Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The Bramson logarithmic delay in the cane toads equations

Authors: Emeric Bouin, Christopher Henderson and Lenya Ryzhik
Journal: Quart. Appl. Math. 75 (2017), 599-634
MSC (2010): Primary 35K57, 45K05, 35C07, 35Q92
DOI: https://doi.org/10.1090/qam/1470
Published electronically: April 18, 2017
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Abstract: We study a non-local reaction-diffusion-mutation equation modelling the spreading of a cane toads population structured by a phenotypical trait responsible for the spatial diffusion rate. When the trait space is bounded, the cane toads equation admits travelling wave solutions as shown in an earlier work of the first author and V. Calvez. Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the travelling waves grows as $ (3/(2\lambda ^*))\log t$. This result relies on a present-time Harnack inequality which allows one to compare solutions of the cane toads equation to those of a Fisher-KPP type equation that is local in the trait variable.

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  • [1] Matthieu Alfaro, Jérôme Coville, and Gaël Raoul, Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait, Comm. Partial Differential Equations 38 (2013), no. 12, 2126-2154. MR 3169773, https://doi.org/10.1080/03605302.2013.828069
  • [2] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607-694. MR 0435594
  • [3] O. Bénichou, V. Calvez, N. Meunier, and R. Voituriez,
    Front acceleration by dynamic selection in fisher population waves,
    Phys. Rev. E 86:041908, 2012.
  • [4] Henri Berestycki, Tianling Jin, and Luis Silvestre, Propagation in a non local reaction diffusion equation with spatial and genetic trait structure, Nonlinearity 29 (2016), no. 4, 1434-1466. MR 3476514, https://doi.org/10.1088/0951-7715/29/4/1434
  • [5] Henri Berestycki and Louis Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), no. 5, 497-572 (English, with English and French summaries). MR 1191008
  • [6] N. Berestycki, C. Mouhot, and G. Raoul, Existence of self-accelerating fronts for a non-local reaction-diffusion equations,
  • [7] Emeric Bouin and Vincent Calvez, Travelling waves for the cane toads equation with bounded traits, Nonlinearity 27 (2014), no. 9, 2233-2253. MR 3266851, https://doi.org/10.1088/0951-7715/27/9/2233
  • [8] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoît Perthame, Gaël Raoul, and Raphaël Voituriez, Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration, C. R. Math. Acad. Sci. Paris 350 (2012), no. 15-16, 761-766 (English, with English and French summaries). MR 2981349, https://doi.org/10.1016/j.crma.2012.09.010
  • [9] E. Bouin, C. Henderson, and L. Ryzhik,
    Super-linear spreading in local and non-local cane toads equations,
    Preprint, 2016.
  • [10] Emeric Bouin and Sepideh Mirrahimi, A Hamilton-Jacobi approach for a model of population structured by space and trait, Commun. Math. Sci. 13 (2015), no. 6, 1431-1452. MR 3351436, https://doi.org/10.4310/CMS.2015.v13.n6.a4
  • [11] Maury D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978), no. 5, 531-581. MR 0494541, https://doi.org/10.1002/cpa.3160310502
  • [12] Maury Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc. 44 (1983), no. 285, iv+190. MR 705746, https://doi.org/10.1090/memo/0285
  • [13] E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), no. 4, 327-338. MR 855753, https://doi.org/10.1007/BF00251802
  • [14] Ming Fang and Ofer Zeitouni, Branching random walks in time inhomogeneous environments, Electron. J. Probab. 17 (2012), no. 67, 18. MR 2968674, https://doi.org/10.1214/EJP.v17-2253
  • [15] Ming Fang and Ofer Zeitouni, Slowdown for time inhomogeneous branching Brownian motion, J. Stat. Phys. 149 (2012), no. 1, 1-9. MR 2981635, https://doi.org/10.1007/s10955-012-0581-z
  • [16] A. Fannjiang, A. Kiselev, and L. Ryzhik, Quenching of reaction by cellular flows, Geom. Funct. Anal. 16 (2006), no. 1, 40-69. MR 2221252, https://doi.org/10.1007/s00039-006-0554-y
  • [17] Grégory Faye and Matt Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach, J. Differential Equations 258 (2015), no. 7, 2257-2289. MR 3306338, https://doi.org/10.1016/j.jde.2014.12.006
  • [18] R. Fisher,
    The wave of advance of advantageous genes,
    Ann. Eugenics 7:355-369, 1937.
  • [19] François Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity, J. Math. Pures Appl. (9) 89 (2008), no. 4, 355-399 (English, with English and French summaries). MR 2401143, https://doi.org/10.1016/j.matpur.2007.12.005
  • [20] François Hamel, James Nolen, Jean-Michel Roquejoffre, and Lenya Ryzhik, A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Netw. Heterog. Media 8 (2013), no. 1, 275-289. MR 3043938, https://doi.org/10.3934/nhm.2013.8.275
  • [21] François Hamel, James Nolen, Jean-Michel Roquejoffre, and Lenya Ryzhik, The logarithmic delay of KPP fronts in a periodic medium, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 3, 465-505. MR 3463416, https://doi.org/10.4171/JEMS/595
  • [22] François Hamel and Lenya Ryzhik, On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds, Nonlinearity 27 (2014), no. 11, 2735-2753. MR 3274582, https://doi.org/10.1088/0951-7715/27/11/2735
  • [23] A. Kolmogorov, I. Petrovskii, and N. Piskunov,
    Étude de l'équation de la chaleurde matière et son application à un problème biologique,
    Bull. Moskov. Gos. Univ. Mat. Mekh. 1:1-25, 1937.
    See [30] pp. 105-130 for an English translation.
  • [24] Ka-Sing Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differential Equations 59 (1985), no. 1, 44-70. MR 803086, https://doi.org/10.1016/0022-0396(85)90137-8
  • [25] Pascal Maillard and Ofer Zeitouni, Slowdown in branching Brownian motion with inhomogeneous variance, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 3, 1144-1160 (English, with English and French summaries). MR 3531703, https://doi.org/10.1214/15-AIHP675
  • [26] Grégoire Nadin, Benoît Perthame, and Min Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris 349 (2011), no. 9-10, 553-557 (English, with English and French summaries). MR 2802923, https://doi.org/10.1016/j.crma.2011.03.008
  • [27] G. Nadin, L. Rossi, L. Ryzhik, and B. Perthame, Wave-like solutions for nonlocal reaction-diffusion equations: a toy model, Math. Model. Nat. Phenom. 8 (2013), no. 3, 33-41. MR 3103010, https://doi.org/10.1051/mmnp/20138304
  • [28] James Nolen, Jean-Michel Roquejoffre, and Lenya Ryzhik, Power-like delay in time inhomogeneous Fisher-KPP equations, Comm. Partial Differential Equations 40 (2015), no. 3, 475-505. MR 3285242, https://doi.org/10.1080/03605302.2014.972744
  • [29] J. R. Norris, Long-time behaviour of heat flow: global estimates and exact asymptotics, Arch. Rational Mech. Anal. 140 (1997), no. 2, 161-195. MR 1482931, https://doi.org/10.1007/s002050050063
  • [30] Pierre Pelcé (ed.), Dynamics of curved fronts, Perspectives in Physics, Academic Press, Inc., Boston, MA, 1988. MR 986791
  • [31] B. L. Phillips, G. P. Brown, J. K. Webb, and R. Shine,
    Invasion and the evolution of speed in toads,
    Nature, 439(7078):803-803, 2006.
  • [32] Matthew I. Roberts, A simple path to asymptotics for the frontier of a branching Brownian motion, Ann. Probab. 41 (2013), no. 5, 3518-3541. MR 3127890, https://doi.org/10.1214/12-AOP753
  • [33] B. Shabani.
    Ph.D. thesis, Stanford University.
    in preparation.
  • [34] C. D. Thomas, E. J. Bodsworth, R. J. Wilson, A. D. Simmons, Z. G. Davis, M. Musche, and L. Conradt,
    Ecological and evolutionary processes at expanding range margins,
    Nature 411:577-581, 2001.
  • [35] Olga Turanova, On a model of a population with variable motility, Math. Models Methods Appl. Sci. 25 (2015), no. 10, 1961-2014. MR 3358450, https://doi.org/10.1142/S0218202515500505
  • [36] Kōhei Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ. 18 (1978), no. 3, 453-508. MR 509494, https://doi.org/10.1215/kjm/1250522506
  • [37] S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math. 20 (1967), 431-455. MR 0208191, https://doi.org/10.1002/cpa.3160200210

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Additional Information

Emeric Bouin
Affiliation: CEREMADE - Université Paris-Dauphine, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
Email: bouin@ceremade.dauphine.fr

Christopher Henderson
Affiliation: Department of Mathematics, the University of Chicago, Chicago, Illinois 60637
Email: henderson@math.uchicago.edu

Lenya Ryzhik
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: ryzhik@math.stanford.edu

DOI: https://doi.org/10.1090/qam/1470
Received by editor(s): October 12, 2016
Received by editor(s) in revised form: March 17, 2017
Published electronically: April 18, 2017
Article copyright: © Copyright 2017 Brown University

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