On the persistence of spatiotemporal oscillations generated by invasion

IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)

Volumn 63, Issue 2, 1999, Pages 199-216

  Kay, Alison L ^a   Sherratt, Jonathan A ^b  

^a UNIVERSITY OF WARWICK   (United Kingdom)

^b HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; CHEMICAL REACTIONS; COMPUTER SIMULATION; DIFFUSION; EQUATIONS OF MOTION; FINITE ELEMENT METHOD; PARAMETER ESTIMATION; WAVEFRONTS;

FINITE DOMAIN; SPATIOTEMPORAL OSCILLATIONS; WAKE OF INVASION; ZERO FLUX BOUNDARY CONDITION;

OSCILLATIONS;

EID: 0033353519     PISSN: 02724960     EISSN: None     Source Type: Journal    
DOI: 10.1093/imamat/63.2.199     Document Type: Article

References (42)

1. BORGHANS, J. A. M., DUPONT, G., & GOLDBETER, A. 1997 Complex intra-cellular calcium oscillations - a theoretical exploration of possible mechanisms. J. Biophys. Chem. 66, 25-41.
2. CROSS, M. C. & HOHENBERG, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851-1112.
3. DAVIDSON, F. A. 1998 Chaotic wakes and other wave-induced behavior in a system of reaction-diffusion equations. Int. J. Bif. Chaos 8, 1303-1313.
4. DUNBAR, S. R. 1986 Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits. SIAM J. Appl. Math. 46, 1057-1077.
5. DUNSTER, T. M. 1996 Asymptotic solutions of second order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. R. Soc. A 452, 1331-1349.
6. ERMENTROUT, G. B. 1980 Small amplitude stable wave-trains in reaction-diffusion systems. Lect. Notes Pure Appl. Math. 54, 217-228.
7. ERMENTROUT, G. B., CHEN, X., & CHEN, Z. 1997 Transition fronts and localised structures in bistable reaction-diffusion systems. Physica D 108, 147-167.
8. HASSELL. M. P. COMINS, H. N., & MAY, R. M. 1991 Spatial structure and chaos in insect population dynamics. Nature 353, 255-258.
9. HOWARD, L. N. & KOPELL, N. 1977 Slowly varying waves and shock structures in reaction-diffusion equations. Stud. Appl. Math. 56, 95-145.
10. KOGA, S. 1993 Coexistence of stably propagating periodic wave trains in intrinsically bistable reaction-diffusion systems. Phys. Lett. 191, 251-256.
11. KOGA, S. 1995 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, 148-161.
12. KOPELL, N. & HOWARD, L. N. 1973a Horizontal bands in the Belousov reaction. Science 180, 1171-1175.
13. KOPELL, N. & HOWARD, L. N. 1973b Plane wave solutions to reaction-diffusion equations. Stud. Appl. Math. 52, 291-328.
14. KURAMOTO, Y. 1984 Chemical Oscillations, Waves and Turbulence. Berlin: Springer.
15. LEACH, J. A., MERKIN, T. H., & SCOTT, S. K. 1994 Oscillations and waves in the Belousov-Zhabotinskii reaction in a finite medium. J. Math. Chem. 16, 115-124.
16. MAGINU, K. 1981 Stability of periodic travelling wave solutions with large spatial periods in reaction-diffusion equations. J. Diff. Eq. 39, 73-99.
17. MAY, R. M. 1981 Stability and Complexity in Model Ecosystems. Princeton: University Press.
18. MAY, R. M. & OSTER, G. F. 1976 Bifurcations and dynamics complexity in simple ecological models. Amer. Nat. 110, 573-599.
19. MERKIN, J. H. & SADIQ, M. A. 1997 The propagation of travelling waves in an open cubic autocatalytic chemical system. IMA J. Appl. Math. 57, 273-309.
20. METZ, J. A. J. & DIEKMANN, O. 1986 A gentle introduction to structured population models: three worked examples. In: The Dynamics of Physiologically Structured Populations, Vol. 68. Berlin: Springer.
21. MURPHY, G. M. 1960 Ordinary Differential Equations and their Solutions. New York: Van Nostrand.
22. MURRAY, J. D. 1989 Mathematical Biology. Berlin: Springer.
23. OTHMER, H. G. 1977 Current problems in pattern formation. Lect. Math. Life Sci. 9, 57-86.
24. PAGOLA, A., Ross, J., & VIDAL, C. 1988 Measurement of dispersion-relation of chemical waves in an oscillatory reacting medium. J. Phys. Chem. 92, 163-166.
25. PASCUAL, M. 1993 Diffusion induced chaos in a spatial predator-prey system. Proc. R. Soc B 251, 1-7.
26. RASMUSSEN, K. E., MAZIN, W. & MOSEKILDE, E. 1996 Wave-splitting in the bistable Gray-Scott model. Int. J. Bif. Chaos 6, 1077-1092.
27. ROHANI, P., LEWIS, T. J., GRUNBAUM, D., & RUXTON, G. D. 1997 Spatial self-organization in ecology: Pretty patterns or robust reality? TREE 12, 70-74
28. ROVINSKY, A. B. 1987 Twinkling patterns and diffusion-induced chaos in a model of the Belousov-Zhabotinsky chemical medium. J. Phys. Chem. 91, 5113-5118
29. SCOTT, S. K. 1994 Oscillations, Waves and Chaos in Chemical Kinetics. Oxford: University Press.
30. SHERRATT, J. A. 1994a Irregular wakes in reaction-diffusion waves. Physica D 70, 370-382.
31. SHERRATT, J. A. 1994b On the speed of amplitude transition waves in reaction-diffusion systems of λ - ω type. IMA J. Appl. Math. 52, 79-92.
32. SHERRATT, J. A. 1995 Unstable wave-trains and chaotic wakes in reaction-diffusion systems of λ - ω type. Physica D 82, 165-179.
33. SHERRATT, J. A. 1997 A comparison of two numerical methods for oscillatory reaction-diffusion equations. Appl. Math. Lett. 10, 1-5.
34. SHERRATT, J. A. 1999 Invading wave fronts and their oscilatory wakes are linked by a modulated travelling phase resetting wave. Physica D to appear.
35. SHERRATT, J. A., EAGAN, B. T., & LEWIS, M. A. 1997 Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality. Phil. Trans. R. Soc. B 352, 21-38.
36. SHERRATT, J. A., LEWIS, M. A., & FOWLER, A. C. 1995 Ecological chaos in the wake of invasion. Proc. Nat. Acad. Sci. USA 92, 2524-2528.
37. SKELLAM, J. C. 1951 Random dispersal in theoretical populations. Biometrika 38, 196-218.
38. SNEYD, J., GIRARD, S., & CLAPHAM, D. 1993 Calcium wave propagation by calcium-induced calcium release: an unusual excitable system. Bull. Math. Biol. 55, 315-344.
39. SNEYD, J., KEIZER, J., & SANDERSON, M. J. 1995 Mechanisms of calcium oscillations and waves: a quantative analysis. FASEB J. 14, 1463-1472.
40. SNEYD, J. & SHERRATT, J. A. 1997 On the propagation of calcium waves in an inhomogeneous medium. SIAM J. Appl. Math. 57, 73-94.
41. STÖSSEL, R. & MÜNSTER, A. F. 1995 Periodic and irregular wave patterns in an open tubular gel reactor. Chem. Phys. Lett. 239, 354-360.
42. TEMME, N. M. 1975 Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function. Math. Comp. 29, 1109-1114.