September  2022, 15(9): 2553-2579. doi: 10.3934/dcdss.2022133

Liouville-Green approximation for linearly coupled systems: Asymptotic analysis with applications to reaction-diffusion systems

Dept of Mathematics, FNSPE, Czech Technical University in Prague, Czech Republic

* Corresponding author: Václav Klika

Received  April 2021 Revised  May 2022 Published  September 2022 Early access  July 2022

Asymptotic analysis has become a common approach in investigations of reaction-diffusion equations and pattern formation, especially when considering generalizations of the original model, such as spatial heterogeneity, where finding an analytic solution even to the linearized equations is generally not possible. The Liouville-Green approximation (also known as WKBJ method), one of the more robust asymptotic approaches for investigating dissipative phenomena captured by linear equations, has recently been applied to the Turing model in a heterogeneous environment. It demonstrated the anticipated modifications to the results obtained in a homogeneous setting, such as localized patterns and local Turing conditions. In this context, we attempt a generalization of the scalar Liouville-Green approximation to multicomponent systems. Our broader mathematical approach results in general approximation theorems for systems of ODEs without turning points. We discuss the cases of exponential and oscillatory behaviour first before treating the general case. Subsequently, we demonstrate the spectral properties utilized in the approximation theorems for a typical Turing system, hence showing that Liouville-Green approximation is plausible for an arbitrary number of coupled species outside of turning points and generally valid for fast growing modes as long as the diffusivities are distinct. Note that our line of approach is via showing that the solution is close (using suitable weight functions for measuring the error) to a linear combination of Airy-like functions.

Citation: Juraj Kováč, Václav Klika. Liouville-Green approximation for linearly coupled systems: Asymptotic analysis with applications to reaction-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (9) : 2553-2579. doi: 10.3934/dcdss.2022133
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, chapter 9.6, 10.4, Applied mathematics series, Dover Publications, 1964, https://books.google.cz/books?id=MtU8uP7XMvoC.

[2]

R. E. Baker, E. A. Gaffney and P. K. Maini, Partial differential equations for self-organization in cellular and developmental biology, Nonlinearity, 21 (2008), R251–R290. doi: 10.1088/0951-7715/21/11/R05.

[3]

W. W. Bell, Special Functions for Scientists and Engineers, Van Nostrand, 1968, https://books.google.cz/books?id=Pz8nAAAAMAAJ.

[4]

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I, Springer-Verlag New York, 1999, https://books.google.cz/books?id=Pz8nAAAAMAAJ. doi: 10.1007/978-1-4757-3069-2.

[5]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112.  doi: 10.1103/RevModPhys.65.851.

[6]

G. Dewel and P. Borckmans, Effects of slow spatial variations on dissipative structures, Physics Letters A, 138 (1989), 189-192.  doi: 10.1016/0375-9601(89)90025-X.

[7]

J. B. Keller and S. I. Rubinow, Asymptotic solution of eigenvalue problems, Annals of Physics, 9 (1960), 24-75.  doi: 10.1016/0003-4916(60)90061-0.

[8]

V. Klika, Significance of non-normality-induced patterns: Transient growth versus asymptotic stability, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27 (2017), 073120, 9 pp. doi: 10.1063/1.4985256.

[9]

V. KlikaM. Kozák and E. A. Gaffney, Domain size driven instability: Self-organization in systems with advection, SIAM J. Appl. Math., 78 (2018), 2298-2322.  doi: 10.1137/17M1138571.

[10]

J. Kováč, Qualitative Analysis of a Reaction-Diffusion System using Weakly Nonlinear Analysis and the WKBJ Method, Master's thesis, 2020, Available on request.

[11]

M. KozákE. A. Gaffney and V. Klika, Pattern formation in reaction-diffusion systems with piecewise kinetic modulation: An example study of heterogeneous kinetics, Physical Review E, 100 (2019), 042220. 

[12]

A. L. KrauseM. A. Ellis and R. A. Van Gorder, Influence of curvature, growth, and anisotropy on the evolution of Turing patterns on growing manifolds, Bull. Math. Biol., 81 (2019), 759-799.  doi: 10.1007/s11538-018-0535-y.

[13]

A. L. Krause, E. A. Gaffney, P. K. Maini and V. Klika, Modern perspectives on near-equilibrium analysis of Turing systems, Philos. Trans. Roy. Soc. A, 379 (2021), Paper No. 20200268, 30 pp. doi: 10.1098/rsta.2020.0268.

[14]

A. L. Krause, V. Klika, T. E. Woolley and E. A. Gaffney, Heterogeneity induces spatiotemporal oscillations in reaction-diffusion systems, Physical Review E, 97 (2018), 052206, 12 pp. doi: 10.1103/physreve.97.052206.

[15]

A. L. KrauseV. KlikaT. E. Woolley and E. A. Gaffney, From one pattern into another: Analysis of Turing patterns in heterogeneous domains via WKBJ, Journal of The Royal Society Interface, 17 (2020), 20190621.  doi: 10.1098/rsif.2019.0621.

[16]

D. Krejčiřík, Geometrical aspects of spectral theory, http://nsa.fjfi.cvut.cz/david/other/gspec19.pdf, Accessed 7 December 2021.

[17]

A. MadzvamuseE. A. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains, J. Math. Biol., 61 (2010), 133-164.  doi: 10.1007/s00285-009-0293-4.

[18]

P. K. MainiR. E. Baker and C.-M. Chuong, The Turing model comes of molecular age, Science, 314 (2006), 1397-1398.  doi: 10.1126/science.1136396.

[19]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edition, Springer, 2003.

[20] F. W. J. Olver, Asymptotics and Special Functions, CRC Press, 1997. 
[21]

K. PageP. K. Maini and N. A. M. Monk, Pattern formation in spatially heterogeneous Turing reaction-diffusion models, Physica D: Nonlinear Phenomena, 181 (2003), 80-101.  doi: 10.1016/S0167-2789(03)00068-X.

[22]

R. SekineT. Shibata and M. Ebisuya, Synthetic mammalian pattern formation driven by differential diffusivity of nodal and lefty, Nature Communications, 9 (2018), 5456.  doi: 10.1038/s41467-018-07847-x.

[23]

L. E. Stephenson and D. J. Wollkind, Weakly nonlinear stability analyses of one-dimensional Turing pattern formation in activator-inhibitor/immobilizer model systems, J. Math. Biol., 33 (1995), 771-815.  doi: 10.1007/BF00187282.

[24]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[25]

R. A. Van Gorder, V. Klika and A. L. Krause, Turing conditions for pattern forming systems on evolving manifolds, J. Math. Biol., 82 (2021), Paper No. 4, 61 pp. doi: 10.1007/s00285-021-01552-y.

[26]

F. VeermanM. Mercker and A. Marciniak-Czochra, Beyond Turing: Far-from-equilibrium patterns and mechano-chemical feedback, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 379 (2021), 20200278.  doi: 10.1098/rsta.2020.0278.

[27]

C. H. Waddington, Principles of Embryology, Allen & Unwin Lond, 1956. doi: 10.4324/9781315665405.

[28]

M. J. WardD. McInerneyP. HoustonD. Gavaghan and P. Maini, The dynamics and pinning of a spike for a reaction-diffusion system, SIAM J. Appl. Math., 62 (2002), 1297-1328.  doi: 10.1137/S0036139900375112.

[29]

R. B. White, Asymptotic Analysis of Differential Equations, Imperial College Press, 2010, https://books.google.sk/books?id=F1OwONpC-N8C. doi: 10.1142/p735.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, chapter 9.6, 10.4, Applied mathematics series, Dover Publications, 1964, https://books.google.cz/books?id=MtU8uP7XMvoC.

[2]

R. E. Baker, E. A. Gaffney and P. K. Maini, Partial differential equations for self-organization in cellular and developmental biology, Nonlinearity, 21 (2008), R251–R290. doi: 10.1088/0951-7715/21/11/R05.

[3]

W. W. Bell, Special Functions for Scientists and Engineers, Van Nostrand, 1968, https://books.google.cz/books?id=Pz8nAAAAMAAJ.

[4]

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I, Springer-Verlag New York, 1999, https://books.google.cz/books?id=Pz8nAAAAMAAJ. doi: 10.1007/978-1-4757-3069-2.

[5]

M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112.  doi: 10.1103/RevModPhys.65.851.

[6]

G. Dewel and P. Borckmans, Effects of slow spatial variations on dissipative structures, Physics Letters A, 138 (1989), 189-192.  doi: 10.1016/0375-9601(89)90025-X.

[7]

J. B. Keller and S. I. Rubinow, Asymptotic solution of eigenvalue problems, Annals of Physics, 9 (1960), 24-75.  doi: 10.1016/0003-4916(60)90061-0.

[8]

V. Klika, Significance of non-normality-induced patterns: Transient growth versus asymptotic stability, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27 (2017), 073120, 9 pp. doi: 10.1063/1.4985256.

[9]

V. KlikaM. Kozák and E. A. Gaffney, Domain size driven instability: Self-organization in systems with advection, SIAM J. Appl. Math., 78 (2018), 2298-2322.  doi: 10.1137/17M1138571.

[10]

J. Kováč, Qualitative Analysis of a Reaction-Diffusion System using Weakly Nonlinear Analysis and the WKBJ Method, Master's thesis, 2020, Available on request.

[11]

M. KozákE. A. Gaffney and V. Klika, Pattern formation in reaction-diffusion systems with piecewise kinetic modulation: An example study of heterogeneous kinetics, Physical Review E, 100 (2019), 042220. 

[12]

A. L. KrauseM. A. Ellis and R. A. Van Gorder, Influence of curvature, growth, and anisotropy on the evolution of Turing patterns on growing manifolds, Bull. Math. Biol., 81 (2019), 759-799.  doi: 10.1007/s11538-018-0535-y.

[13]

A. L. Krause, E. A. Gaffney, P. K. Maini and V. Klika, Modern perspectives on near-equilibrium analysis of Turing systems, Philos. Trans. Roy. Soc. A, 379 (2021), Paper No. 20200268, 30 pp. doi: 10.1098/rsta.2020.0268.

[14]

A. L. Krause, V. Klika, T. E. Woolley and E. A. Gaffney, Heterogeneity induces spatiotemporal oscillations in reaction-diffusion systems, Physical Review E, 97 (2018), 052206, 12 pp. doi: 10.1103/physreve.97.052206.

[15]

A. L. KrauseV. KlikaT. E. Woolley and E. A. Gaffney, From one pattern into another: Analysis of Turing patterns in heterogeneous domains via WKBJ, Journal of The Royal Society Interface, 17 (2020), 20190621.  doi: 10.1098/rsif.2019.0621.

[16]

D. Krejčiřík, Geometrical aspects of spectral theory, http://nsa.fjfi.cvut.cz/david/other/gspec19.pdf, Accessed 7 December 2021.

[17]

A. MadzvamuseE. A. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains, J. Math. Biol., 61 (2010), 133-164.  doi: 10.1007/s00285-009-0293-4.

[18]

P. K. MainiR. E. Baker and C.-M. Chuong, The Turing model comes of molecular age, Science, 314 (2006), 1397-1398.  doi: 10.1126/science.1136396.

[19]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edition, Springer, 2003.

[20] F. W. J. Olver, Asymptotics and Special Functions, CRC Press, 1997. 
[21]

K. PageP. K. Maini and N. A. M. Monk, Pattern formation in spatially heterogeneous Turing reaction-diffusion models, Physica D: Nonlinear Phenomena, 181 (2003), 80-101.  doi: 10.1016/S0167-2789(03)00068-X.

[22]

R. SekineT. Shibata and M. Ebisuya, Synthetic mammalian pattern formation driven by differential diffusivity of nodal and lefty, Nature Communications, 9 (2018), 5456.  doi: 10.1038/s41467-018-07847-x.

[23]

L. E. Stephenson and D. J. Wollkind, Weakly nonlinear stability analyses of one-dimensional Turing pattern formation in activator-inhibitor/immobilizer model systems, J. Math. Biol., 33 (1995), 771-815.  doi: 10.1007/BF00187282.

[24]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[25]

R. A. Van Gorder, V. Klika and A. L. Krause, Turing conditions for pattern forming systems on evolving manifolds, J. Math. Biol., 82 (2021), Paper No. 4, 61 pp. doi: 10.1007/s00285-021-01552-y.

[26]

F. VeermanM. Mercker and A. Marciniak-Czochra, Beyond Turing: Far-from-equilibrium patterns and mechano-chemical feedback, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 379 (2021), 20200278.  doi: 10.1098/rsta.2020.0278.

[27]

C. H. Waddington, Principles of Embryology, Allen & Unwin Lond, 1956. doi: 10.4324/9781315665405.

[28]

M. J. WardD. McInerneyP. HoustonD. Gavaghan and P. Maini, The dynamics and pinning of a spike for a reaction-diffusion system, SIAM J. Appl. Math., 62 (2002), 1297-1328.  doi: 10.1137/S0036139900375112.

[29]

R. B. White, Asymptotic Analysis of Differential Equations, Imperial College Press, 2010, https://books.google.sk/books?id=F1OwONpC-N8C. doi: 10.1142/p735.

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