Invading and Receding Sharp-Fronted Travelling Waves

Abstract

Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher–KPP equation. While the Fisher–KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher–KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the Fisher–Stefan model, which is a generalisation of the Fisher–KPP model obtained by reformulating the Fisher–KPP model as a moving boundary problem. The nondimensional Fisher–Stefan model involves just one parameter, \(\kappa \), which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, c. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher–Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between c and \(\kappa \) so that commonly reported experimental estimates of c can be used to provide estimates of the unknown parameter \(\kappa \). Interestingly, when we reinterpret the Fisher–KPP model as a moving boundary problem, many overlooked features of the classical Fisher–KPP phase plane take on a new interpretation since travelling waves solutions with \(c < 2\) are normally disregarded. This means that our analysis of the Fisher–Stefan model has both practical value and an inherent mathematical value.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  1. Bate AM, Hilker FM (2019) Preytaxis and travelling waves in an eco-epidemiological model. Bull Math Biol 81:995–1030

    MathSciNet  MATH  Google Scholar 

  2. Browning AP, McCue SW, Simpson MJ (2017) A Bayesian computational approach to explore the optimal the duration of a cell proliferation assay. Bull Math Biol 79:188–1906

    MathSciNet  MATH  Google Scholar 

  3. Browning AP, Haridas P, Simpson MJ (2019) A Bayesian sequential learning framework to parameterise continuum models of melanoma invasion into human skin. Bull Math Biol 81:676–698

    MathSciNet  MATH  Google Scholar 

  4. Buenzli PR, Lanaro M, Wong C, McLaughlin MP, Allenby MC, Woodruff MA, Simpson MJ (2020) Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size. Acta Biomater 114:285–295

    Google Scholar 

  5. Byrne HM (2010) Dissecting cancer through mathematics: from the cell to the animal model. Nat Rev Cancer 10:221–230

    Google Scholar 

  6. Cai AQ, Landman KA, Hughes BD (2007) Multi-scale modeling of a wound-healing cell migration assay. J Theor Biol 245:576–594

    MathSciNet  MATH  Google Scholar 

  7. Canosa J (1973) On a nonlinear diffusion equation describing population growth. IBM J Res Dev 17:307–313

    MathSciNet  MATH  Google Scholar 

  8. Chaplain MAJ, Lorenzi T, Mcfarlane FR (2020) Bridging the gap between individual-based and continuum models of growing cell populations. J Math Biol 80:343–371

    MathSciNet  MATH  Google Scholar 

  9. Courchamp F, Berec L, Gascoigne J (2008) Allee effects in ecology and conservation. Oxford University Press, Oxford

    Google Scholar 

  10. Crank J (1987) Free and moving boundary problems. Oxford University Press, Oxford

    Google Scholar 

  11. Curtin L, Hawkins-Daarud A, van der Zee KG, Swanson KR, Owen MR (2020) Speed switch in glioblastoma growth rate due to enhanced hypoxia-induced migration. Bull Math Biol 82:43

    MathSciNet  MATH  Google Scholar 

  12. Dalwadi MP, Waters SL, Byrne HM, Hewitt IJ (2020) A mathematical framework for developing freezing protocols in the cryopreservation of cells. SIAM J Appl Math 80:657–689

    MathSciNet  MATH  Google Scholar 

  13. Du Y, Lin Z (2010) Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J Math Anal 42:377–405

    MathSciNet  MATH  Google Scholar 

  14. Du Y, Lou B (2015) Spreading and vanishing in nonlinear diffusion problems with free boundaries. J Eur Math Soc 17:2673–2724

    MathSciNet  MATH  Google Scholar 

  15. Du Y, Matano H, Wang K (2014a) Regularity and asymptotic behavior of nonlinear Stefan problems. Arch Ration Mech Anal 212:957–1010

    MathSciNet  MATH  Google Scholar 

  16. Du Y, Matsuzawa H, Zhou M (2014b) Sharp estimate of the spreading speed determined by nonlinear free boundary problems. SIAM J Math Anal 46:375–396

    MathSciNet  MATH  Google Scholar 

  17. Edelstein-Keshet L (2005) Mathematical models in biology. SIAM, Philadelphia

    Google Scholar 

  18. El-Hachem M, McCue SW, Jin W, Du Y, Simpson MJ (2019) Revisiting the Fisher–Kolmogorov–Petrovsky–Piskunov equation to interpret the spreading-extinction dichotomy. Proc R Soc A Math Phys Eng Sci 475:20190378

    MathSciNet  Google Scholar 

  19. El-Hachem M, McCue SW, Simpson MJ (2020) A sharp-front moving boundary model for malignant invasion. Phys D Nonlinear Phenomena 412:132639

    MathSciNet  Google Scholar 

  20. Fadai NT, Simpson MJ (2020) Population dynamics with threshold effects give rise to a diverse family of Allee effects. Bull Math Biol 82:74

    MathSciNet  MATH  Google Scholar 

  21. Fife PC (1979) Long time behavior of solutions of bistable nonlinear diffusion equations. Arch Ration Mech Anal 70:31–36

    MathSciNet  MATH  Google Scholar 

  22. Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugenics 7:355–369

    MATH  Google Scholar 

  23. Flegg JA, Menon SN, Byrne HM, McElwain DLS (2020) A current perspective on wound healing and tumour-induced angiogenesis. Bull Math Biol 82:43

    MathSciNet  MATH  Google Scholar 

  24. Gaffney EA, Maini PK (1999) Modelling corneal epithelial wound closure in the presence of physiological electric fields via a moving boundary formalism. IMA J Math Appl Med Biol 16:369–393

    MATH  Google Scholar 

  25. Haridas P, McGovern JA, McElwain DLS, Simpson MJ (2017) Quantitative comparison of the spreading and invasion of radial growth phase and metastatic melanoma cells in a three-dimensional human skin equivalent model. PeerJ 5:e3754

    Google Scholar 

  26. Haridas P, Browning AP, McGovern JA, McElwain DLS, Simpson MJ (2018) Three-dimensional experiments and individual based simulations show that cell proliferation drives melanoma nest formation in human skin tissue. BMC Syst Biol 12:34

    Google Scholar 

  27. Hill JM (1987) One-dimensional Stefan problems: an introduction. Longman Scientific & Technical, Harlow

    Google Scholar 

  28. Horgan FG (2009) Invasion and retreat: shifting assemblages of dung beetles amidst changing agricultural landscapes in central Peru. Biodivers Conserv 18:3519

    Google Scholar 

  29. Ibrahim K, Sourrouille P, Hewitt GM (2000) Are recession populations of the desert locust (Schistocerca gregaria) remnants of past swarms? Mol Ecol 9:783–791

    Google Scholar 

  30. Jin W, Shah ET, Penington CJ, McCue SW, Chopin LK, Simpson MJ (2016) Reproducibility of scratch assays is affected by the initial degree of confluence: experiments, modelling and model selection. J Theor Biol 390:136–145

    MATH  Google Scholar 

  31. Jin W, Shah ET, Penington CJ, McCue SW, Maini PK, Simpson MJ (2017) Logistic proliferation of cells in scratch assays is delayed. Bull Math Biol 79:1028–1050

    MathSciNet  MATH  Google Scholar 

  32. Johnston ST, Shah ET, Chopin LK, McElwain DLS, Simpson MJ (2015) Estimating cell diffusivity and cell proliferation rate by interpreting IncuCyte ZOOM\(^{\rm TM}\) assay data using the Fisher-Kolmogorov model. BMC Syst Biol 9:38

    Google Scholar 

  33. Johnston ST, Ross JV, Biner BJ, McElwain DLS, Haridas P, Simpson MJ (2016) Quantifying the effect of experimental design choices for in vitro scratch assays. J Theor Biol 400:19–31

    Google Scholar 

  34. Johnston ST, Baker RE, McElwain DLS, Simpson MJ (2017) Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves. Sci Rep 7:42134

    Google Scholar 

  35. Keller EF, Segel LA (1971) Model for chemotaxis. J Theor Biol 30:225–234

    MATH  Google Scholar 

  36. Killengreen ST, Ims RA, Yoccoz NG, Bråthen KA, Henden J-A, Schott T (2007) Structural characteristics of a low Arctic tundra ecosystem and the retreat of the Arctic fox. Biol Conserv 135:459–472

    Google Scholar 

  37. Kolmogorov AN, Petrovskii PG, Piskunov NS (1937) A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Moscow Univ Math Bull 1:1–26

    Google Scholar 

  38. Kot M (2003) Elements of mathematical ecology. Cambridge University Press, Cambridge

    Google Scholar 

  39. Lewis MA, Kareiva P (1993) Allee dynamics and the spread of invading organisms. Theor Popul Biol 43:141–158

    MATH  Google Scholar 

  40. Maini PK, McElwain DLS, Leavesley DI (2004a) Traveling wave model to interpret a wound-healing cell migration assay for human peritoneal mesothelial cells. Tissue Eng 10:475–482

    Google Scholar 

  41. Maini PK, McElwain DLS, Leavesley D (2004b) Traveling waves in a wound healing assay. Appl Math Lett 17:575–580

    MathSciNet  MATH  Google Scholar 

  42. McCue SW, Jin W, Moroney TJ, Lo K-Y, Chou SE, Simpson MJ (2019) Hole-closing model reveals exponents for nonlinear degenerate diffusivity functions in cell biology. Phys D Nonlinear Phenomena 398:130–140

    MathSciNet  MATH  Google Scholar 

  43. McCue SW, El-Hachem M, Simpson MJ (2021) Exact sharp-fronted travelling wave solutions of the Fisher-KPP equation. Appl Math Lett. 114:106918. https://doi.org/10.1016/j.aml.2020.106918

    Article  MATH  Google Scholar 

  44. Mitchell SL, O’Brien SBG (2014) Asymptotic and numerical solutions of a free boundary problem for the sorption of a finite amount of solvent into a glassy polymer. SIAM J Appl Math 74:697–723

    MathSciNet  MATH  Google Scholar 

  45. Murray JD (1984) Asymptotic analysis. Springer, New York

    Google Scholar 

  46. Murray JD (2002) Mathematical biology I: an introduction, 3rd edn. Springer, New York

    Google Scholar 

  47. National Cancer Institute (1985) Melanoma

  48. Otto G, Bewick S, Li B, Fagan WF (2018) How phenological variation affects species spreading speeds. Bull Math Biol 80:1476–1513

    MathSciNet  MATH  Google Scholar 

  49. Painter KJ, Sherratt JA (2003) Modelling the movement of interacting cell populations. J Theor Biol 225:327–339

    MathSciNet  MATH  Google Scholar 

  50. Painter KJ, Bloomfield JM, Sherratt JA, Gerish A (2015) A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. J Math Biol 77:1132–1165

    MathSciNet  MATH  Google Scholar 

  51. Sánchez Garduno F, Maini PK (1994) Traveling wave phenomena in some degenerate reaction–diffusion equations. J Differ Equ 117:281–319

    MATH  Google Scholar 

  52. Sengers BG, Please CP, Oreffo ROC (2007) Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration. J R Soc Interface 4:1107–1117

    Google Scholar 

  53. Sherratt JA, Marchant BP (1996) Nonsharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion. Appl Math Lett 9:33–38

    MathSciNet  MATH  Google Scholar 

  54. Sherratt JA, Murray JD (1990) Models of epidermal wound healing. Proc R Soc Lond Ser B 241:29–36

    Google Scholar 

  55. Simpson MJ, Landman KA, Clement TP (2005) Assessment of a non-traditional operator split algorithm for simulation of reactive transport. Math Comput Simul 70:44–60

    MathSciNet  MATH  Google Scholar 

  56. Simpson MJ, Zhang DC, Mariani M, Landman KA, Newgreen DF (2007) Cell proliferation drives neural crest cell invasion of the intestine. Dev Biol 302:553–568

    Google Scholar 

  57. Simpson MJ, Treloar KK, Binder BJ, Haridas P, Manton KJ, Leavesley DI, McElwain DLS, Baker RE (2013) Quantifying the roles of motility and proliferation in a circular barrier assay. J R Soc Interface 10:20130007

    Google Scholar 

  58. Sinkins PA, Otfinowski R (2012) Invasion or retreat? The fate of exotic invaders on the northern prairies, 40 years after cattle grazing. Plant Ecol 213:1251–1262

    Google Scholar 

  59. Skellam JG (1951) Random dispersal in theoretical populations. Biometrika 38:196–218

    MathSciNet  MATH  Google Scholar 

  60. Strobl MAR, Krause AL, Damaghi M, Gillies R, Anderson ARA, Maini PK (2020) Mix and match: phenotypic coexistence as a key facilitator of cancer invasion. Bull Math Biol 82:15

    MathSciNet  MATH  Google Scholar 

  61. Swanson KR, Bridge C, Murray JD, Alvord EC Jr (2003) Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J Neurol Sci 216:1–10

    Google Scholar 

  62. Taylor CM, Hastings A (2005) Allee effects in biological invasions. Ecol Lett 8:895–908

    Google Scholar 

  63. Tsoularis A, Wallace J (2002) Analysis of logistic growth models. Math Biosci 179:21–55

    MathSciNet  MATH  Google Scholar 

  64. Warne DJ, Baker RE, Simpson MJ (2019) Using experimental data and information criteria to guide model selection for reaction–diffusion problems in mathematical biology. Bull Math Biol 81:1760–1804

    MathSciNet  MATH  Google Scholar 

  65. Witelski TP (1995) Merging traveling waves for the Porous–Fisher’s equation. Appl Math Lett 8:57–62

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Stuart Johnston, Sean McElwain and two anonymous referees for helpful suggestions and feedback. This work is supported by the Australian Research Council (DP200100177).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Matthew J. Simpson.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

El-Hachem, M., McCue, S.W. & Simpson, M.J. Invading and Receding Sharp-Fronted Travelling Waves. Bull Math Biol 83, 35 (2021). https://doi.org/10.1007/s11538-021-00862-y

Download citation

Keywords

  • Invasion
  • Reaction–diffusion
  • Partial differential equation
  • Stefan problem
  • Moving boundary problem