Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-13T03:34:43.789Z Has data issue: false hasContentIssue false
  • English
  • Français

Inside Dynamics of Delayed Traveling Waves

Published online by Cambridge University Press:  12 June 2013

O. Bonnefon ,
J. Garnier ,
F. Hamel  and
L. Roques
O. Bonnefon
Affiliation:
UR 546 Biostatistique et Processus Spatiaux, INRA, 84000 Avignon, France
J. Garnier
Affiliation:
UR 546 Biostatistique et Processus Spatiaux, INRA, 84000 Avignon, France Aix Marseille Université, CNRS, Centrale Marseille, LATP, UMR 7353 , 13453 Marseille, France
F. Hamel
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, LATP, UMR 7353 , 13453 Marseille, France Institut Universitaire de France
L. Roques*
Affiliation:
UR 546 Biostatistique et Processus Spatiaux, INRA, 84000 Avignon, France
*
⋆ Corresponding author. E-mail: lionel.roques@avignon.inra.fr
Get access

Abstract

The notion of inside dynamics of traveling waves has been introduced in the recent paper [14]. Assuming that a traveling wave u(t,x) = U(x − c   t) is made of several components υi ≥ 0 (i ∈ I ⊂ N), the inside dynamics of the wave is then given by the spatio-temporal evolution of the densities of the components υi. For reaction-diffusion equations of the form ∂tu(t,x) = ∂xxu(t,x) + f(u(t,x)), where f is of monostable or bistable type, the results in [14] show that traveling waves can be classified into two main classes: pulled waves and pushed waves. Using the same framework, we study the pulled/pushed nature of the traveling wave solutions of delay equations

             ∂tu(t,x) = ∂xxu(t,x) + F(u(t −τ,x),u(t,x))

We begin with a review of the latest results on the existence of traveling wave solutions of such equations, for several classical reaction terms. Then, we give analytical and numerical results which describe the inside dynamics of these waves. From a point of view of population ecology, our study shows that the existence of a non-reproductive and motionless juvenile stage can slightly enhance the genetic diversity of a species colonizing an empty environment.

Keywords

traveling wavesdelayed equationpulled and pushed solutionsKobayashi’s equationHutchinson’s equation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 42 - 59
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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