Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion

Author(s)
Muhammad Shakeel

ABSTRACT

In this paper we study one-dimensional Fisher-Kolmogorov equation with density dependent non-linear diffusion. We choose the diffusion as a function of cell density such that it is high in highly cell populated areas and it is small in the regions of fewer cells. The Fisher equation with non-linear diffusion is known as modified Fisher equation. We study the travelling wave solution of modified Fisher equation and find the approximation of minimum wave speed analytically, by using the eigenvalues of the stationary states, and numerically by using COMSOL (a commercial finite element solver). The results reveal that the minimum wave speed depends on the parameter values involved in the model. We observe that when diffusion is moderately non-linear, the eigenvalue method correctly predicts the minimum wave speed in our numerical calculations, but when diffusion is strongly non-linear the eigenvalues method gives the wrong answer.

In this paper we study one-dimensional Fisher-Kolmogorov equation with density dependent non-linear diffusion. We choose the diffusion as a function of cell density such that it is high in highly cell populated areas and it is small in the regions of fewer cells. The Fisher equation with non-linear diffusion is known as modified Fisher equation. We study the travelling wave solution of modified Fisher equation and find the approximation of minimum wave speed analytically, by using the eigenvalues of the stationary states, and numerically by using COMSOL (a commercial finite element solver). The results reveal that the minimum wave speed depends on the parameter values involved in the model. We observe that when diffusion is moderately non-linear, the eigenvalue method correctly predicts the minimum wave speed in our numerical calculations, but when diffusion is strongly non-linear the eigenvalues method gives the wrong answer.

Cite this paper

M. Shakeel, "Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion,"*Applied Mathematics*, Vol. 4 No. 8, 2013, pp. 148-160. doi: 10.4236/am.2013.48A021.

M. Shakeel, "Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion,"

References

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[24] F. Sánchez-Garduno and P. K. Maini, “Existence and Uni queness of a Sharp Traveling Wave in Degenerate Non Linear Diffusion Fisher-KPP Equations,” Journal of Ma thematical Biology, Vol. 33, No. 2, 1994, pp. 163-192.

[25] R. Luther, “Propagation of Chemical Reactions in Space,” Zeitschrift für Elektrochemie und Angewandte Physika lische Chemie, Vol. 12, No. 32, 1906, p. 596. doi:10.1002/bbpc.19060123208

[26] M. J. Ablowitz and A. Zeppetella, “Explicit Solutions of Fisher’s Equation for a Special Wave Speed,” Bulletin of Mathematical Biology, Vol. 41, No. 6, 1979, pp. 835-840.

[27] A. M. Wazwaz, “The tanh Method for Traveling Wave Solutions of Nonlinear Equations,” Applied Mathematics and Computation, Vol. 154, No. 3, 2004, pp. 713-723.

[28] M. B. A. Mansour, “Accurate Computation of Traveling Wave Solutions of Some Nonlinear Diffusion Equations,” Wave Motion, Vol. 44, No. 3, 2007, pp. 222-230.

[29] Y. Tan, H. Xu and S. J. Liao, “Explicit Series Solution of Travelling Waves with a Front of Fisher Equation,” Chaos, Solitons & Fractals, Vol. 31, No. 2, 2007, pp. 462-472.

[30] J. Q. Mo, W. J. Zhang and M. He, “Asymptotic Method of Travelling Wave Solutions for a Class of Nonlinear Reaction Diffusion Equation,” Acta Mathematica Scientia, Vol. 27, No. 4, 2007, pp. 777-780.

[31] Z. Feng, S. Zheng and D. Y. Gao, “Traveling Wave Solu tions to a Reaction Diffusion Equation,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 60, No. 4, 2009, pp. 756-773.

[32] X. Hou, Y. Li and K. R. Meyer, “Traveling Wave Solu tions for a Reaction Diffusion Equation with Double De generate Nonlinearities,” Discrete and Continuous Dy namical Systems, Vol. 26, No. 1, 2010, pp. 265-290.

[33] M. Shakeel, P. C. Matthews, S. L. Waters and R. Graham, “Continuum Modeling of Cell Growth and Nutrient Trans port in a Perfusion Bioreactor,” 2011, pp. 90-117.

[34] M. Shakeel, “Continuum Modelling of Cell Growth and Nutrient Transport in a Perfusion Bioreactor,” Ph.D. Thesis, The University of Nottingham, Nottingham, 2011.

[35] F. Coletti, S. Macchietto and N. Elvassore, “Mathematical Modeling of Three Dimensional Cell Cultures in Perfu sion Bioreactors,” Industrial & Engineering Chemistry Research, Vol. 45, No. 24, 2006, pp. 8158-8169.

[36] F. Rothe, “Convergence to Pushed Fronts,” Rocky Moun tain Journal of Mathematics, Vol. 11, No. 4, 1981, pp. 617-634.

[37] W. Van Saarloos, “Front Propagation into Unstable States,” Physics Reports, Vol. 386, No. 2-6, 2003, pp. 29-222.

[38] A. N. Stokes, “On Two Types of Moving Front in Qua silinear Diffusion, Mathematical Biosciences, Vol. 31, No. 3, 1976, pp. 307-315.

[1] A. Okubo, “Diffusion and Ecological Problems: Mathe matical Models,” Springer-Verlag, Berlin, 1980.

[2] J. D. Murray, “Mathematical Biology,” Biomathematics Texts, Vol. 19, Springer Verlag, New York, 1989.

[3] J. G. Skellam, “Random Dispersal in Theoretical Popula tions,” Bulletin of Mathematical Biology, Vol. 53, No. 1, 1991, pp. 135-165.

[4] J. A. Sherratt and J. D. Murray, “Models of Epidermal Wound Healing,” Proceedings: Biological Sciences, Vol. 241, No. 1300, 1990, pp. 29-36.

[5] M. A. J. Chaplain and A. M. Stuart, “A Model Mecha nism for the Chemotactic Response of Endothelial Cells to Tumour Angiogenesis Factor,” Mathematical Medicine and Biology, Vol. 10, No. 3, 1993, pp. 149-168.

[6] J. A. Sherratt, “Wavefront Propagation in a Competition Equation with a New Motility Term Modelling Contact Inhibition between Cell Populations,” Proceedings of the Royal Society of London. Series A: Mathematical, Physi cal and Engineering Sciences, Vol. 456, No. 2002, 2000, pp. 2365-2386.

[7] M. A. J. Chaplain and A. M. Stuart, “A Mathematical Model for the Diffusion of Tumour Angiogenesis Factor into the Surrounding Host Tissue,” Mathematical Medi cine and Biology, Vol. 8, No. 3, 1991, pp. 191-220.

[8] M. Abercrombie, “Contact Inhibition in Tissue Culture,” In Vitro Cellular & Developmental Biology-Plant, Vol. 6, No. 2, 1970, pp. 128-142.

[9] P. M. Kareiva, “Local Movement in Herbivorous Insects: Applying a Passive Diffusion Model to Mark-Recapture Field Experiments,” Oecologia, Vol. 57, No. 3, 1983, pp. 322-327.

[10] D. Tilman and P. M. Kareiva, “Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Inter actions,” Princeton University Press, Princeton, 1997.

[11] R. A. Fisher, “The Wave of Advance of Advantageous Genes,” Annals of Eugenics, Vol. 7, No. 4, 1937, pp. 353-369.

[12] A. Kolmogorov, I. Petrovsy and N. Piskounov, “Study of the Diffusion Equation with Growth of the Quantity of Matter and Its Applications to a Biological Problem,” Moscow University Mathematics Bulletin, 1989, p. 105.

[13] D. A. Frank-Kamenetskii, “Diffusion and Heat Exchange in Chemical Kinetics,” Princeton University Press, Prin ceton, 1955.

[14] J. Canosa, “Diffusion in Non-Linear Multiplicative Media,” Journal of Mathematical Physics, Vol. 10, No. 10, 1969, pp. 1862-1868.

[15] H. Cohen, “Non-Linear Diffusion Problems,” In: A. H. Taub, Ed., Studies in Applied Mathematics, The Mathe matical Association of America, 1971, pp. 27-64.

[16] P. C. Fife and J. B. McLeod, “The Approach of Solutions of Non-Linear Diffusion Equations to Travelling Front Solutions,” Archive for Rational Mechanics and Analysis, Vol. 65, No. 4, 1977, pp. 335-361.

[17] J. D. Murray, “Lectures on Non-Linear Differential-Equa tion Models in Biology,” Oxford University Press, Oxford, 1977.

[18] H. C. Tuckwell, “Introduction to Theoretical Neurobiol ogy: Non-Linear and Stochastic Theories,” Cambridge University Press, Cambridge, 1988.

[19] E. A. Carl, “Population Control in Arctic Ground Squirrels,” Ecology, Vol. 52, No. 3, 1971, pp. 395-413.

[20] J. H. Myers and C. J. Krebs, “Population Cycles in Rodents,” Scientific American, Vol. 230, No. 6, 1974, p. 38.

[21] W. S. Gurney and R. M. Nisbet, “The Regulation of Inhomogeneous Populations,” Journal of Theoretical Biology, Vol. 52, No. 2, 1975, pp. 441-457.

[22] W. S. Gurney and R. M. Nisbet, “A Note on Non-Linear Population Transport,” Journal of Theoretical Biology, Vol. 56, No. 1, 1976, pp. 249-251.

[23] E. W. Montroll and B. J. West, “On an Enriched Collec tion of Stochastic Processes,” Fluctuation Phenomena, Vol. 66, 1979, p. 61. doi:10.1016/B978-0-444-85248-9.50005-4

[24] F. Sánchez-Garduno and P. K. Maini, “Existence and Uni queness of a Sharp Traveling Wave in Degenerate Non Linear Diffusion Fisher-KPP Equations,” Journal of Ma thematical Biology, Vol. 33, No. 2, 1994, pp. 163-192.

[25] R. Luther, “Propagation of Chemical Reactions in Space,” Zeitschrift für Elektrochemie und Angewandte Physika lische Chemie, Vol. 12, No. 32, 1906, p. 596. doi:10.1002/bbpc.19060123208

[26] M. J. Ablowitz and A. Zeppetella, “Explicit Solutions of Fisher’s Equation for a Special Wave Speed,” Bulletin of Mathematical Biology, Vol. 41, No. 6, 1979, pp. 835-840.

[27] A. M. Wazwaz, “The tanh Method for Traveling Wave Solutions of Nonlinear Equations,” Applied Mathematics and Computation, Vol. 154, No. 3, 2004, pp. 713-723.

[28] M. B. A. Mansour, “Accurate Computation of Traveling Wave Solutions of Some Nonlinear Diffusion Equations,” Wave Motion, Vol. 44, No. 3, 2007, pp. 222-230.

[29] Y. Tan, H. Xu and S. J. Liao, “Explicit Series Solution of Travelling Waves with a Front of Fisher Equation,” Chaos, Solitons & Fractals, Vol. 31, No. 2, 2007, pp. 462-472.

[30] J. Q. Mo, W. J. Zhang and M. He, “Asymptotic Method of Travelling Wave Solutions for a Class of Nonlinear Reaction Diffusion Equation,” Acta Mathematica Scientia, Vol. 27, No. 4, 2007, pp. 777-780.

[31] Z. Feng, S. Zheng and D. Y. Gao, “Traveling Wave Solu tions to a Reaction Diffusion Equation,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 60, No. 4, 2009, pp. 756-773.

[32] X. Hou, Y. Li and K. R. Meyer, “Traveling Wave Solu tions for a Reaction Diffusion Equation with Double De generate Nonlinearities,” Discrete and Continuous Dy namical Systems, Vol. 26, No. 1, 2010, pp. 265-290.

[33] M. Shakeel, P. C. Matthews, S. L. Waters and R. Graham, “Continuum Modeling of Cell Growth and Nutrient Trans port in a Perfusion Bioreactor,” 2011, pp. 90-117.

[34] M. Shakeel, “Continuum Modelling of Cell Growth and Nutrient Transport in a Perfusion Bioreactor,” Ph.D. Thesis, The University of Nottingham, Nottingham, 2011.

[35] F. Coletti, S. Macchietto and N. Elvassore, “Mathematical Modeling of Three Dimensional Cell Cultures in Perfu sion Bioreactors,” Industrial & Engineering Chemistry Research, Vol. 45, No. 24, 2006, pp. 8158-8169.

[36] F. Rothe, “Convergence to Pushed Fronts,” Rocky Moun tain Journal of Mathematics, Vol. 11, No. 4, 1981, pp. 617-634.

[37] W. Van Saarloos, “Front Propagation into Unstable States,” Physics Reports, Vol. 386, No. 2-6, 2003, pp. 29-222.

[38] A. N. Stokes, “On Two Types of Moving Front in Qua silinear Diffusion, Mathematical Biosciences, Vol. 31, No. 3, 1976, pp. 307-315.