Validating Numerical to Theoretical Solutions in a Reaction-Diffusion with Linear Cross-Diffusion Systems.
Keywords:
Cross-diffusion driven instability, parameter space, spatial paterns, pattern formation, validation, numerical simulation, Turing theory, Finite difference method.
Abstract
In this paper, we consider a reaction diffusion system with linear cross-diffusion. We carry out the analytical study in detail and find out that, when the diffusion coefficient is unity, Turing instability does not occur, but with the introduction of cross-diffusion, the system exhibit Turing instability. The numerical results reveal that, on increasing the value of gamma, there is an occurrence of spatial patterns which conforms with the theoretical results.
The cross-diffusion coefficients really plays a vital role on the parameter spaces and spatial patterns of our system.
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References
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[28] McAfree, M.S. and Annunziata, O. (2013). Cross-diffusion in a colloidpolymer aqueous system. Fluid Phase Equilibria. 356:46-55.
[29] McDougall T. and Turner J.S. (1982). Influence of cross-diffusion on inger double-diffusive convection. Nature. 299:812-814.
[30] Morton K.W. and Mayers D.F. (1994). Numerical Solution of Partial Differential Equations. Cambridge University Press.
[31] Murray J.D. (1982). Parameter space for Turing instability in reaction diffusion mechanisms: A comparison of models. J. Theor. Bio. 98:143-163.
[32] Murray J.D. (2003). Mathematical Biology II: Spatial models and biomedical applications. Third Edition. Springer.
[33] Prigogine, I. and Lefever, R. (1968). Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695-1700.
[34] Ruiz-Baier, R. and Tian, C. (2013). Mathematical analysis and numerical simulation of pattern formation under cross-diffusion. Nonlinear Analysis: Real World Applications. 14:601-612.
[35] Schnakenberg, J. (1979). Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol., 81:389-400.
[36] Tian, Lin, Z. and Pedersen, M. (2010). Instability induced by cross-diffusion in reaction-diffusion systems. Nonlinear Analysis: Real World Applications. 11:1036-1045.
[37] Turing, A. (1952). On the chemical basis of morphogenesis. Phil. Trans. Royal Soc. B. 237:37-72
[38] Uduak, Z.G. (2011). A Numerical approach to the studying cell dynamics. D.Phil Thesis, University of Sussex.
[39] Vanag, V.K. and Epstein, I.R. (2009). Cross-diffusion and pattern formation in reaction diffusion systems. Physical Chemistry Chemical Physics. 11:897-912.
[2] Bard, J. and Lauder,I. (1974). "How well does Turing's Theory of morphogenesis work?" J.Theor. Bio. 45:501-531.
[3] Barreira R., Elliott C.M., Madzvamuse A. (2011). The surface finite element method for pattern formation on evolving biological surfaces, Journal of Math. Bio. 63, 1095-1119.
[4] Barrio R.A., Varea C., Aragon J.L., Maini P.K. (1999). A Two-dimensional Numerical study of Spatial Pattern Formation in Interacting Turing systems Bulletin of Mathematical Biology, 61, 483-505.
[5] Bentil D.E and Murray J.D. (1993). On mechanical theory for biological pattern formation. Physica D, 63:161-190.
[6] Chattopadhyay J. and Chatterjee S. (2001). Cross-diffusional effect in a Lotka-Volterra competitive system. Nonlinear phenom complex syst, 4: 364-369.
[7] Chattopadhyay J. and Tapaswi P.K (1993). Order and disorder in biological systems through negative cross-diffusion of mitotic inhibitor. Math. Comp. Model, 17: 105-112.
[8] Crampin E.J.Hackborn, W.W. and Maini,P.K. (2002). Pattern formation in reaction-diffusion models with nonuniform domain growth. Bulletin of Mathematical Biology,64:746-769.
[9] Epstein, I.R. and Pojman, J.A. (1998). An introduction to nonlinear chemical dynamics (Topics in Physical Chemistry). Oxford University Press New York.
[10] Federico R, Vanag K.V, Enzo T. and Epstein I. R. (2010). Quaternary cross diffusion in water-in oil microemulsions.
[11] Gambino, G. Lombardo, M.C. and Sammartino, M. (2012). Turing instability and traveling fronts for nonlinear reaction-diffusion system with cross-diffusion. Maths. Comp. in Sim. 82:1112-1132.
[12] Gambino, G. Lombardo, M.C. and Sammartino, M. (2013). Pattern formation driven by cross-diffusion in 2-D domain. Nonlinear Analysis: Real World Applications. 14:1755-1779.
[13] Gierer, A. and Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik. 12:30-39.
[14] Gray,P. and Scott, S.K.(1990). Chemical oscillations and instabilities. Nonlinear chemical kinetics. Oxford University Press, New York.
[15] Gurtin M.E. (1974). Some Mathematical models for population dynamics that lead segregation. Quart J. Appl. Math, 32: 1-8
[16] Hillen, T. and Painter, K.J.(2009). A user's guide to PDE models for chemotaxis. J. Maths.Biol,58:183-217.
[17] Iida, M. and Mimura, M. (2006). Diffusion, cross-diffusion an competitive interaction. J.Math. Biol. 53:617-641.
[18] Kerner E.H(1959). Further consideration on the statistical mechanics of biological associations. Bull. Math. Bio, 21: 217-255
[19] Kovacs, S. (2004). Turing bifurcation in a system with cross-diffusion. Nonlinear Analysis. 59:567-581.
[20] Li L., Jin Z and Sun G. (2008). Spatial pattern of an epidemic model with cross-diffusion. Chin. Phys. Lett. 25:3500.
[21] Madzvamuse A. (2000). A Numerical approach to the study of spatial pattern formation D.Phil Thesis, University of Oxford.
[22] Madzvamuse A., Gaffney, E.A. and Maini,P.K.(2010). Stability Analysis of non-autonomous reaction-diffusion systems: the effects of growing domains.J.Maths. Biol. 61(1):133-164
[23] Madzvamuse A; Maini, P.K. and Wathen,A.J.(2003) A moving grid finite element method applied to a model biological pattern generator.J.Comp. Phys. 190:478-500.
[24] Madzvamuse A, Roger K. Thomas., Philip K. Maini., Andrew J. Wathen. (2002). A Numerical Approach to the Study of Spatial Pattern Formation in the Ligaments of Arcoid Bivalves. Bulletin of Mathematical Biology. 64:501-530.
[25] Madzvamuse A, Ndakwo H.S. and Barreira R. (2014). Cross-diffusion-driven instability for reaction-diffusion systems: Analysis and simulations. J. Maths. Biol.
[26] Marten S., Steve C., Jonathan A. F., Carl F. and Brain W. (2001). Catastrophic shifts in ecosystems, Nature. 413:591-596.
[27] Max R., Stefan C. D., Peter. C. De R, and Johan V. De K. (2004). Self-organized patchiness and Catastrophic shifts in ecosystems, Science. 305:1926-1929.
[28] McAfree, M.S. and Annunziata, O. (2013). Cross-diffusion in a colloidpolymer aqueous system. Fluid Phase Equilibria. 356:46-55.
[29] McDougall T. and Turner J.S. (1982). Influence of cross-diffusion on inger double-diffusive convection. Nature. 299:812-814.
[30] Morton K.W. and Mayers D.F. (1994). Numerical Solution of Partial Differential Equations. Cambridge University Press.
[31] Murray J.D. (1982). Parameter space for Turing instability in reaction diffusion mechanisms: A comparison of models. J. Theor. Bio. 98:143-163.
[32] Murray J.D. (2003). Mathematical Biology II: Spatial models and biomedical applications. Third Edition. Springer.
[33] Prigogine, I. and Lefever, R. (1968). Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695-1700.
[34] Ruiz-Baier, R. and Tian, C. (2013). Mathematical analysis and numerical simulation of pattern formation under cross-diffusion. Nonlinear Analysis: Real World Applications. 14:601-612.
[35] Schnakenberg, J. (1979). Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol., 81:389-400.
[36] Tian, Lin, Z. and Pedersen, M. (2010). Instability induced by cross-diffusion in reaction-diffusion systems. Nonlinear Analysis: Real World Applications. 11:1036-1045.
[37] Turing, A. (1952). On the chemical basis of morphogenesis. Phil. Trans. Royal Soc. B. 237:37-72
[38] Uduak, Z.G. (2011). A Numerical approach to the studying cell dynamics. D.Phil Thesis, University of Sussex.
[39] Vanag, V.K. and Epstein, I.R. (2009). Cross-diffusion and pattern formation in reaction diffusion systems. Physical Chemistry Chemical Physics. 11:897-912.
Published
2020-07-05
How to Cite
Ndakwo, H., Abdulkarim, U., & Sulayman, B. (2020). Validating Numerical to Theoretical Solutions in a Reaction-Diffusion with Linear Cross-Diffusion Systems. Journal of Progressive Research in Mathematics, 16(3), 2986-3000. Retrieved from https://scitecresearch.com/journals/index.php/jprm/article/view/1874
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