Speed-up of reaction-diffusion fronts by a line of fast diffusion

Berestycki, Henri; Coulon, Anne-Charline; Roquejoffre, Jean-Michel; Rossi, Luca

Séminaire Laurent Schwartz — EDP et applications (2013-2014), Exposé no. 19, 25 p.

In these notes, we discuss a new model, proposed by H. Berestycki, J.-M. Roquejoffre and L. Rossi, to describe biological invasions in the plane when a strong diffusion takes place on a line. This model seems relevant to account for the effects of roads on the spreading of invasive species. In what follows, the diffusion on the line will either be modelled by the Laplacian operator, or the fractional Laplacian of order less than 1. Of interest to us is the asymptotic speed of spreading in the direction of the line, but also in the plane. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. When the diffusion is the Laplacian, the global asymptotic speed of spreading on the line grows as the square root of the diffusion. In the other directions, the line of strong diffusion influences the spreading up to a critical angle, from which one recovers the classical spreading velocity. When the diffusion is the fractional Laplacian, the spreading on the line is exponential in time, and propagation in the plane is equivalent to that of a one-dimensional infinite planar front parallel to the line.

[1] D. G. Aronson and H. F. Weinberger. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), volume 446, pages 5–49. Springer, Berlin, 1975. | MR | Zbl

[2] D. G. Aronson and H. F. Weinberger. Multidimensional nonlinear diffusion arising in population genetics. Adv. in Math., 30(1):33–76, 1978. | MR | Zbl

[3] H. Berestycki, A.-C. Coulon, J.-M. Roquejoffre, and L. Rossi. Exponential front propagation in a Fisher-KPP model driven by a line of integral diffusion. in preparation, 2014.

[4] H. Berestycki and F. Hamel. Generalized transition waves and their properties. Comm. Pure Appl. Math., 65:592–648, 2012. | MR | Zbl

[5] H. Berestycki and F. Hamel. Reaction-diffusion equations and propagation phenomena. Applied Mathematical Sciences, 2014.

[6] H. Berestycki, F. Hamel, and N. Nadin. Asymptotic spreading in heterogeneous diffusive excitable media. J. Funct. Analysis, 255:2146–2189, 2008. | MR | Zbl

[7] H. Berestycki, F. Hamel, and N. Nadirashvili. The speed of propagation for KPP type problems. I. Periodic framework. J. European Math. Soc., 7:173–213, 2005. | MR | Zbl

[8] H. Berestycki, F. Hamel, and N. Nadirashvili. The speed of propagation for KPP type problems. II. General domains. J. Amer. Math. Soc., 23(1):1–34, 2010. | MR | Zbl

[9] H. Berestycki and N. Nadin. Spreading speeds for one-dimensional monostable reaction-diffusion equations. J. Math. Phys., 53(11), 2012. | MR | Zbl

[10] H. Berestycki, J.-M. Roquejoffre, and L. Rossi. The influence of a line with fast diffusion on Fisher-KPP propagation. J. Math. Biol., 66(4-5):743–766, 2013. | MR | Zbl

[11] H. Berestycki, J.-M. Roquejoffre, and L. Rossi. The shape of expansion induced by a line with fast diffusion in Fisher-KPP propagation. 2014. | arXiv | MR

[12] H. Berestycki and L. Rossi. Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains. Comm. Pure Appl. Math., 2014.

[13] R. M. Blumenthal and R. K. Getoor. Some theorems on stable processes. Trans. Amer. Math. Soc., 95:263–273, 1960. | MR | Zbl

[14] J.-M. Bony, P. Courrège, and P. Priouret. Semi-groupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum. Ann. Inst. Fourier (Grenoble), 18(fasc. 2):369–521 (1969), 1968. | Numdam | MR | Zbl

[15] M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves. Memoirs of the Amer. Math. Soc., 44, 1983. | MR | Zbl

[16] X. Cabré, A.-C. Coulon, and J.-M. Roquejoffre. Propagation in Fisher-KPP type equations with fractional diffusion in periodic media. C. R. Math. Acad. Sci. Paris, 350(19-20):885–890, 2012. | MR | Zbl

[17] X. Cabré and J.-M. Roquejoffre. Front propagation in Fisher-KPP equations with fractional diffusion. C. R. Math. Acad. Sci. Paris, 347:1361–1366, 2009. | MR | Zbl

[18] X. Cabré and J.-M. Roquejoffre. The influence of fractional diffusion in Fisher-KPP equations. Comm. Math. Phys., 320(3):679–722, 2013. | MR

[19] L. Caffarelli, S. Salsa, and L. Silvestre. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math., 171(2):425–461, 2008. | MR | Zbl

[20] L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations, 32(7-9):1245–1260, 2007. | MR | Zbl

[21] P. Constantin and V. Vicol. Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal., 22:1289–1321, 2012. | MR | Zbl

[22] A.-C. Coulon. Fast front propagation in reaction-diffusion models with integral diffusion. Toulouse University PhD thesis, 2014.

[23] A.-C. Coulon. The heat kernel in a parabolic model with a line of fast diffusion. in preparation, 2014.

[24] R.A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics, 7:355–369, 1937.

[25] J. Garnier. Accelerating solutions in integro-differential equations. SIAM J. Math. Anal., 43(4):1955–1974, 2011. | MR | Zbl

[26] J. Gartner. Location of wave fronts for the multi-dimensional KPP equation and Brownian first exit densities. Math. Nachr., 105:317–351, 1982. | MR | Zbl

[27] J. Gartner and M. I. Freidlin. The propagation of concentration waves in periodic and random media. Dokl. Akad. Nauk SSSR, 249:521–525, 1979. | MR | Zbl

[28] F. Hamel, J. Nolen, J.-M. Roquejoffre, and L. Ryzhik. A short proof of the logarithmic bramson correction in Fisher-KPP equations. Netw. Heterog. Media, pages 275–289, 2013. | MR | Zbl

[29] F. Hamel and L. Roques. Fast propagation for KPP equations with slowly decaying initial conditions. J. Differential Equations, 249(7):1726–1745, 2010. | MR | Zbl

[30] M.V. Hirsch. Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math., 383:1–53, 1988. | MR | Zbl

[31] A.N. Kolmogorov, I.G. Petrovskii, and N.S. Piskunov. Etude de l’équation de diffusion avec accroissement de la quantité de matière, et son application à un problème biologique. Bjul. Moskowskogo Gos. Univ., 17:1–26, 1937. | Zbl

[32] R. Mancinelli, D. Vergni, and A. Vulpiani. Front propagation in reactive systems with anomalous diffusion. Phys. D, 185(3-4):175–195, 2003. | MR | Zbl

[33] H. Matano. Existence of nontrivial unstable sets for equilibriums of strongly order preserving systems. J. Fac. Sci. Tokyo, 1984. | MR | Zbl

[34] H.W. McKenzie, E.H. Merrill, R.J. Spiteri, and M.A. Lewis. How linear features alter predator movement and the functional response. Interface focus, 2(2):205–216, 2012.

[35] G. Polya. On the zeros of an integral function represented by Fourier’s integral. Messenger of Math., 52:185–188, 1923.

[36] A. Siegfried. Itinéraires des contagions, épidémies et idéologies. A. Colin, Paris, 1960.

[37] H.F. Weinberger. On spreading speeds and traveling waves for growth and migration in periodic habitat. J. Math. Biol., 45:511–548, 2002. | MR | Zbl

[38] F.A. Williams. Combustion theory. Benjamin Cummings, 1985.