Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition

Peng, Rui; Wu, Chang-Hong; Zhou, Maolin

doi:10.1016/j.anihpc.2020.07.006

Peng, Rui ¹ ; Wu, Chang-Hong ² ; Zhou, Maolin ³

¹ a School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, People's Republic of China
² b Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan, Republic of China
³ c Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, People's Republic of China

Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 507-547.

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Résumé

This paper is concerned with the classical two-species Lotka-Volterra diffusion system with strong competition. The sharp dynamical behavior of the solution is established in two different situations: either one species is an invasive one and the other is a native one or both are invasive species. Our results seem to be the first that provide a precise spreading speed and profile for such a strong competition system. Among other things, our analysis relies on the construction of new types of supersolution and subsolution, which are optimal in certain sense.

Reçu le : 2019-08-14
Révisé le : 2020-05-25
Accepté le : 2020-07-13

DOI : 10.1016/j.anihpc.2020.07.006

Classification : 35K57, 35K45, 92D25
Mots-clés : Lotka-Volterra diffusion system, Strong competition, Traveling waves, Long-time behavior, Spreading speed and profile

Affiliations des auteurs :

Peng, Rui ¹ ; Wu, Chang-Hong ² ; Zhou, Maolin ³

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     author = {Peng, Rui and Wu, Chang-Hong and Zhou, Maolin},
     title = {Sharp estimates for the spreading speeds of the {Lotka-Volterra} diffusion system with strong competition},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {507--547},
     publisher = {Elsevier},
     volume = {38},
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Peng, Rui; Wu, Chang-Hong; Zhou, Maolin. Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition. Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 507-547. doi : 10.1016/j.anihpc.2020.07.006. http://www.numdam.org/articles/10.1016/j.anihpc.2020.07.006/

Bibliographie
Cité par

[1] Aronson, D.G.; Weinberger, H.F. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, Lecture Notes in Math., vol. 446, Springer, Berlin, 1975, pp. 5-49 | MR | Zbl

[2] Aronson, D.G.; Weinberger, H.F. Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., Volume 30 (1978), pp. 33-76 | DOI | MR | Zbl

[3] Bao, X.; Wang, Z.-C. Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differ. Equ., Volume 255 (2013), pp. 2402-2435 | DOI | MR | Zbl

[4] Bramson, M. Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Am. Math. Soc., Volume 44 (1983) | MR | Zbl

[5] Cantrell, R.S.; Cosner, C. Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, UK, 2003 | MR | Zbl

[6] Carrere, C. Spreading speeds for a two-species competition-diffusion system, J. Differ. Equ., Volume 264 (2018), pp. 2133-2156 | DOI | MR

[7] Chen, X. Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differ. Equ., Volume 2 (1997), pp. 125-160 | MR | Zbl

[8] Chen, X.; Tsai, J.-C.; Wu, Y. Longtime behavior of solutions of a SIS epidemiological model, SIAM J. Math. Anal., Volume 49 (2017), pp. 3925-3950 | DOI | MR

[9] Du, Y.; Lou, B. Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., Volume 17 (2015), pp. 2673-2724 | DOI | MR

[10] Du, Y.; Matsuzawa, H.; Zhou, M. Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., Volume 103 (2015), pp. 741-787 | DOI | MR | Zbl

[11] Du, Y.; Wang, M.X.; Zhou, M. Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., Volume 107 (2017), pp. 253-287 | DOI | MR

[12] Du, Y.; Wu, C.-H. Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differ. Equ., Volume 57 (2018), p. 52 | DOI | MR

[13] Ducrot, A.; Giletti, T.; Matano, H. Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations, Trans. Am. Math. Soc., Volume 366 (2014), pp. 5541-5566 | DOI | MR | Zbl

[14] Ducrot, A.; Giletti, T.; Matano, H. Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differ. Equ., Volume 58 (2019), p. 137 | DOI | MR

[15] Fang, J.; Zhao, X.-Q. Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc., Volume 17 (2015), pp. 2243-2288 | DOI | MR

[16] Faye, G.; Holzer, M. Asymptotic stability of the critical pulled front in a Lotka-Volterra competition model | arXiv | DOI | MR

[17] Fife, P.C.; McLeod, J.B. The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., Volume 65 (1977), pp. 335-361 | DOI | MR | Zbl

[18] Fisher, R.A. The wave of advance of advantageous genes, Annu. Eugen., Volume 7 (1937), pp. 335-369 | JFM

[19] Friedman, A. Parabolic Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964 | MR | Zbl

[20] Gardner, R.A. Existence and stability of traveling wave solutions of competition models: a degree theoretic, J. Differ. Equ., Volume 44 (1982), pp. 343-364 | DOI | MR | Zbl

[21] Giletti, T.; Matano, H. Existence and uniqueness of propagating terraces, Commun. Contemp. Math., Volume 22 (2020) | DOI | MR

[22] Girardin, L. The effect of random dispersal on competitive exclusion-a review, Math. Biosci., Volume 318 (2019) | DOI | MR

[23] Girardin, L.; Lam, K.-Y. Invasion of an empty habitat by two competitors: spreading properties of monostable two-species competition-diffusion systems, Proc. Lond. Math. Soc., Volume 119 (2019), pp. 1279-1335 | DOI | MR

[24] Girardin, L.; Nadin, G. Travelling waves for diffusive and strongly competitive systems: relative motility and invasion speed, Eur. J. Appl. Math., Volume 26 (2015), pp. 521-534 | DOI | MR

[25] Guo, J.-S.; Lin, Y.-C. The sign of the wave speed for the Lotka-Volterra competition-diffusion system, Commun. Pure Appl. Anal., Volume 12 (2013), pp. 2083-2090 | DOI | MR | Zbl

[26] Guo, J.-S.; Wu, C.-H. Entire solutions originating from traveling fronts for a two-species competition-diffusion system, Nonlinearity, Volume 32 (2019), p. 3234 | DOI | MR

[27] Hamel, F.; Nolen, J.; Roquejoffre, J.-M.; Ryzhik, L. A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Netw. Heterog. Media, Volume 8 (2013), pp. 275-289 | DOI | MR | Zbl

[28] Iida, M.; Lui, R.; Ninomiya, H. Stacked fronts for cooperative systems with equal diffusion coefficients, SIAM J. Math. Anal., Volume 43 (2011), pp. 1369-1389 | DOI | MR | Zbl

[29] Kan-On, Y. Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., Volume 26 (1995), pp. 340-363 | DOI | MR | Zbl

[30] Kan-on, Y.; Fang, Q. Stability of monotone travelling waves for competition-diffusion equations, Jpn. J. Ind. Appl. Math., Volume 13 (1996), pp. 343-349 | DOI | MR | Zbl

[31] Kaneko, Y.; Matsuzawa, H. Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., Volume 428 (2015), pp. 43-76 | DOI | MR

[32] Kolmogorov, A.N.; Petrovsky, I.G.; Piskunov, N.S. Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem, Bull. Univ. Etat. Moscow Ser. Int. Math. Mec. Sect. A, Volume 1 (1937), pp. 1-29

[33] Lam, K.-Y.; Salako, R.B.; Wu, Q. Entire solutions of diffusive Lotka-Volterra system (preprint) | arXiv | MR

[34] Lau, K.-S. On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differ. Equ., Volume 59 (1985), pp. 44-70 | DOI | MR | Zbl

[35] Lewis, M.A.; Li, B.; Weinberger, H.F. Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol., Volume 45 (2002), pp. 219-233 | DOI | MR | Zbl

[36] Li, B.; Weinberger, H.F.; Lewis, M.A. Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., Volume 196 (2005), pp. 82-98 | DOI | MR | Zbl

[37] Liang, X.; Zhao, X.-Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math., Volume 60 (2007), pp. 1-40 | DOI | MR | Zbl

[38] Lin, G.; Li, W.-T. Asymptotic spreading of competition diffusion systems: the role of interspecific competitions, Eur. J. Appl. Math., Volume 23 (2012), pp. 669-689 | DOI | MR | Zbl

[39] Liu, Q.; Liu, S.; Lam, K.-Y. Asymptotic spreading of interacting species with multiple fronts I: a geometric optics approach, Discrete Contin. Dyn. Syst., Ser. A, Volume 40 (2020), pp. 3683-3714 | DOI | MR

[40] Q. Liu, S. Liu, K.-Y. Lam, Asymptotic spreading of interacting species with multiple fronts II: Exponentially decaying initial data, preprint. | MR

[41] Ma, M.; Huang, Z.; Ou, C. Speed of the traveling wave for the bistable Lotka-Volterra competition model, Nonlinearity, Volume 32 (2019), pp. 3143-3162 | DOI | MR

[42] Murray, J.D. Mathematical Biology, Springer-Verlag, Berlin, 1993 | DOI | MR | Zbl

[43] Morita, Y.; Tachibana, K. An entire solution for wave fronts to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., Volume 40 (2009), pp. 2217-2240 | DOI | MR | Zbl

[44] Poláčik, P. Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on $R$ , Mem. Am. Math. Soc., Volume 264 (2020) no. 1278 | MR

[45] Protter, M.H.; Weinberger, H.F. Maximum Principles in Differential Equations, Springer-Verlag, 1984 | DOI | MR | Zbl

[46] Rodrigo, M.; Mimura, M. Exact solutions of a competition-diffusion system, Hiroshima Math. J., Volume 30 (2000), pp. 257-270 | DOI | MR | Zbl

[47] Rothe, F. Convergence to travelling fronts in semilinear parabolic equations, Proc. R. Soc. Edinb., Sect. A, Volume 80 (1978), pp. 213-234 | DOI | MR | Zbl

[48] Sattinger, D. On the stability of waves of nonlinear parabolic systems, Adv. Math., Volume 22 (1976), pp. 312-355 | DOI | MR | Zbl

[49] Shigesada, N.; Kawasaki, K. Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford UP, Oxford, 1997

[50] Tsai, J.-C. Asymptotic stability of traveling wave fronts in the buffered bistable system, SIAM J. Math. Anal., Volume 39 (2007), pp. 138-159 | DOI | MR | Zbl

[51] Uchiyama, K. The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., Volume 18 (1978), pp. 453-508 | MR | Zbl

[52] Volpert, A.I.; Volpert, V.A.; Volpert, V.A. Traveling Wave Solutions of Parabolic Systems, Amer. Math. Soc., Providence, 1994 | DOI | MR | Zbl

[53] Weinberger, H.F.; Lewis, M.A.; Li, B. Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol., Volume 45 (2002), pp. 183-218 | DOI | MR | Zbl

[54] Zhao, X.-Q. Dynamical Systems in Population Biology, Springer, New York, 2003 | MR | Zbl

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