October  2012, 17(7): 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

Interface oscillations in reaction-diffusion systems above the Hopf bifurcation

1. 

Department of Mathematics and Statistics, Dalhousie University, Halifax, N.S. B3H 3J5

2. 

Department of Mathematics, Dalhousie University, Halifax, N.S. B3H 3J5

3. 

Department of Mathematics and Computing Science, Saint Mary's University, Canada

Received  October 2011 Revised  May 2012 Published  July 2012

We consider a reaction-diffusion system of the form \[ \left\{ \begin{array} \ u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\ \tau w_{t}=Dw_{xx}+g(u,w) \end{array} \right. \] with Neumann boundary conditions on a finite interval. Under certain generic conditions on the nonlinearities $f,g$ and in the singular limit $\varepsilon\ll1$ such a system may admit a steady state solution where $u$ has sharp interfaces. It is also known that such interfaces may be destabilized due to a Hopf bifurcation [Y. Nishiura and M. Mimura. SIAM J.Appl. Math., 49:481--514, 1989], as $\tau$ is increased beyond a certain threshold $\tau_{h}$. In this paper, we study what happens for $\tau>\tau _{h},$ or even $\tau\rightarrow\infty,$ for a solution that consists of either one or two interfaces. Under the additional assumption $D\gg1,$ using singular perturbation theory, we determine the existence of another threshold $\tau _{c}>\tau_{h}$ (where $\tau_{c}$ is allowed to be infinite) such that if $\tau_{h}<\tau<\tau_{c}$ then the system admits a solution consisting of periodically oscillating interfaces. On the other hand if $\tau>\tau_{c},$ the extent of the oscillation eventually exceeds the spatial domain size, even though very long transient dynamics can preceed this occurence. We make use of recently developed numerical software (that employs adaptive error control in space and time) to accurately compute an approximate solution. Excellent agreement with the analytical theory is observed.
Citation: Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523
References:
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W. Chen and M. J. Ward, Oscillatory instabilities anddynamics of multi-spike patterns for the one-dimensional Gray-Scott model,, European Journal of Applied Mathematics, 20 (2009), 187.   Google Scholar

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W. Chen and M. J. Ward, The stability and dynamics oflocalized spot patterns in the two-dimensional Gray-Scott model,, SIAM J. Appl. Dynam. Systems, 10 (2011), 582.   Google Scholar

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A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains,, Mem. Amer. Math. Soc., 199 (2009).   Google Scholar

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S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dyn. Diff. Eq., 14 (2002), 85.   Google Scholar

[8]

S.-I. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions ina specific reaction-diffusion system in one dimension,, Japan J. Indust. Appl. Math., 25 (2008), 117.   Google Scholar

[9]

R. E. Goldstein, D. J. Muraki and D. M. Petrich, Interface proliferation and the growth of labyrinths in a reaction-diffusion system,, Phys. Rev. E (3), 53 (1996), 3933.   Google Scholar

[10]

S. V. Gurevich, Sh. Amiranashvili and H.-G. Purwins, Breathing dissipative solitons in three component reaction-diffusion system,, Physical Review E (3), 74 (2006).   Google Scholar

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A. Hagberg and E. Meron, Pattern formation nongradient reaction-diffusion systems: The effect of front bifurcations,, Nonlinearity, 7 (1994), 805.   Google Scholar

[12]

H. Hardway and Y.-X. Li, Stationary and oscillatory fronts in a two-component genetic regulatory network model,, Physica D, 239 (2010), 1650.   Google Scholar

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P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a three-component system: Stability and bifurcations,, Physica D, 237 (2008), 3335.   Google Scholar

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D. Haim, G. Li, Q. Ouyang, W. D. McCormick, H. L. Swinney, A. Hagberg and E. Meron, Breathing spots in a reaction-diffusion system,, Phys. Rev. Lett., 77 (1996), 190.   Google Scholar

[15]

T. Ikeda and Y. Nishiura, Pattern selection for two breathers,, SIAM J. Appl. Math., 54 (1994), 195.   Google Scholar

[16]

H. Ikeda and T. Ikeda, Bifurcaiton phenomena from standingpulse solutions in some reaction-diffusion systems,, J. Dyn. Diff. Eqns., 12 (2000), 117.   Google Scholar

[17]

A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device,, Angewandte Chemie, 45 (2006).   Google Scholar

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R. Kapral and K. Showalter, eds., "Chemical Waves and Patterns,", Kluwer, (1995).   Google Scholar

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B. S. Kerner and V. V. Osipov, "Autosolitons: A New Approach to Problems of Self-Organization and Turbulence,", Fundamental Theories of Physics, 61 (1994).   Google Scholar

[20]

A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures,, Rev. Modern Physics, 66 (1994), 1481.   Google Scholar

[21]

S. Koga and Y. Kuramoto, Localized patterns in reaction-diffusion systems,, Progress of Theoretical Physics, 63 (1980), 106.   Google Scholar

[22]

T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction-diffusion system,, Physical Review Letters, 98 (2007).   Google Scholar

[23]

T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability,, Physica D, 214 (2006), 63.   Google Scholar

[24]

T. Kolokolnikov, M. Ward and J. Wei, Self-replication of mesa patterns in reaction-diffusion systems,, Physica D, 236 (2007), 104.   Google Scholar

[25]

T. Kolokolnikov, M. J. Ward and J. Wei, Slow transitional instabilities of spike patterns in the one-dimensional Gray-Scott model,, Interfaces and Free Boundaries, 8 (2006), 185.   Google Scholar

[26]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984).   Google Scholar

[27]

I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system,, Science, 251 (1991), 650.   Google Scholar

[28]

Y.-X. Li, Tango waves in a bidomain model of fertilization calcium waves,, Physica D, 186 (2003), 27.   Google Scholar

[29]

H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982).   Google Scholar

[30]

R. C. McKay and T. Kolokolnikov, Instability thresholds and dynamics of mesa patterns in reaction-diffusion systems,, Discrete and Continuous Dynamical Systems - B, (2011).   Google Scholar

[31]

C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reaction-diffusion system,, Phys. Rev. E, 54 (1996), 4860.   Google Scholar

[32]

C. B. Muratov, Synchronization, chaos, and breakdown of collective domain oscillations in reaction-diffusion systems,, Phys. Rev. E, 55 (1997), 1463.   Google Scholar

[33]

C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems,, Phys. Rev. E (3), 53 (1996), 3101.   Google Scholar

[34]

J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989).   Google Scholar

[35]

M. Nagayama, K.-I. Ueda and M. Yadome, Numerical approachto transient dynamics of oscillatory pulses in a bistable reaction-diffision system,, Japan J. Indust. Appl. Math., 27 (2010), 295.   Google Scholar

[36]

W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction,, Trans. AMS, 357 (2005), 3953.   Google Scholar

[37]

Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726.   Google Scholar

[38]

Y. Nishiura and M. Mimura, Layer oscillations inreaction-diffusion systems,, SIAM J. Appl. Math., 49 (1989), 481.   Google Scholar

[39]

J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189.   Google Scholar

[40]

M. Suzuki, T. Ohta, M. Mimura and H. Sakaguchi, Breathing and wiggling motions in three-species laterally inhibitory systems,, Physical Review E, 52 (1995), 3645.   Google Scholar

[41]

M. Taki, M. Tlidi and T. Kolokolnikov, eds., Dissipative localized structures in extended systems,, Chaos Focus Issue, 17 (2007).   Google Scholar

[42]

V. K. Vanag and I. R. Epstein, Localized patterns inreaction-diffusion systems,, Chaos, 17 (2007).   Google Scholar

[43]

V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology,, Physics of Life Reviews, 6 (2009), 267.  doi: 10.1016/j.plrev.2009.10.002.  Google Scholar

[44]

R. Wang, P. Keast and P. Muir, A comparison of adaptive software for 1-D parabolic PDEs,, J. Comput. Appl. Math., 169 (2004), 127.   Google Scholar

[45]

R. Wang, P. Keast and P. Muir, A high-order global spatially adaptive collocation method for 1-D parabolic PDEs,, Applied Numerical Mathematics, 50 (2004), 239.   Google Scholar

[46]

R. Wang, P. Keast and P. Muir, BACOL: B-Spline Adaptive COLlocation Software for 1-D Parabolic PDEs,, ACM Transacations on Mathematical Software, 30 (2004), 454.   Google Scholar

[47]

M. J. Ward and J. Wei, Hopf bifurcation and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model,, J. Nonlinear Science, 13 (2003), 209.   Google Scholar

[48]

H. Yizhaq and E. Gilad, Localized structures in dry land vegetation: Forms and functions,, Chaos, 17 (2007).   Google Scholar

show all references

References:
[1]

M. Banerjee and S. Petrovski, Self-organised spatialpatterns and chaos in a ratio-dependent predator-prey system,, Theor. Ecol., 4 (2011), 37.   Google Scholar

[2]

C. M. Bender and S. A. Orszag, "Advanced Mathematical Methods for Scientists and Engineers. I. Asymptotic Methods and Perturbation Theory,", Reprint of the 1978 original, (1978).   Google Scholar

[3]

K. E. Brenan, S. L.Campbell and L. R. Petzold, "Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations,", Classics in Applied Mathematics, (1996).   Google Scholar

[4]

W. Chen and M. J. Ward, Oscillatory instabilities anddynamics of multi-spike patterns for the one-dimensional Gray-Scott model,, European Journal of Applied Mathematics, 20 (2009), 187.   Google Scholar

[5]

W. Chen and M. J. Ward, The stability and dynamics oflocalized spot patterns in the two-dimensional Gray-Scott model,, SIAM J. Appl. Dynam. Systems, 10 (2011), 582.   Google Scholar

[6]

A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains,, Mem. Amer. Math. Soc., 199 (2009).   Google Scholar

[7]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dyn. Diff. Eq., 14 (2002), 85.   Google Scholar

[8]

S.-I. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions ina specific reaction-diffusion system in one dimension,, Japan J. Indust. Appl. Math., 25 (2008), 117.   Google Scholar

[9]

R. E. Goldstein, D. J. Muraki and D. M. Petrich, Interface proliferation and the growth of labyrinths in a reaction-diffusion system,, Phys. Rev. E (3), 53 (1996), 3933.   Google Scholar

[10]

S. V. Gurevich, Sh. Amiranashvili and H.-G. Purwins, Breathing dissipative solitons in three component reaction-diffusion system,, Physical Review E (3), 74 (2006).   Google Scholar

[11]

A. Hagberg and E. Meron, Pattern formation nongradient reaction-diffusion systems: The effect of front bifurcations,, Nonlinearity, 7 (1994), 805.   Google Scholar

[12]

H. Hardway and Y.-X. Li, Stationary and oscillatory fronts in a two-component genetic regulatory network model,, Physica D, 239 (2010), 1650.   Google Scholar

[13]

P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a three-component system: Stability and bifurcations,, Physica D, 237 (2008), 3335.   Google Scholar

[14]

D. Haim, G. Li, Q. Ouyang, W. D. McCormick, H. L. Swinney, A. Hagberg and E. Meron, Breathing spots in a reaction-diffusion system,, Phys. Rev. Lett., 77 (1996), 190.   Google Scholar

[15]

T. Ikeda and Y. Nishiura, Pattern selection for two breathers,, SIAM J. Appl. Math., 54 (1994), 195.   Google Scholar

[16]

H. Ikeda and T. Ikeda, Bifurcaiton phenomena from standingpulse solutions in some reaction-diffusion systems,, J. Dyn. Diff. Eqns., 12 (2000), 117.   Google Scholar

[17]

A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device,, Angewandte Chemie, 45 (2006).   Google Scholar

[18]

R. Kapral and K. Showalter, eds., "Chemical Waves and Patterns,", Kluwer, (1995).   Google Scholar

[19]

B. S. Kerner and V. V. Osipov, "Autosolitons: A New Approach to Problems of Self-Organization and Turbulence,", Fundamental Theories of Physics, 61 (1994).   Google Scholar

[20]

A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures,, Rev. Modern Physics, 66 (1994), 1481.   Google Scholar

[21]

S. Koga and Y. Kuramoto, Localized patterns in reaction-diffusion systems,, Progress of Theoretical Physics, 63 (1980), 106.   Google Scholar

[22]

T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction-diffusion system,, Physical Review Letters, 98 (2007).   Google Scholar

[23]

T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability,, Physica D, 214 (2006), 63.   Google Scholar

[24]

T. Kolokolnikov, M. Ward and J. Wei, Self-replication of mesa patterns in reaction-diffusion systems,, Physica D, 236 (2007), 104.   Google Scholar

[25]

T. Kolokolnikov, M. J. Ward and J. Wei, Slow transitional instabilities of spike patterns in the one-dimensional Gray-Scott model,, Interfaces and Free Boundaries, 8 (2006), 185.   Google Scholar

[26]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984).   Google Scholar

[27]

I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system,, Science, 251 (1991), 650.   Google Scholar

[28]

Y.-X. Li, Tango waves in a bidomain model of fertilization calcium waves,, Physica D, 186 (2003), 27.   Google Scholar

[29]

H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982).   Google Scholar

[30]

R. C. McKay and T. Kolokolnikov, Instability thresholds and dynamics of mesa patterns in reaction-diffusion systems,, Discrete and Continuous Dynamical Systems - B, (2011).   Google Scholar

[31]

C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reaction-diffusion system,, Phys. Rev. E, 54 (1996), 4860.   Google Scholar

[32]

C. B. Muratov, Synchronization, chaos, and breakdown of collective domain oscillations in reaction-diffusion systems,, Phys. Rev. E, 55 (1997), 1463.   Google Scholar

[33]

C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems,, Phys. Rev. E (3), 53 (1996), 3101.   Google Scholar

[34]

J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989).   Google Scholar

[35]

M. Nagayama, K.-I. Ueda and M. Yadome, Numerical approachto transient dynamics of oscillatory pulses in a bistable reaction-diffision system,, Japan J. Indust. Appl. Math., 27 (2010), 295.   Google Scholar

[36]

W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction,, Trans. AMS, 357 (2005), 3953.   Google Scholar

[37]

Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726.   Google Scholar

[38]

Y. Nishiura and M. Mimura, Layer oscillations inreaction-diffusion systems,, SIAM J. Appl. Math., 49 (1989), 481.   Google Scholar

[39]

J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189.   Google Scholar

[40]

M. Suzuki, T. Ohta, M. Mimura and H. Sakaguchi, Breathing and wiggling motions in three-species laterally inhibitory systems,, Physical Review E, 52 (1995), 3645.   Google Scholar

[41]

M. Taki, M. Tlidi and T. Kolokolnikov, eds., Dissipative localized structures in extended systems,, Chaos Focus Issue, 17 (2007).   Google Scholar

[42]

V. K. Vanag and I. R. Epstein, Localized patterns inreaction-diffusion systems,, Chaos, 17 (2007).   Google Scholar

[43]

V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology,, Physics of Life Reviews, 6 (2009), 267.  doi: 10.1016/j.plrev.2009.10.002.  Google Scholar

[44]

R. Wang, P. Keast and P. Muir, A comparison of adaptive software for 1-D parabolic PDEs,, J. Comput. Appl. Math., 169 (2004), 127.   Google Scholar

[45]

R. Wang, P. Keast and P. Muir, A high-order global spatially adaptive collocation method for 1-D parabolic PDEs,, Applied Numerical Mathematics, 50 (2004), 239.   Google Scholar

[46]

R. Wang, P. Keast and P. Muir, BACOL: B-Spline Adaptive COLlocation Software for 1-D Parabolic PDEs,, ACM Transacations on Mathematical Software, 30 (2004), 454.   Google Scholar

[47]

M. J. Ward and J. Wei, Hopf bifurcation and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model,, J. Nonlinear Science, 13 (2003), 209.   Google Scholar

[48]

H. Yizhaq and E. Gilad, Localized structures in dry land vegetation: Forms and functions,, Chaos, 17 (2007).   Google Scholar

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