Invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave

Physica D: Nonlinear Phenomena

Volumn 117, Issue 1-4, 1998, Pages 145-166

  Sherratt, Jonathan A ^a  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

Oscillatory systems; Reaction diffusion; Travelling waves; Wave trains

Indexed keywords

EID: 0001218738     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(97)00317-5     Document Type: Article

References (40)

1. R. Kapral, K. Showalter, Chemical Waves and Patterns, Kluwer Academic Publishers, Dordrecht, 1995.
2. J. Sneyd, J. Keizer, M.J. Sanderson, Mechanisms of calcium oscillations and waves: A quantitative analysis, FASEB J. 14 (1995) 1463-1472.
3. E.E. Holmes, M.A. Lewis, J. Banks, R.R. Veit, Partial differential equations in ecology: Spatial interactions and populations dynamics, Ecology 75 (1994) 17-29.
4. N. Kopell, L.N. Howard, Plane wave solutions to reaction-diffusion equations, Stud. Appl. Math. 52 (1973) 291-328.
5. H.G. Othmer, Current problems in pattern formation, Lect. Math. Life Sci. 9 (1977) 57-86.
6. G.B. Ermentrout, Small amplitude stable wave trains in reaction-diffusion systems, Lect. Notes Pure Appl. Math. 54 (1980) 217-228.
7. K. Maginu, Stability of periodic travelling wave solutions with large spatial periods in reaction-diffusion systems, J. Diff. Eqns. 39 (1981) 73-99.
8. T.M. Kapitula, Stability of weak shocks in λ-ω systems, Indiana Univ. Math. J. 40 (1991) 1193-1219.
9. S. Koga, Coexistence of stably propagating periodic wave trains in intrinsically bistable reaction-diffusion systems, Phys. Lett. A 191 (1994) 251-256.
10. S. Koga, A variety of stable persistent waves in intrinsically bistable reaction-diffusion systems - from one-dimensional periodic-waves to one-armed and 2-armed rotating spiral waves, Physica D 84 (1995) 148-161.
11. B. Ermentrout, X. Chen, Z. Chen, Transition fronts and localized structures in bistable reaction-diffusion systems, Physica D 108 (1997) 147-167.
12. J.A. Sherratt, Irregular wakes in reaction-diffusion waves, Physica D 70 (1994) 370-382.
13. A. Kolmogorov, I. Petrovsky, N. Piscounov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moscow Univ. Bull. Math. 1 (1937) 1-25.
14. R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937) 353-369.
15. J.D. Murray, Mathematical Biology, Springer, Berlin, 1989.
16. R.M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1981.
17. A. Okubo, Dynamical aspects of animal grouping: swarms, schools, flocks and herds, Adv. Biophys. 22 (1986) 1-94.
18. P.D. Dale, P.K. Maini, J.A. Sherratt, Mathematical modelling of corneal epithelial wound healing, Math. Biosci. 124 (1994) 127-147.
19. 4, Trans. AMS 286 (1984) 557-594.
20. J.A. Sherratt, B.T. Eagan, M.A. Lewis, Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?, Phil. Trans. Roy. Soc. B 352 (1997) 21-38.
21. J. Sneyd, J.A. Sherratt, On the propagation of calcium waves in an inhomogeneous medium, SIAM J. Appl. Math. 57 (1997) 73-94.
22. J.A. Sherratt, Unstable wave trains and chaotic wakes in reaction-diffusion systems of λ-ω type. Physica D 82 (1995) 165-179.
23. R.J. Field, M. Burger (Eds.), Oscillations and Travelling Waves in Chemical Systems, Wiley, New York, 1985.
24. J.A. Sherratt, On the evolution of periodic plane waves in reaction-diffusion equations of λ-ω type. SIAM J. Appl. Math. 54 (1994) 1374-1385.
25. J.A. Sherratt, A comparison of two numerical methods for oscillatory reaction-diffusion equations, Appl. Math. Lett. 10 (1997) 1-5.
26. L.N. Howard, N. Kopell, Slowly varying waves and shock structures in reaction-diffusion equations, Stud. Appl. Math. 56 (1977) 95-145.
27. J.A. Sherratt, On the speed of amplitude transition waves in reaction-diffusion equations of λ-ω type, IMA J. Appl. Math. 52 (1994) 79-92.
28. M.I. Freidlin, On wavefront propagation in multicomponent media, Trans. AMS 276 (1983) 181-191; see also the erratum, Trans. AMS 289 (1985) 429.
29. M. Freidlin, Coupled reaction-diffusion equations, Ann. Prob. 19 (1991) 29-57.
30. J. Billingham, D.J. Needham, The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. III: Large time development in quadratic autocatalysis, Quart. Appl. Math. 50 (1992) 343-372.
31. S.R. Dunbar, Travelling waves in diffusive predator-prey equations - periodic orbits and point-to-periodic heteroclinic connections, SIAM J. Appl. Math. 46 (1986) 1057-1078.
32. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970.
33. F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Roy. Soc. Edin. A 80 (1978) 213-234.
34. D. Rand, Dynamics and symmetry. Predictions for modulated waves in rotating fluids, Arch. Rat. Mech. Anal. 79 (1982) 1-37.
35. M. Myers, P. Holmes, J. Elezgaray, G. Berkooz, Wavelet projections of the Kuramoto-Sivashinsky equation. 1. Heteroclinic cycles and modulated waves for short systems, Physica D 86 (1995) 396-427.
36. S.M. Cox, Mode interactions in Rayleigh-Benard convection, Physica D 95 (1996) 50-61.
37. R. Mantel, D. Barkley, Periodic forcing of spiral waves in excitable media, Phys. Rev. E 54 (1996) 4791-4802.
38. D.E. Clapham, J. Sneyd, Intracellular calcium waves, Adv. Second Mess. Phosphoprotein Res. 30 (1995) 1-24.
39. Y.H. Tang, J.L. Stephenson, H.G. Othmer, Simplification and analysis of models of calcium dynamics based on IP3-sensitive calcium channel kinetics, Biophys. J. 70 (1996) 246-263.
40. J.H. Merkin, M.A. Sadiq, The propagation of travelling waves in an open cubic autocatalytic chemical system, IMA J. Appl. Math. 57 (1996) 273-309.