Inside dynamics of delayed traveling waves

Mathematical Modelling of Natural Phenomena

Volumn 8, Issue 3, 2013, Pages 42-59

  Bonnefon, O ^a   Garnier, J ^a,b   Hamel, F ^b,c   Roques, L ^a  

^a INRA   (France)

^b AIX MARSEILLE UNIVERSITY   (France)

^c INSTITUT UNIVERSITAIRE DE FRANCE   (France)

Author keywords

Delayed equation; Hutchinson's equation; Kobayashi's equation; Pulled and pushed solutions; Traveling waves

Indexed keywords

CLASSICAL REACTIONS; DELAYED EQUATION; HUTCHINSON'S EQUATION; KOBAYASHI'S EQUATION; REACTION DIFFUSION EQUATIONS; SPATIOTEMPORAL EVOLUTION; TRAVELING WAVE; TRAVELING WAVE SOLUTION;

DYNAMICS; MATHEMATICAL MODELS;

LINEAR EQUATIONS;

EID: 84879303344     PISSN: 09735348     EISSN: 17606101     Source Type: Journal    
DOI: 10.1051/mmnp/20138305     Document Type: Article

References (49)

1. M. Aguerrea, S. Trofimchuk, G. Valenzuela. Uniqueness of fast travelling fronts in reaction-diffusion equations with delay. Proc. Royal Soc. A, 464 (2008), 2591-2608.
2. S. Ai. Traveling wave fronts for generalized Fisher equations with spatio-temporal delays. J. Diff. Equations, 232 (2007), 104-133. (Pubitemid 44597103)
3. D. G. Aronson, H. G. Weinberger. Nonlinear diffusion in population genetics, combustion and nerve propagation. In Partial Differential Equations and Related Topics, volume 446 of Lectures Notes Math, 5-49. Springer, New York, 1975.
4. D. G. Aronson, H. G. Weinberger. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math., 30 (1978), 33-76.
5. F. Austerlitz, S. Mariette, N. Machon, P. H. Gouyon, B. Godelle. Effects of colonization processes on genetic diversity: differences between annual plants and tree species. Genetics, 154 (2000), 1309-1321. (Pubitemid 30140715)
6. L. Berec, E. Angulo, F. Courchamp. Multiple Allee effects and population management. Trends Ecol. Evol., 22 (2007), 185-191. (Pubitemid 46437275)
7. J. Billingham, D. J. Needham. The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form of traveling waves. Phil. Trans. Royal Soc. A, 334 (1991), 1-24.
8. M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc., 44 (1983).
9. N. F. Britton. Reaction-Diffusion Equations and their Applications to Biology. Academic Press, London, 1986.
10. J. P. Eckmann, C. E. Wayne. The nonlinear stability of front solutions for parabolic partial differential equations. Comm. Math. Phys., 161 (1994), 323-334.
11. T. Faria, S. Trofimchuk. Nonmonotone travelling waves in a single species reaction-diffusion equation with delay. J. Diff. Equations, 228 (2006), 357-376. (Pubitemid 44041151)
12. P. C. Fife. Mathematical Aspects of Reacting and Diffusing Systems, volume 28 of Lecture Notes in Biomathematics. Springer-Verlag, 1979.
13. P. C. Fife, J. McLeod. The approach of solutions of nonlinear diffusion equations to traveling front solutions. Arch. Ration. Mech. Anal., 65 (1977), 335-361. (Pubitemid 8577794)
14. J. Garnier, T. Giletti, F. Hamel, L. Roques. Inside dynamics of pulled and pushed fronts. J. Math. Pures Appl., 98 (2012), 428-449.
15. J. Garnier, L. Roques, F. Hamel. Success rate of a biological invasion in terms of the spatial distribution of the founding population. B. Math. Biol., 74 (2012), 453-473.
16. A. Gomez, S. Trofimchuk. Monotone traveling wavefronts of the KPP-Fisher delayed equation. J. Diff. Equations, 250 (2011), 1767-1787.
17. P. Grindrod. Theory and Applications of Reaction-Diffusion Equations. Clarendon Press, 1996.
18. O. Hallatschek, D. R. Nelson. Gene surfing in expanding populations. Theor. Popul. Biol., 73 (2008), 158-170.
19. F. Hamel, J. Nolen, J.-M. Roquejoffre, L. Ryzhik. A short proof of the logarithmic Bramson correction in Fisher-KPP equations. Netw. Heterog. Media, In press (2013).
20. K. Hasik, S. Trofimchuk. Slowly oscillating wavefronts of the KPP-Fisher delayed equation. arXiv, arXiv:1206.0484 (2012).
21. A. L. Hodgkin, A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiology, 117 (1952), 500-544.
22. G. E. Hutchinson. Circular causal systems in ecology. Ann. New York Acad. Sci., 50 (1948), 221-246.
23. J. I. Kanel. Certain problems of burning-theory equations. Sov. Math. Doklady, 2 (1961), 48-51.
24. K. Kobayashi. On the semilinear heat equations with time-lag. Hiroshima Math. J., 7 (1977), 459-472.
25. A. N. Kolmogorov, I. G. Petrovsky, N. S. Piskunov. Atude de l'quation de la diffusion avec croissance de la quantité de matïre et son application à un probl̈me biologique. Bull. Univ. Atat Moscou, Sér. Int. A, 1 (1937), 1-26.
26. M. K. Kwong, J. Ou. Existence and nonexistence of monotone traveling waves for the delayed Fisher equation. J. Diff. Equations, 249 (2010), 728-745.
27. K.-S. Lau. On the nonlinear diffusion equation of Kolmogorov, Petrovsky and Piscounov. J. Diff. Equations, 59 (1985), 44-70.
28. M. A. Lewis, P. Kareiva. Allee dynamics and the spread of invading organisms. Theor. Popul. Biol., 43 (1993), 141-158.
29. M. A. Lewis, P. Van Den Driessche. Waves of extinction from sterile insect release. Math. Biosci., 116 (1993), 221-247.
30. X. Liang, X.-Q. Zhao. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm. Pure Appl. Math., 60 (2007), 1-40. (Pubitemid 44832314)
31. S. Ma. Traveling waves for non-local delayed diffusion equations via auxiliary equations. J. Diff. Equations, 237 (2007), 259-277.
32. J. D. Murray. Mathematical Biology. Third Edition. Interdisciplinary Applied Mathematics 17, Springer-Verlag, New York, 2002.
33. S. Pan. Asymptotic behavior of travelling fronts of the delayed Fisher equation. Nonlinear Anal. Real World Appl., 10 (2009), 1173-1182.
34. L. Roques, J. Garnier, F. Hamel, E K. Klein. Allee effect promotes diversity in traveling waves of colonization. Proc. Natl. Acad. Sci. USA, 109 (2012), 8828-8833.
35. L. Roques, F. Hamel, J. Fayard, B. Fady, E K. Klein. Recolonisation by diffusion can generate increasing rates of spread. Theor. Popul. Biol., 77 (2010), 205-212.
36. D. H. Sattinger. On the stability of waves of nonlinear parabolic systems. Adv. Math., 22 (1976), 312-355.
37. D. H. Sattinger. Weighted norms for the stability of traveling waves. J. Diff. Equations, 25 (1977), 130-144.
38. K. W. Schaaf. Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans. Amer. Math. Soc., 302 (1987), 587-615.
39. N. Shigesada, K. Kawasaki. Biological Invasions: Theory and Practice. Oxford Series in Ecology and Evolution, Oxford: Oxford University Press, 1997.
40. A. N. Stokes. On two types of moving front in quasilinear diffusion. Math. Biosci., 31 (1976), 307-315.
41. E. Trofimchuk, P. Alvarado, S. Trofimchuk. On the geometry of wave solutions of a delayed reaction-diffusion equation. J. Diff. Equations, 246 (2009), 1422-1444.
42. E. Trofimchuk, M. Pinto, S. Trofimchuk. Pushed traveling fronts in monostable equations with monotone delayed reaction. arXiv:1111.5161v1.
43. K. Uchiyama. The behaviour of solutions of some non-linear diffusion equations for large time. J. Math. Kyoto Univ., 18 (1978), 453-508.
44. M. O. Vlad, L. L. Cavalli-Sforza, J. Ross. Enhanced (hydrodynamic) transport induced by population growth in reaction-diffusion systems with application to population genetics. Proc. Natl. Acad. Sci. USA, 101 (2004), 10249-10253. (Pubitemid 38924911)
45. Z.-C. Wang, W.-T. Li, S. Ruan. Travelling wave fronts of reaction-diffusion systems with spatio-temporal delays. J. Diff. Equations, 222 (2006), 185-232. (Pubitemid 43192430)
46. Z.-C. Wang, W.-T. Li, S. Ruan. Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay. J. Diff. Equations, 238 (2007), 153-200.
47. J. Wu. Theory and Applications of Partial Functional Differential Equations, volume 119 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.
48. J. Wu, X. Zou. Traveling wave fronts of reaction-diffusion systems with delay. J. Dyn. Diff. Equations, 13 (2001), 651-687.
49. X. Zou. Delay induced traveling wave fronts in reaction diffusion equations of fisher-kpp type. J. Comput. Appl. Math., 146 (2002), 309-321.