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May  2016, 21(3): 959-975. doi: 10.3934/dcdsb.2016.21.959

Oscillations of many interfaces in the near-shadow regime of two-component reaction-diffusion systems

Shuangquan Xie 1, and  Theodore Kolokolnikov 1,

1. 

Dalhousie University, Halifax, NS, Canada, Canada

Received  August 2014 Revised  October 2015 Published  January 2015

We consider the general class of two-component reaction-diffusion systems on a finite domain that admit interface solutions in one of the components, and we study the dynamics of $n$ interfaces in one dimension. In the limit where the second component has large diffusion, we fully characterize the possible behaviour of $n$ interfaces. We show that after the transients die out, the motion of $n$ interfaces is described by the motion of a single interface on the domain that is $1/n$ the size of the original domain. Depending on parameter regime and initial conditions, one of the following three outcomes results: (1) some interfaces collide; (2) all $n$ interfaces reach a symmetric steady state; (3) all $n$ interfaces oscillate indefinitely. In the latter case, the oscillations are described by a simple harmonic motion with even-numbered interfaces oscillating in phase while odd-numbered interfaces are oscillating in anti-phase. This extends a recent work by [McKay, Kolokolnikov, Muir, DCDS B(17), 2012] from two to any number of interfaces.
Keywords: Pattern formation, reaction-diffusion systems., interface oscillation.
Mathematics Subject Classification: Primary: 35K57, 35B36; Secondary: 35B2.
Citation: Shuangquan Xie, Theodore Kolokolnikov. Oscillations of many interfaces in the near-shadow regime of two-component reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 959-975. doi: 10.3934/dcdsb.2016.21.959
References:
[1]

M. Banerjee and S. Petrovski, Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system, Theor Ecol, 4 (2011), 37-53. doi: 10.1007/s12080-010-0073-1.  Google Scholar

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W. Chen and M. J. Ward, Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional gray-scott model, European Journal of Applied Mathematics, 20 (2009), 187-214. doi: 10.1017/S0956792508007766.  Google Scholar

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S. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions in a specific reaction-diffusion system in one dimension, Japan J. Indust. Appl. Math., 25 (2008), 117-147. doi: 10.1007/BF03167516.  Google Scholar

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S. Gurevich, S. Amiranashvili and H.-G. Purwins, Breathing dissipative solitons in three-component reaction-diffusion system, Physical Review E, 74 (2006), 066201, 7pp. doi: 10.1103/PhysRevE.74.066201.  Google Scholar

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[6]

D. Haim, G. Li, Q. Ouyang, W. D. McCormick, H. L. Swinney, A. Hagberg and E. Meron, Breathing spots in a reaction-diffusion system, Physical review letters, 77 (1996), p190. Google Scholar

[7]

H. Hardway and Y.-X. Li, Stationary and oscillatory fronts in a two-component genetic regulatory network model, Physica D, 239 (2010), 1650-1661. doi: 10.1016/j.physd.2010.04.009.  Google Scholar

[8]

H. Ikeda and T. Ikeda, Bifurcation phenomena from standing pulse solutions in some reaction-diffusion systems, Journal of Dynamics and Differential Equations, 12 (2000), 117-167. doi: 10.1023/A:1009098719440.  Google Scholar

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T. Ikeda and M. Mimura, An interfacial approach to regional segregation of two competing species mediated by a predator, Journal of Mathematical Biology, 31 (1993), 215-240. doi: 10.1007/BF00166143.  Google Scholar

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T. Ikeda and Y. Nishiura, Pattern selection for two breathers, SIAM Journal on Applied Mathematics, 54 (1994), 195-230. doi: 10.1137/S0036139992237250.  Google Scholar

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A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device, Angewandte Chemie International Edition, 45 (2006), 3087-3089. Google Scholar

[12]

S. Koga and Y. Kuramoto, Localized patterns in reaction-diffusion systems, Progress of Theoretical Physics, 63 (1980), 106-121. doi: 10.1143/PTP.63.106.  Google Scholar

[13]

T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional belousov-zhabotinski reaction in a water-in-oil microemulsion, Physical review letters, 98 (2007), 188303. doi: 10.1103/PhysRevLett.98.188303.  Google Scholar

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I. Lengyel and I. Epstein, Modeling of turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652. doi: 10.1126/science.251.4994.650.  Google Scholar

[15]

Y.-X. Li, Tango waves in a bidomain model of fertilization calcium waves, Physica D, 186 (2003), 27-49. doi: 10.1016/S0167-2789(03)00237-9.  Google Scholar

[16]

R. McKay, T. Kolokolnikov and P. Muir, Interface oscillations in reaction-diffusion systems above the hopf bifurcation,, Discrete and Continuous Dynamical Systems. Series BA Journal Bridging Mathematics and Sciences., ().   Google Scholar

[17]

E. Meron, H. Yizhaq and E. Gilad, Localized structures in dryland vegetation: forms and functions, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 037109. doi: 10.1063/1.2767246.  Google Scholar

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C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems, Phys, Rev. E, 53 (1996), 3101-3116. doi: 10.1103/PhysRevE.53.3101.  Google Scholar

[19]

M. Nagayama, K.-i. Ueda and M. Yadome, Numerical approach to transient dynamics of oscillatory pulses in a bistable reaction-diffusion system, Japan journal of industrial and applied mathematics, 27 (2010), 295-322. doi: 10.1007/s13160-010-0015-8.  Google Scholar

[20]

Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J.Appl. Math, 49 (1989), 481-514. doi: 10.1137/0149029.  Google Scholar

[21]

T. Ohta, A. Ito and A. Tetsuka, Self-organization in an excitable reaction-diffusion system: synchronization of oscillatory domains in one dimension, Physical Review A, 42 (1990), 3225. doi: 10.1103/PhysRevA.42.3225.  Google Scholar

[22]

M. Suzuki, T. Ohta, M. Mimura and H. Sakaguchi, Breathing and wiggling motions in three-species laterally inhibitory systems, Physical Review E, 52 (1995), 3645-3655. doi: 10.1103/PhysRevE.52.3645.  Google Scholar

[23]

P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a three-component system: Stability and bifurcations, Physica D, 237 (2008), 3335-3368. doi: 10.1016/j.physd.2008.07.014.  Google Scholar

[24]

V. K. Vanag and I. R. Epstein, Localized patterns in reaction-diffusion systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 037110, 11pp. doi: 10.1063/1.2752494.  Google Scholar

[25]

V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology, Physics of Life Reviews, 6 (2009), 267-310. doi: 10.1016/j.plrev.2009.10.002.  Google Scholar

[26]

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), 209-264. doi: 10.1007/s00332-002-0531-z.  Google Scholar

show all references

References:
[1]

M. Banerjee and S. Petrovski, Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system, Theor Ecol, 4 (2011), 37-53. doi: 10.1007/s12080-010-0073-1.  Google Scholar

[2]

W. Chen and M. J. Ward, Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional gray-scott model, European Journal of Applied Mathematics, 20 (2009), 187-214. doi: 10.1017/S0956792508007766.  Google Scholar

[3]

S. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions in a specific reaction-diffusion system in one dimension, Japan J. Indust. Appl. Math., 25 (2008), 117-147. doi: 10.1007/BF03167516.  Google Scholar

[4]

S. Gurevich, S. Amiranashvili and H.-G. Purwins, Breathing dissipative solitons in three-component reaction-diffusion system, Physical Review E, 74 (2006), 066201, 7pp. doi: 10.1103/PhysRevE.74.066201.  Google Scholar

[5]

A. Hagberg and E. Meron, Pattern formation in non-gradient reaction-diffusion systems: the effect of front bifurcations, Nonlinearity, 7 (1994), 805-835. doi: 10.1088/0951-7715/7/3/006.  Google Scholar

[6]

D. Haim, G. Li, Q. Ouyang, W. D. McCormick, H. L. Swinney, A. Hagberg and E. Meron, Breathing spots in a reaction-diffusion system, Physical review letters, 77 (1996), p190. Google Scholar

[7]

H. Hardway and Y.-X. Li, Stationary and oscillatory fronts in a two-component genetic regulatory network model, Physica D, 239 (2010), 1650-1661. doi: 10.1016/j.physd.2010.04.009.  Google Scholar

[8]

H. Ikeda and T. Ikeda, Bifurcation phenomena from standing pulse solutions in some reaction-diffusion systems, Journal of Dynamics and Differential Equations, 12 (2000), 117-167. doi: 10.1023/A:1009098719440.  Google Scholar

[9]

T. Ikeda and M. Mimura, An interfacial approach to regional segregation of two competing species mediated by a predator, Journal of Mathematical Biology, 31 (1993), 215-240. doi: 10.1007/BF00166143.  Google Scholar

[10]

T. Ikeda and Y. Nishiura, Pattern selection for two breathers, SIAM Journal on Applied Mathematics, 54 (1994), 195-230. doi: 10.1137/S0036139992237250.  Google Scholar

[11]

A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device, Angewandte Chemie International Edition, 45 (2006), 3087-3089. Google Scholar

[12]

S. Koga and Y. Kuramoto, Localized patterns in reaction-diffusion systems, Progress of Theoretical Physics, 63 (1980), 106-121. doi: 10.1143/PTP.63.106.  Google Scholar

[13]

T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional belousov-zhabotinski reaction in a water-in-oil microemulsion, Physical review letters, 98 (2007), 188303. doi: 10.1103/PhysRevLett.98.188303.  Google Scholar

[14]

I. Lengyel and I. Epstein, Modeling of turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652. doi: 10.1126/science.251.4994.650.  Google Scholar

[15]

Y.-X. Li, Tango waves in a bidomain model of fertilization calcium waves, Physica D, 186 (2003), 27-49. doi: 10.1016/S0167-2789(03)00237-9.  Google Scholar

[16]

R. McKay, T. Kolokolnikov and P. Muir, Interface oscillations in reaction-diffusion systems above the hopf bifurcation,, Discrete and Continuous Dynamical Systems. Series BA Journal Bridging Mathematics and Sciences., ().   Google Scholar

[17]

E. Meron, H. Yizhaq and E. Gilad, Localized structures in dryland vegetation: forms and functions, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 037109. doi: 10.1063/1.2767246.  Google Scholar

[18]

C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems, Phys, Rev. E, 53 (1996), 3101-3116. doi: 10.1103/PhysRevE.53.3101.  Google Scholar

[19]

M. Nagayama, K.-i. Ueda and M. Yadome, Numerical approach to transient dynamics of oscillatory pulses in a bistable reaction-diffusion system, Japan journal of industrial and applied mathematics, 27 (2010), 295-322. doi: 10.1007/s13160-010-0015-8.  Google Scholar

[20]

Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J.Appl. Math, 49 (1989), 481-514. doi: 10.1137/0149029.  Google Scholar

[21]

T. Ohta, A. Ito and A. Tetsuka, Self-organization in an excitable reaction-diffusion system: synchronization of oscillatory domains in one dimension, Physical Review A, 42 (1990), 3225. doi: 10.1103/PhysRevA.42.3225.  Google Scholar

[22]

M. Suzuki, T. Ohta, M. Mimura and H. Sakaguchi, Breathing and wiggling motions in three-species laterally inhibitory systems, Physical Review E, 52 (1995), 3645-3655. doi: 10.1103/PhysRevE.52.3645.  Google Scholar

[23]

P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a three-component system: Stability and bifurcations, Physica D, 237 (2008), 3335-3368. doi: 10.1016/j.physd.2008.07.014.  Google Scholar

[24]

V. K. Vanag and I. R. Epstein, Localized patterns in reaction-diffusion systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 037110, 11pp. doi: 10.1063/1.2752494.  Google Scholar

[25]

V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology, Physics of Life Reviews, 6 (2009), 267-310. doi: 10.1016/j.plrev.2009.10.002.  Google Scholar

[26]

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), 209-264. doi: 10.1007/s00332-002-0531-z.  Google Scholar

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