# American Institute of Mathematical Sciences

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

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.
Citation: Shuangquan Xie, Theodore Kolokolnikov. Oscillations of many interfaces in the near-shadow regime of two-component reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 959-975. doi: 10.3934/dcdsb.2016.21.959
##### References:
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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. [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. [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. [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. [10] T. Ikeda and Y. Nishiura, Pattern selection for two breathers, SIAM Journal on Applied Mathematics, 54 (1994), 195-230. doi: 10.1137/S0036139992237250. [11] A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device, Angewandte Chemie International Edition, 45 (2006), 3087-3089. [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. [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. [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. [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. [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. [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. [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. [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. [20] Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J.Appl. Math, 49 (1989), 481-514. doi: 10.1137/0149029. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] T. Ikeda and Y. Nishiura, Pattern selection for two breathers, SIAM Journal on Applied Mathematics, 54 (1994), 195-230. doi: 10.1137/S0036139992237250. [11] A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device, Angewandte Chemie International Edition, 45 (2006), 3087-3089. [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. [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. [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. [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. [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. [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. [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. [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. [20] Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J.Appl. Math, 49 (1989), 481-514. doi: 10.1137/0149029. [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. [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. [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. [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. [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. [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.
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