Feuilleter
no. 5
Sharp exponent of acceleration in general non-local equations with a weak Allee effect
[Exposant d’accéleration dans les équations non locales générales avec un faible effet Allee]
Emeric Bouin1 ; Jérôme Coville2 ; Guillaume Legendre1
1 CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Université PSL, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16 (France)
2 UR 546 Biostatistique et Processus Spatiaux, INRAE – Centre de Recherche PACA, 228 route de l’Aérodrome, CS 40509, Domaine Saint Paul – Site Agroparc, 84914 Avignon cedex 9 (France)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 5, pp. 1475-1540

We study an acceleration phenomenon arising in monostable integro-differential equations with a weak Allee effect. Previous works have shown its occurrence and have given correct upper bounds on the rate of expansion in some particular cases, but precise lower bounds were still missing. In this paper, we provide a sharp lower bound for this acceleration rate, which is valid for a large class of dispersion operators. Our results manage to cover fractional Laplace operators and standard convolutions in a unified way, which is new in the literature. An important result of the paper is a general flattening estimate of independent interest: this phenomenon appears regularly in acceleration situations, but getting quantitative estimates is in general an open question. With this estimate at hand, we construct a subsolution that captures the expected behaviour of the accelerating solution (rates of expansion and flattening) and identifies several regimes that appear in the dynamics depending on the parameters of the problem.

Nous étudions un phénomène d’accélération survenant dans les équations intégro-différentielles monostables avec un faible effet Allee. De précédents travaux avaient établi son existence et fourni des bornes supérieures correctes sur le taux d’expansion pour des classes particulières d’opérateur de diffusion, mais des bornes inférieures précises faisaient encore défaut. Nous fournissons ici une borne inférieure précise pour ce taux, valable pour une large classe d’opérateurs de dispersion. Nos résultats sont génériques dans la classe d’opérateurs considérée et couvrent de manière unifiée aussi bien les opérateurs de type laplacien fractionnaire que les opérateurs de convolution standards, ce qui est une nouveauté. Un résultat important de l’article, et d’intérêt indépendant, est une estimation générale de l’aplatissement des solutions. Ce phénomène apparaît dans les situations d’accélération, mais l’obtention d’estimations quantitatives reste en général une question ouverte. Cette estimation est fondamentale dans notre construction d’une sous-solution qui capture la comportement attendu de la solution accélérée (taux d’expansion et d’aplatissement) en identifiant plusieurs régimes qui apparaissent dans la dynamique en fonction des paramètres du problème.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1838
Classification : 35B40, 35S10, 60J60, 60K50
Keywords: semilinear evolution problem, generic non-local dispersion operators, fractional Laplace operator, convolution operator, positive solution, acceleration, level lines
Mots-clés : problème d’évolution semi-linéaire, opérateurs non locaux génériques de dispersion, laplacien fractionaire, opérateur de convolution, solution positive, accélération, lignes de niveau

Emeric Bouin 1 ; Jérôme Coville 2 ; Guillaume Legendre 1

1 CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Université PSL, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16 (France)
2 UR 546 Biostatistique et Processus Spatiaux, INRAE – Centre de Recherche PACA, 228 route de l’Aérodrome, CS 40509, Domaine Saint Paul – Site Agroparc, 84914 Avignon cedex 9 (France)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AFST_2025_6_34_5_1475_0,
     author = {Emeric Bouin and J\'er\^ome Coville and Guillaume Legendre},
     title = {Sharp exponent of acceleration in general non-local equations with a weak {Allee} effect},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1475--1540},
     year = {2025},
     publisher = {Universit\'e de Toulouse, Toulouse},
     volume = {Ser. 6, 34},
     number = {5},
     doi = {10.5802/afst.1838},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1838/}
}
TY  - JOUR
AU  - Emeric Bouin
AU  - Jérôme Coville
AU  - Guillaume Legendre
TI  - Sharp exponent of acceleration in general non-local equations with a weak Allee effect
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2025
SP  - 1475
EP  - 1540
VL  - 34
IS  - 5
PB  - Université de Toulouse, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1838/
DO  - 10.5802/afst.1838
LA  - en
ID  - AFST_2025_6_34_5_1475_0
ER  - 
%0 Journal Article
%A Emeric Bouin
%A Jérôme Coville
%A Guillaume Legendre
%T Sharp exponent of acceleration in general non-local equations with a weak Allee effect
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2025
%P 1475-1540
%V 34
%N 5
%I Université de Toulouse, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1838/
%R 10.5802/afst.1838
%G en
%F AFST_2025_6_34_5_1475_0
Emeric Bouin; Jérôme Coville; Guillaume Legendre. Sharp exponent of acceleration in general non-local equations with a weak Allee effect. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 5, pp. 1475-1540. doi: 10.5802/afst.1838

[1] Franz Achleitner; Christian Kuehn Traveling waves for a bistable equation with nonlocal diffusion, Adv. Differ. Equ., Volume 20 (2015) no. 9-10, pp. 887-936 | DOI | Zbl | MR

[2] Giovanni Alberti; Giovanni Bellettini A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., Volume 310 (1998) no. 3, pp. 527-560 | DOI | Zbl | MR

[3] Matthieu Alfaro Fujita blow up phenomena and hair trigger effect: the role of dispersal tails, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 34 (2017) no. 5, pp. 1309-1327 | DOI | Zbl | Numdam | MR

[4] Matthieu Alfaro Slowing Allee effect versus accelerating heavy tails in monostable reaction diffusion equations, Nonlinearity, Volume 30 (2017) no. 2, pp. 687-702 | DOI | Zbl | MR

[5] Matthieu Alfaro; Jérôme Coville Propagation phenomena in monostable integro-differential equations: acceleration or not?, J. Differ. Equations, Volume 263 (2017) no. 9, pp. 5727-5758 | DOI | Zbl | MR

[6] Matthieu Alfaro; Thomas Giletti Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions (2017) | arXiv

[7] Matthieu Alfaro; Thomas Giletti When fast diffusion and reactive growth both induce accelerating invasions, Commun. Pure Appl. Anal., Volume 18 (2019) no. 6, pp. 3011-3034 | DOI | Zbl | MR

[8] Warder C. Allee The social life of animals, W. W. Norton & Company, Inc., 1938 | DOI

[9] David Applebaum Lévy processes and stochastic calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, 2009 | DOI | Zbl | MR

[10] Donald G. Aronson; Hans F. Weinberger Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., Volume 30 (1978) no. 1, pp. 33-76 | DOI | Zbl

[11] Uri M. Ascher; Steven J. Ruuth; Raymond J. Spiteri Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., Volume 25 (1997) no. 2-3, pp. 151-167 | DOI | Zbl | MR

[12] Martin T. Barlow; Richard F. Bass; Zhen-Qing Chen; Moritz Kassmann Non-local Dirichlet forms and symmetric jump processes, Trans. Am. Math. Soc., Volume 361 (2009) no. 4, pp. 1963-1999 | Zbl | MR | DOI

[13] Luděk Berec; Elena Angulo; Franck Courchamp Multiple Allee effects and population management, Trends Ecol. Evol., Volume 22 (2007) no. 4, pp. 185-191 | DOI

[14] Jean Bertoin Lévy processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, 1998 | Zbl | MR

[15] Robert M. Blumenthal; Ronald K. Getoor Some theorems on stable processes, Trans. Am. Math. Soc., Volume 95 (1960) no. 2, pp. 263-273 | DOI | Zbl | MR

[16] Krzysztof Bogdan; Tomasz Grzywny; Michał Ryznar Density and tails of unimodal convolution semigroups, J. Funct. Anal., Volume 266 (2014) no. 6, pp. 3543-3571 | DOI | Zbl | MR

[17] Krzysztof Bogdan; Paweł Sztonyk; Victoria Knopova Heat kernel of anisotropic nonlocal operators, Doc. Math., Volume 25 (2020), pp. 1-54 | DOI | Zbl | MR

[18] Emeric Bouin; Jérôme Coville; Guillaume Legendre Acceleration in integro-differential combustion equations (2021) | arXiv

[19] Emeric Bouin; Jérôme Coville; Guillaume Legendre Sharp exponent of acceleration in integro-differential equations with weak Allee effect (2021) (https://hal.science/hal-03452141v1)

[20] Emeric Bouin; Jérôme Coville; Guillaume Legendre Sharp exponent of acceleration in integro-differential equations with weak Allee effect (2021) | arXiv

[21] Emeric Bouin; Jimmy Garnier; Christopher Henderson; Florian Patout Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels, SIAM J. Math. Anal., Volume 50 (2018) no. 3, pp. 3365-3394 | DOI | Zbl | MR

[22] Julien Brasseur; Jérôme Coville Propagation phenomena with nonlocal diffusion in presence of an obstacle, J. Dyn. Differ. Equations, Volume 35 (2023) no. 1, pp. 237-301 | DOI | Zbl | MR

[23] Xavier Cabré; Anne-Charline Coulon; Jean-Michel Roquejoffre Propagation in Fisher–KPP type equations with fractional diffusion in periodic media, C. R. Math., Volume 350 (2012) no. 19-20, pp. 885-890 | DOI | Zbl | Numdam

[24] Xavier Cabré; Jean-Michel Roquejoffre Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire, C. R. Math., Volume 347 (2009) no. 23-24, pp. 1361-1366 | Numdam | DOI | Zbl

[25] Xavier Cabré; Jean-Michel Roquejoffre The influence of fractional diffusion in Fisher-KPP equations, Commun. Math. Phys., Volume 320 (2013) no. 3, pp. 679-722 | DOI | Zbl | MR

[26] Jack Carr; Adam Chmaj Uniqueness of travelling waves for nonlocal monostable equations, Proc. Am. Math. Soc., Volume 132 (2004) no. 8, pp. 2433-2439 | DOI | Zbl | MR

[27] Emmanuel Chasseigne; Manuela Chaves; Julio D. Rossi Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., Volume 86 (2006) no. 3, pp. 271-291 | DOI | Zbl | MR

[28] Xinfu Chen Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differ. Equ., Volume 2 (1997) no. 1, pp. 125-160 | DOI | Zbl | MR

[29] Zhen-Qing Chen; Panki Kim; Takashi Kumagai Weighted Poincaré inequality and heat kernel estimates for finite range jump processes, Math. Ann., Volume 342 (2008) no. 4, pp. 833-883 | DOI | Zbl | MR

[30] Zhen-Qing Chen; Panki Kim; Takashi Kumagai Global heat kernel estimates for symmetric jump processes, Trans. Am. Math. Soc., Volume 363 (2011) no. 9, pp. 5021-5055 | DOI | Zbl | MR

[31] Adam Chmaj Existence of traveling waves in the fractional bistable equation, Arch. Math., Volume 100 (2013) no. 5, pp. 473-480 | DOI | Zbl

[32] Anne-Charline Coulon; Jean-Michel Roquejoffre Transition between linear and exponential propagation in Fisher-KPP type reaction-diffusion equations, Commun. Math. Phys., Volume 37 (2012) no. 11, pp. 2029-2049 | Zbl | DOI

[33] Franck Courchamp; Luděk Berec; Joanna Gascoigne Allee effects in ecology and conservation, Oxford University Press, 2008 | DOI

[34] Jérôme Coville On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl. (4), Volume 185 (2006) no. 3, pp. 461-485 | DOI | Zbl | MR

[35] Jérôme Coville Travelling fronts in asymmetric nonlocal reaction diffusion equations: the bistable and ignition cases (2007) (preprint, https://hal.science/hal-00696208v1)

[36] Jérôme Coville; Juan Dávila; Salomé Martínez Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differ. Equations, Volume 244 (2008) no. 12, pp. 3080-3118 | DOI | Zbl | MR

[37] Jérôme Coville; Louis Dupaigne Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., Theory Methods Appl., Volume 60 (2005) no. 5, pp. 797-819 | DOI | Zbl | MR

[38] Jérôme Coville; Louis Dupaigne On a non-local equation arising in population dynamics, Proc. R. Soc. Edinb., Sect. A, Math., Volume 137 (2007) no. 4, pp. 727-755 | DOI | Zbl | MR

[39] Jérôme Coville; Changfeng Gui; Mingfeng Zhao Propagation acceleration in reaction diffusion equations with anomalous diffusions, Nonlinearity, Volume 34 (2021) no. 3, pp. 1544-1576 | DOI | Zbl | MR

[40] Wojciech Cygan; Tomasz Grzywny; Bartosz Trojan Asymptotic behavior of densities of unimodal convolution semigroups, Trans. Am. Math. Soc., Volume 369 (2017) no. 8, pp. 5623-5644 | DOI | Zbl | MR

[41] Brian Dennis Allee effects: population growth, critical density, and the chance of extinction, Nat. Resour. Model., Volume 3 (1989) no. 4, pp. 481-538 | Zbl | DOI | MR

[42] Serena Dipierro; Edoardo Proietti Lippi; Enrico Valdinoci The role of Allee effects for Gaussian and Lévy dispersals in an environmental niche, J. Math. Biol., Volume 89 (2024), 19, 39 pages | DOI | Zbl | MR

[43] Siwei Duo; Hans Werner van Wyk; Yanzhi Zhang A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Comput. Phys., Volume 355 (2018), pp. 233-252 | MR | DOI | Zbl

[44] Jian Fang; Xiao-Qiang Zhao Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc., Volume 17 (2015) no. 9, pp. 2243-2288 | DOI | Zbl | MR

[45] Ronald A. Fisher The wave of advance of advantageous genes, Ann. Eugenics, Volume 7 (1937) no. 4, pp. 335-369 | DOI | Zbl

[46] Jimmy Garnier Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., Volume 43 (2011) no. 4, pp. 1955-1974 | DOI | Zbl | MR

[47] Tomasz Grzywny; Michał Ryznar; Bartosz Trojan Asymptotic behaviour and estimates of slowly varying convolution semigroups, Int. Math. Res. Not., Volume 2019 (2019) no. 23, pp. 7193-7258 | DOI | Zbl | MR

[48] Changfeng Gui; Tingting Huan Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, Calc. Var. Partial Differ. Equ., Volume 54 (2015) no. 1, pp. 251-273 | DOI | Zbl | MR

[49] François Hamel; Lionel Roques Fast propagation for KPP equations with slowly decaying initial conditions, J. Differ. Equations, Volume 249 (2010) no. 7, pp. 1726-1745 | DOI | Zbl | MR

[50] Yanghong Huang; Adam Oberman Numerical methods for the fractional Laplacian: a finite difference-quadrature approach, SIAM J. Numer. Anal., Volume 52 (2014) no. 6, pp. 3056-3084 | DOI | Zbl | MR

[51] Kamil Kaleta; Paweł Sztonyk Spatial asymptotics at infinity for heat kernels of integro-differential operators, Trans. Am. Math. Soc., Volume 371 (2019) no. 9, pp. 6627-6663 | DOI | Zbl | MR

[52] John R. King; Philip M. McCabe On the Fisher-KPP equation with fast nonlinear diffusion, Proc. R. Soc. Lond., Ser. A, Volume 459 (2003) no. 2038, pp. 2529-2546 | DOI | Zbl | MR

[53] Victoria Knopova; Alexei Kulik; René L. Schilling Construction and heat kernel estimates of general stable-like Markov processes, Diss. Math., Volume 569 (2021), pp. 1-86 | DOI | Zbl | MR

[54] Andreĭ N. Kolmogorov; Ivan G. Petrovskiĭ; Nikolaĭ S. Piscounov 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., Sect. A: Math. et Mécan., Volume 4 (1937) no. 6, pp. 1-25 | Zbl

[55] Vassili Kolokoltsov Symmetric stable laws and stable-like jump-diffusions, Proc. Lond. Math. Soc. (3), Volume 80 (2000) no. 3, pp. 725-768 | DOI | Zbl | MR

[56] Frithjof Lutscher; Elizaveta Pachepsky; Marc A. Lewis The effect of dispersal patterns on stream populations, SIAM Rev., Volume 47 (2005) no. 4, pp. 749-772 | DOI | Zbl | MR

[57] Jan Medlock; Mark Kot Spreading disease: integro-differential equations old and new, Math. Biosci., Volume 184 (2003) no. 2, pp. 201-222 | DOI | Zbl | MR

[58] Sylvie Méléard; Sepideh Mirrahimi Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity, Commun. Partial Differ. Equations, Volume 40 (2015) no. 5, pp. 957-993 | DOI | Zbl | MR

[59] Victor Minden; Lexing Ying A simple solver for the fractional Laplacian in multiple dimensions, SIAM J. Sci. Comput., Volume 42 (2020) no. 2, p. A878-A900 | DOI | Zbl | MR

[60] Ran Nathan; Etienne Klein; Juan J. Robledo-Arnuncio; Eloy Revilla Dispersal kernels: review, Dispersal ecology and evolution, Oxford University Press, 2012, pp. 187-210 | DOI

[61] Andrew P. Nield; Ran Nathan; Neal J. Enright; Philip G. Ladd; George L. W. Perry The spatial complexity of seed movement: animal-generated seed dispersal patterns in fragmented landscapes revealed by animal movement models, J. Ecol., Volume 108 (2020) no. 2, pp. 687-701 | DOI

[62] Amnon Pazy Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer, 1983 | DOI | Zbl | MR

[63] Sergei Petrovskii; Alla Mashanova; Vincent A. A. Jansen Variation in individual walking behavior creates the impression of a Lévy flight, Proc. Natl. Acad. Sci. USA, Volume 108 (2011) no. 21, pp. 8704-8707 | DOI

[64] George Pólya On the zeros of an integral function represented by Fourier’s integral, Messenger Math., Volume 52 (1923), pp. 185-188 | Zbl

[65] Ken-Iti Sato Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, 2013 | Zbl | MR

[66] Konrad Schumacher Travelling-front solutions for integro-differential equations. I, J. Reine Angew. Math., Volume 1980 (1980) no. 316, pp. 54-70 | DOI | Zbl | MR

[67] Diana Stan; Juan L. Vázquez The Fisher-KPP equation with nonlinear fractional diffusion, SIAM J. Math. Anal., Volume 46 (2014) no. 5, pp. 3241-3276 | Zbl | MR | DOI

[68] Xiaochuan Tian; Qiand Du Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J. Numer. Anal., Volume 51 (2013) no. 6, pp. 3458-3482 | DOI | Zbl | MR

[69] Hans F. Weinberger Long-time behavior of a class of biological models, SIAM J. Math. Anal., Volume 13 (1982) no. 3, pp. 353-396 | DOI | Zbl | MR

[70] Hiroki Yagisita Existence and nonexistence of travelling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., Volume 45 (2009) no. 4, pp. 925-953 | DOI | Zbl | MR

[71] Guo-Bao Zhang; Wan-Tong Li; Zhi-Cheng Wang Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differ. Equations, Volume 252 (2012) no. 9, pp. 5096-5124 | DOI | Zbl | MR

[72] Yuming P. Zhang; Andrej Zlatoš Optimal estimates on the propagation of reactions with fractional diffusion, Arch. Ration. Mech. Anal., Volume 247 (2023), 93, 33 pages | DOI | Zbl | MR

[73] Andrej Zlatoš Quenching and propagation of combustion without ignition temperature cutoff, Nonlinearity, Volume 18 (2005) no. 4, pp. 1463-1476 | DOI | Zbl | MR

Cité par Sources :