards
  • Bronwyn H. Bradshaw-Hajek
  • María Jesús Munoz-Lopez
  • Peter M. Waterhouse
  • Robert S. AnderssenEmail author
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 28)

Abstract

Lie group analysis is one of the most useful techniques for analyzing the analytic structure of the solutions of differential equations. Here, reaction–diffusion (RD) modelling of biological invasion is used to illustrate this fact in terms of identifying the conditions that the diffusion and reaction terms must satisfy for their solutions to have compact support. Biological invasion, such as the spread of viruses on the leaves of plants and the invasive spread of animals and weeds into new environments, has a well-defined progressing compactly supported spatial \(\mathbb {R}^2\) structure. There are two distinct ways in which such progressing compact structure can be modelled mathematically; namely, cellular automata modelling and reaction–diffusion (RD) equation modelling. The goal in this paper is to review the extensive literature on RD equations to investigate the extent to which RD equations are known to have compactly supported solutions. Though the existence of compactly supported solutions of nonlinear diffusion equations, without reaction, is well documented, the conditions that the reaction terms should satisfy in conjunction with such nonlinear diffusion equations, for the compact support to be retained, has not been examined in specific detail. A possible partial connection relates to the results of Arrigo, Hill, Goard and Broadbridge, who examined, under various symmetry analysis assumptions, situations where the diffusion and reaction terms are connected by explicit relationships. However, it was not investigated whether the reaction terms generated by these relationships are such that the compact support of the solutions is maintained. Here, results from a computational analysis for the addition of different reaction terms to power law diffusion are presented and discussed. It appears that whether or not the reaction term is zero, as a function of its argument at zero, is an important consideration. In addition, it is confirmed algebraically and graphically that the shapes of compactly supported solutions are strongly controlled by the choice of the reaction term.

Keywords

Biological invasion Plant viruses Gene silencing Reaction–diffusion Compactly supported solutions Symmetry analysis 

Notes

Acknowledgements

The authors would like to acknowledge the discussion with Rick Loy that led to introducing the condition that q(0) must equal zero in order to guarantee compactly supported solutions. RSA, BH and ME greatly appreciate the financial support of the Institute for Mathematics and Its Applications (IMIA) at the University of Wollongong which has underpinned their collaboration related to the research reported here. They also acknowledge the important mentoring that they have received over the years from Phil Broadbridge.

References

  1. 1.
    R.S. Anderssen, P.M. Waterhouse, Antiviral resistance in plants in Modelling Antiviral Resistance in Plants - Methods in Molecular 894, ed. by J. Watson, M.-B. Wang (Humana Press, 2012), pp. 139–140Google Scholar
  2. 2.
    M. Arim, S.R. Abades, P.E. Neill, M. Lima, P.A. Marquet, Spread dynamics of invasive species. PNAS 103, 374–378 (2006)CrossRefGoogle Scholar
  3. 3.
    D.J. Arrigo, J.M. Hill, Nonclassical symmetries for nonlinear diffusion and absorption. Stud. Appl. Math. 94, 21–39 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    B. Basse, M. Plank, Modelling biological invasions over homogeneous and inhomogeneous landscapes using level set methods. Biol. Invasions 10, 157–167 (2008)CrossRefGoogle Scholar
  5. 5.
    G.W. Bluman, S. Kumei, Symmetries and Differential Equations (Springer, Berlin, 1989)Google Scholar
  6. 6.
    G.W. Bluman, J.D. Cole, General similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    P. Broadbridge, B.H. Bradshaw-Hajek, Exact solutions for logistic reaction-diffusion in biology. Zeitschrift Angew. Mathematik Phys. 67, Article 93 (2016)Google Scholar
  8. 8.
    B.L. Cheeseman, D.F. Newgreen, K.A. Landman, Spatial and temporal dynamics of cell generations within an invasion wave: a link to cell lineage tracing. J. Theor. Biol. 363, 344–356 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Eamens, M.-B. Wang, N.A. Smith, P.M. Waterhouse, RNA silencing in plants: yesterday, today, and tomorrow. Plant Physiol. 147, 456–468 (2008)CrossRefGoogle Scholar
  10. 10.
    M.P. Edwards, P.M. Waterhouse, M.J. Munoz-Lopez, R.S. Anderssen, Nonlinear diffusion and viral spread through the leaf of a plant. Zeitschrift Angew. Mathematik Phys. 67, Article 112 (2016)Google Scholar
  11. 11.
    W.F. Fagan, M.A. Lewis, M.G. Neubert, P. Van Den Driessche, Invasion theory and biological control. Ecol. Lett. 5, 148–157 (2002)CrossRefGoogle Scholar
  12. 12.
    R.A. Fisher, The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)CrossRefzbMATHGoogle Scholar
  13. 13.
    J. Goard, P. Broadbridge, Nonclassical symmetry analysis of nonlinear reaction-diffusion equations in two spatial dimensions. Nonlinear Anal. Theory Methods Appl. 26, 735–754 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M.A.C. Groenenboom, P. Hogeweg, RNA silencing can explain chlorotic infection patterns on plant leaves. BMC Syst. Biol. 2, Article 105 (2008)Google Scholar
  15. 15.
    D.A. Herms, D.G. McCullough, Emerald ash borer invasion of North America: history, biology, ecology, impacts, and management. Annu. Rev. Entomol. 59, 13–30 (2014)CrossRefGoogle Scholar
  16. 16.
    J.M. Hill, Differential Equations and Group Methods for Scientists and Engineers (CRC Press, Boca Raton, 1992)Google Scholar
  17. 17.
    N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws (CRC Press, Boca Raton, 1994)Google Scholar
  18. 18.
    J.R. King, Exact solutions to some nonlinear diffusion equations. Q. J. Mech. Appl. Math. 42, 407–409 (1989)MathSciNetGoogle Scholar
  19. 19.
    J.C. Larkin, N. Young, M. Prigge, M.D. Marks, The control of trichome spacing and number in Arabidopsis. Development 122, 997–1005 (1996)Google Scholar
  20. 20.
    S. Lie, Uber die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen. Arch. Math. 6, 328–368 (1881)zbMATHGoogle Scholar
  21. 21.
    X. Li, A.G.-O. Yeh, Neural-network-based cellular automata for simulating multiple land use changes using GIS. Int. J. Geogr. Inform. Sci. 16, 323–343 (2002)CrossRefGoogle Scholar
  22. 22.
    D.G. Mallet, L.G. De Pillis, A cellular automata model of tumor-immune system interactions. J. Theor. Biol. 239, 334–350 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    P.J. Olver, Applications of Lie Groups to differential equations (Springer, New York, 1993)Google Scholar
  24. 24.
    R.E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient. Q. J. Mech. Appl. Math. 12, 407–409 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    J.R. Philip, J.H. Knight, Redistribution of soil water from plane, line, and point sources. Irrig. Sci. 12, 169–180 (1991)CrossRefGoogle Scholar
  26. 26.
    E.P. Rybicki, A Top Ten list for economically important plant viruses. Arch. Virol. 160, 17–20 (2015)CrossRefGoogle Scholar
  27. 27.
    G.C. Sander, R.D. Braddock, Analytical solutions to the transient, unsaturated transport of water and contaminants through horizontal porous media. Adv. Water Resour. 28, 1102–1111 (2005)CrossRefGoogle Scholar
  28. 28.
    N. Shigesada, K. Kawaasaki, Biological Invasion: Theory and Practice (Oxford University Press, 1997)Google Scholar
  29. 29.
    Ya.B. Zeldovich, A.S. Kompaneets, On the theory of propagation of heat with the heat conductivity depending upon the temperature, in Collection in Honor of the Seventieth Birthday of Academician, ed. by A.F. Ioffe (1950), pp. 61–71Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Maureen P. Edwards
    • 1
  • Bronwyn H. Bradshaw-Hajek
    • 2
  • María Jesús Munoz-Lopez
    • 3
  • Peter M. Waterhouse
    • 4
  • Robert S. Anderssen
    • 5
    • 6
    Email author
  1. 1.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia
  2. 2.School of Phenomics and Bioinformatics Research Centre, School of Information Technology and Mathematical SciencesUniversity of South AustraliaMawson LakesAustralia
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  4. 4.Centre for Tropical Crops and BiocommoditiesQueensland University of TechnologyBrisbaneAustralia
  5. 5.CSIRO Data61CanberraAustralia
  6. 6.Department of Mathematics and StatisticsLa Trobe UniversityMelbourneAustralia

Personalised recommendations