Loading [Contrib]/a11y/accessibility-menu.js
Search Advanced

Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 275-289. doi: 10.3934/nhm.2013.8.275
Previous Article Next Article

A short proof of the logarithmic Bramson correction in Fisher-KPP equations

  • 1.
    Université d'Aix-Marseille, LATP, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13
  • 2.
    Department of Mathematics, Duke University, Box 90320, Durham, NC, 27708-0320
  • 3.
    Institut de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 4
  • 4.
    Department of Mathematics, Stanford University, Stanford, CA 94305
  • Received: 01 May 2012 Revised: 01 November 2012

  • Primary: 35K57, 35C07; Secondary: 35B40.

  • In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions $u(t,x)$ of Fisher-KPP reaction-diffusion equations in $\mathbb{R}$, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of $u$ to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions $u$ along their level sets to the profile of the minimal travelling front.

    Citation: François Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik. A short proof of the logarithmic Bramson correction in Fisher-KPP equations[J]. Networks and Heterogeneous Media, 2013, 8(1): 275-289. doi: 10.3934/nhm.2013.8.275

    Related Papers:

    [1] François Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik . A short proof of the logarithmic Bramson correction in Fisher-KPP equations. Networks and Heterogeneous Media, 2013, 8(1): 275-289. doi: 10.3934/nhm.2013.8.275
    [2] John R. King . Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity. Networks and Heterogeneous Media, 2013, 8(1): 343-378. doi: 10.3934/nhm.2013.8.343
    [3] Benjamin Contri . Fisher-KPP equations and applications to a model in medical sciences. Networks and Heterogeneous Media, 2018, 13(1): 119-153. doi: 10.3934/nhm.2018006
    [4] Henri Berestycki, Guillemette Chapuisat . Traveling fronts guided by the environment for reaction-diffusion equations. Networks and Heterogeneous Media, 2013, 8(1): 79-114. doi: 10.3934/nhm.2013.8.79
    [5] James Nolen . A central limit theorem for pulled fronts in a random medium. Networks and Heterogeneous Media, 2011, 6(2): 167-194. doi: 10.3934/nhm.2011.6.167
    [6] Maria Colombo, Gianluca Crippa, Stefano Spirito . Logarithmic estimates for continuity equations. Networks and Heterogeneous Media, 2016, 11(2): 301-311. doi: 10.3934/nhm.2016.11.301
    [7] Tong Li . Qualitative analysis of some PDE models of traffic flow. Networks and Heterogeneous Media, 2013, 8(3): 773-781. doi: 10.3934/nhm.2013.8.773
    [8] Andrea Corli, Lorenzo di Ruvo, Luisa Malaguti, Massimiliano D. Rosini . Traveling waves for degenerate diffusive equations on networks. Networks and Heterogeneous Media, 2017, 12(3): 339-370. doi: 10.3934/nhm.2017015
    [9] Hantaek Bae . On the local and global existence of the Hall equations with fractional Laplacian and related equations. Networks and Heterogeneous Media, 2022, 17(4): 645-663. doi: 10.3934/nhm.2022021
    [10] Gary Bunting, Yihong Du, Krzysztof Krakowski . Spreading speed revisited: Analysis of a free boundary model. Networks and Heterogeneous Media, 2012, 7(4): 583-603. doi: 10.3934/nhm.2012.7.583
  • In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions $u(t,x)$ of Fisher-KPP reaction-diffusion equations in $\mathbb{R}$, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of $u$ to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions $u$ along their level sets to the profile of the minimal travelling front.


    [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5
    [2] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations, in Honor of H. Brezis", Amer. Math. Soc., Contemp. Math., (2007), 101-123. doi: 10.1090/conm/446/08627
    [3] M. D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math., 31 (1978), 531-581. doi: 10.1002/cpa.3160310502
    [4] M. D. Bramson, "Convergence of Solutions of the Kolmogorov Equation to Travelling Waves," Mem. Amer. Math. Soc., 1983.
    [5] E. Brunet and B. Derrida, Shift in the velocity of a front due to a cutoff, Phys. Rev. E, 56 (1997), 2597-2604. doi: 10.1103/PhysRevE.56.2597
    [6] C. Cuesta and J. King, Front propagation in a heterogeneous Fisher equation: The homogeneous case is non-generic, Quart. J. Mech. Appl. Math., 63 (2010), 521-571. doi: 10.1093/qjmam/hbq017
    [7] J.-P. Eckmann and T. Gallay, Front solutions for the Ginzburg-Landau equation, Comm. Math. Phys., 152 (1993), 221-248. doi: 10.1007/BF02098298
    [8] U. Ebert and W. van Saarloos, Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Phys. D, 146 (2000), 1-99. doi: 10.1016/S0167-2789(00)00068-3
    [9] U. Ebert, W. van Saarloos and B. Peletier, Universal algebraic convergence in time of pulled fronts: The common mechanism for difference-differential and partial differential equations, European J. Appl. Math., 13 (2002), 53-66. doi: 10.1017/S0956792501004673
    [10] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, Springer Verlag, 1979.
    [11] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. doi: 10.1111/j.1469-1809.1937.tb02153.x
    [12] preprint.
    [13] K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovskii and Piskunov, J. Diff. Eqs., 59 (1985), 44-70. doi: 10.1016/0022-0396(85)90137-8
    [14] A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, 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. Inter. A, 1 (1937), 1-26.
    [15] J. D. Murray, "Mathematical Biology," Springer-Verlag, 2003. doi: 10.1007/b98869
    [16] F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Royal Soc. Edinburgh A, 80 (1978), 213-234. doi: 10.1017/S0308210500010258
    [17] D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Diff. Eqs., 25 (1977), 130-144. doi: 10.1016/0022-0396(77)90185-1
    [18] K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.
    [19] J. Xin, "An Introduction to Fronts in Random Media," Surveys and Tutorials in the Applied Mathematical Sciences, Springer, New York, 2009. doi: 10.1007/978-0-387-87683-2

  • This article has been cited by:

    1. Thomas Giletti, François Hamel, Sharp thresholds between finite spread and uniform convergence for a reaction–diffusion equation with oscillating initial data, 2017, 262, 00220396, 1461, 10.1016/j.jde.2016.10.014
    2. O. Bonnefon, J. Garnier, F. Hamel, L. Roques, Elaine Crooks, Fordyce Davidson, Bogdan Kazmierczak, Gregoire Nadin, Je-Chiang Tsai, Inside Dynamics of Delayed Traveling Waves, 2013, 8, 0973-5348, 42, 10.1051/mmnp/20138305
    3. Arnaud Ducrot, On the large time behaviour of the multi-dimensional Fisher–KPP equation with compactly supported initial data, 2015, 28, 0951-7715, 1043, 10.1088/0951-7715/28/4/1043
    4. Grégory Faye, Matt Holzer, Arnd Scheel, Linear spreading speeds from nonlinear resonant interaction, 2017, 30, 0951-7715, 2403, 10.1088/1361-6544/aa6c74
    5. Julien Berestycki, Éric Brunet, Simon C. Harris, Piotr Miłoś, Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift, 2017, 273, 00221236, 2107, 10.1016/j.jfa.2017.06.006
    6. Julien Berestycki, Éric Brunet, Cole Graham, Leonid Mytnik, Jean-Michel Roquejoffre, Lenya Ryzhik, The distance between the two BBM leaders, 2022, 35, 0951-7715, 1558, 10.1088/1361-6544/ac4a8e
    7. Rafael D Benguria, Abraham Solar, An estimation of level sets for non local KPP equations with delay, 2019, 32, 0951-7715, 777, 10.1088/1361-6544/aaedd7
    8. Jane Allwright, Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain, 2022, 65, 0013-0915, 53, 10.1017/S0013091521000754
    9. Julien Berestycki, Éric Brunet, Bernard Derrida, Exact solution and precise asymptotics of a Fisher–KPP type front, 2018, 51, 1751-8113, 035204, 10.1088/1751-8121/aa899f
    10. James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik, Refined long-time asymptotics for Fisher–KPP fronts, 2019, 21, 0219-1997, 1850072, 10.1142/S0219199718500724
    11. Julien Berestycki, Nathanaël Berestycki, Jason Schweinsberg, Critical branching Brownian motion with absorption: Particle configurations, 2015, 51, 0246-0203, 10.1214/14-AIHP613
    12. Bernard Derrida, Cross-Overs of Bramson’s Shift at the Transition Between Pulled and Pushed Fronts, 2023, 190, 0022-4715, 10.1007/s10955-023-03077-8
    13. Emeric Bouin, Christopher Henderson, Lenya Ryzhik, The Bramson delay in the non-local Fisher-KPP equation, 2020, 37, 0294-1449, 51, 10.1016/j.anihpc.2019.07.001
    14. Leonid Mytnik, Jean-Michel Roquejoffre, Lenya Ryzhik, Fisher-KPP equation with small data and the extremal process of branching Brownian motion, 2022, 396, 00018708, 108106, 10.1016/j.aim.2021.108106
    15. Thomas Giletti, Monostable pulled fronts and logarithmic drifts, 2022, 29, 1021-9722, 10.1007/s00030-022-00766-3
    16. Julien Berestycki, Nathanaël Berestycki, Jason Schweinsberg, Critical branching Brownian motion with absorption: survival probability, 2014, 160, 0178-8051, 489, 10.1007/s00440-013-0533-9
    17. Hong Gu, Fisher-KPP equation with advection on the half-line, 2016, 39, 01704214, 344, 10.1002/mma.3485
    18. Junfeng He, Yanxia Wu, Yaping Wu, Large time behavior of solutions for degenerate p-degree Fisher equation with algebraic decaying initial data, 2017, 448, 0022247X, 1, 10.1016/j.jmaa.2016.10.037
    19. Montie Avery, Arnd Scheel, Asymptotic Stability of Critical Pulled Fronts via Resolvent Expansions Near the Essential Spectrum, 2021, 53, 0036-1410, 2206, 10.1137/20M1343476
    20. Rui Peng, Chang-Hong Wu, Maolin Zhou, Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition, 2021, 38, 0294-1449, 507, 10.1016/j.anihpc.2020.07.006
    21. Emeric Bouin, Christopher Henderson, Lenya Ryzhik, The Bramson logarithmic delay in the cane toads equations, 2017, 75, 0033-569X, 599, 10.1090/qam/1470
    22. Zhiguo Wang, Qian Qin, Jianhua Wu, Spreading Speed and Profile for the Lotka–Volterra Competition Model with Two Free Boundaries, 2022, 1040-7294, 10.1007/s10884-022-10222-6
    23. Yihong Du, Bendong Lou, Rui Peng, Maolin Zhou, The Fisher-KPP equation over simple graphs: varied persistence states in river networks, 2020, 80, 0303-6812, 1559, 10.1007/s00285-020-01474-1
    24. Montie Avery, Arnd Scheel, Universal selection of pulled fronts, 2022, 2, 2692-3688, 172, 10.1090/cams/8
    25. H. Matano, P. Poláčik, Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities, 2020, 45, 0360-5302, 483, 10.1080/03605302.2019.1700273
    26. Weiwei Ding, Hiroshi Matano, Dynamics of Time-Periodic Reaction-Diffusion Equations with Front-Like Initial Data on $\mathbb{R}$, 2020, 52, 0036-1410, 2411, 10.1137/19M1268987
    27. Henri Berestycki, Grégoire Nadin, Asymptotic Spreading for General Heterogeneous Fisher-KPP Type Equations, 2022, 280, 0065-9266, 10.1090/memo/1381
    28. Vincent Calvez, Christopher Henderson, Sepideh Mirrahimi, Olga Turanova, Thierry Dumont, Non-local competition slows down front acceleration during dispersal evolution, 2022, 5, 2644-9463, 1, 10.5802/ahl.117
    29. Yihong Du, Fernando Quirós, Maolin Zhou, Logarithmic corrections in Fisher–KPP type porous medium equations, 2020, 136, 00217824, 415, 10.1016/j.matpur.2019.12.008
    30. Henri Berestycki, Anne-Charline Coulon, Jean-Michel Roquejoffre, Luca Rossi, Speed-up of reaction-diffusion fronts by a line of fast diffusion, 2014, 2266-0607, 1, 10.5802/slsedp.62
    31. Chang-Hong Wu, Dongyuan Xiao, Maolin Zhou, Sharp estimates for the spreading speeds of the Lotka-Volterra competition-diffusion system: The strong-weak type with pushed front, 2023, 172, 00217824, 236, 10.1016/j.matpur.2023.02.004
    32. Dongyuan Xiao, Ryunosuke Mori, Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers, 2021, 38, 0294-1449, 911, 10.1016/j.anihpc.2020.09.007
    33. Arnaud Ducrot, Spreading speed for a KPP type reaction-diffusion system with heat losses and fast decaying initial data, 2021, 270, 00220396, 217, 10.1016/j.jde.2020.07.044
    34. Montie Avery, Louis Garénaux, Spectral stability of the critical front in the extended Fisher-KPP equation, 2023, 74, 0044-2275, 10.1007/s00033-023-01960-8
    35. Éric Brunet, Bernard Derrida, An Exactly Solvable Travelling Wave Equation in the Fisher–KPP Class, 2015, 161, 0022-4715, 801, 10.1007/s10955-015-1350-6
    36. Ningkui Sun, Bendong Lou, Maolin Zhou, Fisher–KPP equation with free boundaries and time-periodic advections, 2017, 56, 0944-2669, 10.1007/s00526-017-1165-1
    37. Cole Graham, Precise asymptotics for Fisher–KPP fronts, 2019, 32, 0951-7715, 1967, 10.1088/1361-6544/aaffe8
    38. Montie Avery, Cedric Dedina, Aislinn Smith, Arnd Scheel, Instability in large bounded domains—branched versus unbranched resonances, 2021, 34, 0951-7715, 7916, 10.1088/1361-6544/ac2a15
    39. François Hamel, Luca Rossi, Spreading sets and one-dimensional symmetry for reaction-diffusion equations, 2022, 2266-0607, 1, 10.5802/slsedp.150
    40. Matthieu Alfaro, Dongyuan Xiao, Lotka–Volterra competition-diffusion system: the critical competition case, 2023, 0360-5302, 1, 10.1080/03605302.2023.2169936
    41. Matt Holzer, A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations, 2015, 36, 1078-0947, 2069, 10.3934/dcds.2016.36.2069
    42. François Hamel, Frithjof Lutscher, Mingmin Zhang, Propagation and blocking in a two-patch reaction-diffusion model, 2022, 168, 00217824, 213, 10.1016/j.matpur.2022.11.006
    43. Ningkui Sun, A time-periodic reaction–diffusion–advection equation with a free boundary and sign-changing coefficients, 2020, 51, 14681218, 102952, 10.1016/j.nonrwa.2019.06.002
    44. Alex D. O. Tisbury, David J. Needham, Alexandra Tzella, The evolution of traveling waves in a KPP reaction–diffusion model with cut‐off reaction rate. II. Evolution of traveling waves, 2021, 146, 0022-2526, 330, 10.1111/sapm.12352
    45. Yonggang Zhao, Mingxin Wang, A Reaction–Diffusion–Advection Equation with Mixed and Free Boundary Conditions, 2018, 30, 1040-7294, 743, 10.1007/s10884-017-9571-9
    46. James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik, Power-Like Delay in Time Inhomogeneous Fisher-KPP Equations, 2015, 40, 0360-5302, 475, 10.1080/03605302.2014.972744
    47. François Hamel, Luca Rossi, Admissible speeds of transition fronts for nonautonomous monostable equations, 2015, 47, 0036-1410, 3342, 10.1137/140995519
    48. James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik, Convergence to a single wave in the Fisher-KPP equation, 2017, 38, 0252-9599, 629, 10.1007/s11401-017-1087-4
    49. Hong Gu, Bendong Lou, Spreading in advective environment modeled by a reaction diffusion equation with free boundaries, 2016, 260, 00220396, 3991, 10.1016/j.jde.2015.11.002
    50. François Hamel, Luca Rossi, Transition fronts for the Fisher-KPP equation, 2016, 368, 0002-9947, 8675, 10.1090/tran/6609
    51. Christopher Henderson, Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data, 2016, 29, 0951-7715, 3215, 10.1088/0951-7715/29/11/3215
    52. Yihong Du, Lei Wei, Ling Zhou, Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary, 2018, 30, 1040-7294, 1389, 10.1007/s10884-017-9614-2
    53. Julien Berestycki, Éric Brunet, Simon C. Harris, Matt Roberts, Vanishing Corrections for the Position in a Linear Model of FKPP Fronts, 2017, 349, 0010-3616, 857, 10.1007/s00220-016-2790-9
    54. Jens D. M. Rademacher, Lars Siemer, Domain Wall Motion in Axially Symmetric Spintronic Nanowires, 2021, 20, 1536-0040, 2204, 10.1137/20M1382696
    55. Anton Bovier, Lisa Hartung, Branching Brownian Motion with Self-Repulsion, 2023, 24, 1424-0637, 931, 10.1007/s00023-022-01223-8
    56. Emeric Bouin, Christopher Henderson, Lenya Ryzhik, Super-linear spreading in local and non-local cane toads equations, 2017, 108, 00217824, 724, 10.1016/j.matpur.2017.05.015
    57. Montie Avery, Arnd Scheel, Sharp Decay Rates for Localized Perturbations to the Critical Front in the Ginzburg–Landau Equation, 2021, 1040-7294, 10.1007/s10884-021-10093-3
    58. Yihong Du, Hiroshi Matsuzawa, Maolin Zhou, Sharp Estimate of the Spreading Speed Determined by Nonlinear Free Boundary Problems, 2014, 46, 0036-1410, 375, 10.1137/130908063
    59. Pasha Tkachov, On stability of traveling wave solutions for integro-differential equations related to branching Markov processes, 2020, 26, 1350-7265, 10.3150/19-BEJ1159
    60. Jean-Michel Roquejoffre, Violaine Roussier-Michon, Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations, 2018, 31, 0951-7715, 3284, 10.1088/1361-6544/aaba3b
    61. Emeric Bouin, Christopher Henderson, The Bramson delay in a Fisher–KPP equation with log-singular nonlinearity, 2021, 213, 0362546X, 112508, 10.1016/j.na.2021.112508
    62. Hong Gu, Bendong Lou, Maolin Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, 2015, 269, 00221236, 1714, 10.1016/j.jfa.2015.07.002
    63. François Hamel, Frithjof Lutscher, Mingmin Zhang, Propagation Phenomena in Periodic Patchy Landscapes with Interface Conditions, 2022, 1040-7294, 10.1007/s10884-022-10134-5
    64. Xinfu Chen, Je-Chiang Tsai, Yaping Wu, Longtime Behavior of Solutions of a SIS Epidemiological Model, 2017, 49, 0036-1410, 3925, 10.1137/16M1108741
    65. Julien Berestycki, Éric Brunet, Sarah Penington, Global existence for a free boundary problem of Fisher–KPP type, 2019, 32, 0951-7715, 3912, 10.1088/1361-6544/ab25af
    66. Xinfu Chen, Junfeng He, Xuefeng Wang, Asymptotic Propagation Speeds of the Fisher-KPP Equation with an Effective Boundary Condition on a Road, 2023, 247, 0003-9527, 10.1007/s00205-023-01858-9
    67. Yuan He, Guo Lin, Shuo Zhang, Spreading speeds in an asymptotic autonomous system with application to an epidemic model, 2024, 47, 0170-4214, 9621, 10.1002/mma.10086
    68. Montie Avery, Front Selection in Reaction–Diffusion Systems via Diffusive Normal Forms, 2024, 248, 0003-9527, 10.1007/s00205-024-01961-5
    69. Jing An, Christopher Henderson, Lenya Ryzhik, Quantitative Steepness, Semi-FKPP Reactions, and Pushmi-Pullyu Fronts, 2023, 247, 0003-9527, 10.1007/s00205-023-01924-2
    70. Éric Brunet, Ahead of the Fisher–KPP front, 2023, 36, 0951-7715, 5541, 10.1088/1361-6544/acf268
    71. Christophe Besse, Grégory Faye, Jean-Michel Roquejoffre, Mingmin Zhang, The logarithmic Bramson correction for Fisher-KPP equations on the lattice ℤ, 2023, 0002-9947, 10.1090/tran/9007
    72. Montie Avery, Matt Holzer, Arnd Scheel, Pushed-to-Pulled Front Transitions: Continuation, Speed Scalings, and Hidden Monotonicty, 2023, 33, 0938-8974, 10.1007/s00332-023-09957-3
    73. François Hamel, Mingmin Zhang, KPP transition fronts in a one-dimensional two-patch habitat, 2024, 1477-8599, 10.1093/imammb/dqae011
    74. Jingjing Li, Ningkui Sun, Dynamical behavior of solutions of a reaction–diffusion model in river network, 2024, 75, 14681218, 103989, 10.1016/j.nonrwa.2023.103989
    75. Ningkui Sun, , Dynamical behavior of solutions of a reaction–diffusion–advection model with a free boundary, 2024, 75, 0044-2275, 10.1007/s00033-023-02183-7
    76. Abraham Solar, Stability of Solutions to Functional KPP-Fisher Equations, 2023, 1040-7294, 10.1007/s10884-023-10297-9
    77. Yihong Du, Wenjie Ni, The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, part 2: Sharp estimates, 2024, 287, 00221236, 110649, 10.1016/j.jfa.2024.110649
    78. Xueqi Fan, Ningkui Sun, Di Zhang, Propagation dynamics of a free boundary problem in advective environments, 2024, 153, 08939659, 109082, 10.1016/j.aml.2024.109082
    79. Jing An, Christopher Henderson, Lenya Ryzhik, Voting models and semilinear parabolic equations, 2023, 36, 0951-7715, 6104, 10.1088/1361-6544/ad001c
    80. Ge Tian, Guo‐Bao Zhang, Propagation dynamics of a discrete diffusive equation with non‐local delay, 2023, 46, 0170-4214, 14072, 10.1002/mma.9305
    81. Qing Li, Xinfu Chen, King-Yeung Lam, Yaping Wu, Propagation phenomena for a nonlocal reaction-diffusion model with bounded phenotypic traits, 2024, 411, 00220396, 794, 10.1016/j.jde.2024.08.032
    82. Thomas Giletti, Propagating fronts and terraces in multistable reaction-diffusion equations, 2024, 2118-9366, 1, 10.5802/jedp.677
    83. Nathanaël Boutillon, Large deviations and the emergence of a logarithmic delay in a nonlocal linearised Fisher–KPP equation, 2024, 240, 0362546X, 113465, 10.1016/j.na.2023.113465
    84. Jean-Michel Roquejoffre, 2024, Chapter 5, 978-3-031-77771-4, 123, 10.1007/978-3-031-77772-1_5
    85. Jean-Michel Roquejoffre, 2024, Chapter 4, 978-3-031-77771-4, 87, 10.1007/978-3-031-77772-1_4
    86. Léo Girardin, Persistence, Extinction, and Spreading Properties of Noncooperative Fisher–KPP Systems in Space-Time Periodic Media, 2025, 57, 0036-1410, 233, 10.1137/23M1576086
    87. Jingjing Cai, Shizhao Ma, Dongyuan Xiao, Lotka-Volterra competition system with critical competition case in high-dimensional space, 2025, 0, 1531-3492, 0, 10.3934/dcdsb.2025031
  • Reader Comments
  • © 2013 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Networks and Heterogeneous Media
1.2 1.8

Metrics

Article views(5270) PDF downloads(164) Cited by(87)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog