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On a reaction-diffusion model for sterile insect release method with release on the boundary
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 |
References:
[1] |
M. Banerjee and S. Petrovski, Self-organised spatialpatterns and chaos in a ratio-dependent predator-prey system, Theor. Ecol., 4 (2011), 37-53. |
[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, Springer-Verlag, New York, 1999. |
[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, Vol. 14, SIAM, Philadelphia, PA, 1996. |
[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), 187-214. |
[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), 582-666. |
[6] |
A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains, Mem. Amer. Math. Soc., 199 (2009). |
[7] |
S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dyn. Diff. Eq., 14 (2002), 85-137. |
[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), 117-147. |
[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), 3933-3957. |
[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), 7 pp. |
[11] |
A. Hagberg and E. Meron, Pattern formation nongradient reaction-diffusion systems: The effect of front bifurcations, Nonlinearity, 7 (1994), 805-835. |
[12] |
H. Hardway and Y.-X. Li, Stationary and oscillatory fronts in a two-component genetic regulatory network model, Physica D, 239 (2010), 1650-1661. |
[13] |
P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a three-component system: Stability and bifurcations, Physica D, 237 (2008), 3335-3368. |
[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), 190-193. |
[15] |
T. Ikeda and Y. Nishiura, Pattern selection for two breathers, SIAM J. Appl. Math., 54 (1994), 195-230. |
[16] |
H. Ikeda and T. Ikeda, Bifurcaiton phenomena from standingpulse solutions in some reaction-diffusion systems, J. Dyn. Diff. Eqns., 12 (2000), 117-167. |
[17] |
A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device, Angewandte Chemie, 45 (2006), 3087. |
[18] |
R. Kapral and K. Showalter, eds., "Chemical Waves and Patterns," Kluwer, Dordrecht, 1995. |
[19] |
B. S. Kerner and V. V. Osipov, "Autosolitons: A New Approach to Problems of Self-Organization and Turbulence," Fundamental Theories of Physics, 61, Kluwer Academic Publishers Group, Dordrecht, 1994. |
[20] |
A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures, Rev. Modern Physics, 66 (1994), 1481-1507. |
[21] |
S. Koga and Y. Kuramoto, Localized patterns in reaction-diffusion systems, Progress of Theoretical Physics, 63 (1980), 106-121. |
[22] |
T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction-diffusion system, Physical Review Letters, 98 (2007), art. 188303. |
[23] |
T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability, Physica D, 214 (2006), 63-77. |
[24] |
T. Kolokolnikov, M. Ward and J. Wei, Self-replication of mesa patterns in reaction-diffusion systems, Physica D, 236 (2007), 104-122. |
[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), 185-222. |
[26] |
Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence," Springer Series in Synergetics, 19, Springer-Verlag, Berlin, 1984. |
[27] |
I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652. |
[28] |
Y.-X. Li, Tango waves in a bidomain model of fertilization calcium waves, Physica D, 186 (2003), 27-49. |
[29] |
H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, London, 1982. |
[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), submitted. |
[31] |
C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reaction-diffusion system, Phys. Rev. E, 54 (1996), 4860-4879. |
[32] |
C. B. Muratov, Synchronization, chaos, and breakdown of collective domain oscillations in reaction-diffusion systems, Phys. Rev. E, 55 (1997), 1463-1477. |
[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), 3101-3116. |
[34] |
J. D. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 1989. |
[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), 295-322. |
[36] |
W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. AMS, 357 (2005), 3953-3969. |
[37] |
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770. |
[38] |
Y. Nishiura and M. Mimura, Layer oscillations inreaction-diffusion systems, SIAM J. Appl. Math., 49 (1989), 481-514. |
[39] |
J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. |
[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), 3645-3655. |
[41] |
M. Taki, M. Tlidi and T. Kolokolnikov, eds., Dissipative localized structures in extended systems, Chaos Focus Issue, 17 (2007). |
[42] |
V. K. Vanag and I. R. Epstein, Localized patterns inreaction-diffusion systems, Chaos, 17 (2007), 037110, 11 pp. |
[43] |
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. |
[44] |
R. Wang, P. Keast and P. Muir, A comparison of adaptive software for 1-D parabolic PDEs, J. Comput. Appl. Math., 169 (2004), 127-150. |
[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), 239-260. |
[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), 454-470. |
[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), 209-264. |
[48] |
H. Yizhaq and E. Gilad, Localized structures in dry land vegetation: Forms and functions, Chaos, 17 (2007), 037109. |
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), 37-53. |
[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, Springer-Verlag, New York, 1999. |
[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, Vol. 14, SIAM, Philadelphia, PA, 1996. |
[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), 187-214. |
[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), 582-666. |
[6] |
A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains, Mem. Amer. Math. Soc., 199 (2009). |
[7] |
S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dyn. Diff. Eq., 14 (2002), 85-137. |
[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), 117-147. |
[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), 3933-3957. |
[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), 7 pp. |
[11] |
A. Hagberg and E. Meron, Pattern formation nongradient reaction-diffusion systems: The effect of front bifurcations, Nonlinearity, 7 (1994), 805-835. |
[12] |
H. Hardway and Y.-X. Li, Stationary and oscillatory fronts in a two-component genetic regulatory network model, Physica D, 239 (2010), 1650-1661. |
[13] |
P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a three-component system: Stability and bifurcations, Physica D, 237 (2008), 3335-3368. |
[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), 190-193. |
[15] |
T. Ikeda and Y. Nishiura, Pattern selection for two breathers, SIAM J. Appl. Math., 54 (1994), 195-230. |
[16] |
H. Ikeda and T. Ikeda, Bifurcaiton phenomena from standingpulse solutions in some reaction-diffusion systems, J. Dyn. Diff. Eqns., 12 (2000), 117-167. |
[17] |
A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion memory device, Angewandte Chemie, 45 (2006), 3087. |
[18] |
R. Kapral and K. Showalter, eds., "Chemical Waves and Patterns," Kluwer, Dordrecht, 1995. |
[19] |
B. S. Kerner and V. V. Osipov, "Autosolitons: A New Approach to Problems of Self-Organization and Turbulence," Fundamental Theories of Physics, 61, Kluwer Academic Publishers Group, Dordrecht, 1994. |
[20] |
A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures, Rev. Modern Physics, 66 (1994), 1481-1507. |
[21] |
S. Koga and Y. Kuramoto, Localized patterns in reaction-diffusion systems, Progress of Theoretical Physics, 63 (1980), 106-121. |
[22] |
T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction-diffusion system, Physical Review Letters, 98 (2007), art. 188303. |
[23] |
T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability, Physica D, 214 (2006), 63-77. |
[24] |
T. Kolokolnikov, M. Ward and J. Wei, Self-replication of mesa patterns in reaction-diffusion systems, Physica D, 236 (2007), 104-122. |
[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), 185-222. |
[26] |
Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence," Springer Series in Synergetics, 19, Springer-Verlag, Berlin, 1984. |
[27] |
I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652. |
[28] |
Y.-X. Li, Tango waves in a bidomain model of fertilization calcium waves, Physica D, 186 (2003), 27-49. |
[29] |
H. Meinhardt, "Models of Biological Pattern Formation," Academic Press, London, 1982. |
[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), submitted. |
[31] |
C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reaction-diffusion system, Phys. Rev. E, 54 (1996), 4860-4879. |
[32] |
C. B. Muratov, Synchronization, chaos, and breakdown of collective domain oscillations in reaction-diffusion systems, Phys. Rev. E, 55 (1997), 1463-1477. |
[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), 3101-3116. |
[34] |
J. D. Murray, "Mathematical Biology," Biomathematics, 19, Springer-Verlag, Berlin, 1989. |
[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), 295-322. |
[36] |
W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. AMS, 357 (2005), 3953-3969. |
[37] |
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770. |
[38] |
Y. Nishiura and M. Mimura, Layer oscillations inreaction-diffusion systems, SIAM J. Appl. Math., 49 (1989), 481-514. |
[39] |
J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. |
[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), 3645-3655. |
[41] |
M. Taki, M. Tlidi and T. Kolokolnikov, eds., Dissipative localized structures in extended systems, Chaos Focus Issue, 17 (2007). |
[42] |
V. K. Vanag and I. R. Epstein, Localized patterns inreaction-diffusion systems, Chaos, 17 (2007), 037110, 11 pp. |
[43] |
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. |
[44] |
R. Wang, P. Keast and P. Muir, A comparison of adaptive software for 1-D parabolic PDEs, J. Comput. Appl. Math., 169 (2004), 127-150. |
[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), 239-260. |
[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), 454-470. |
[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), 209-264. |
[48] |
H. Yizhaq and E. Gilad, Localized structures in dry land vegetation: Forms and functions, Chaos, 17 (2007), 037109. |
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