# The influence of a line with fast diffusion on Fisher-KPP propagation

- 751 Downloads
- 19 Citations

## Abstract

We propose here a new model to describe biological invasions in the plane when a strong diffusion takes place on a line. We establish the main properties of the system, and also derive the asymptotic speed of spreading in the direction of the line. 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. It is shown here that the global asymptotic speed of spreading in the plane, in the direction of the line, grows as the square root of the diffusion on the line. The model is much relevant to account for the effects of fast diffusion lines such as roads on spreading of invasive species.

## Keywords

KPP equations Reaction-diffusion system Fast diffusion on a line Asymptotic speed of propagation## Mathematics Subject Classification

35K57 92D25 35B40 35K40 35B53## Notes

### Acknowledgments

This study was supported by the French “gence Nationale de la Recherche” through the project PREFERED (ANR 08-BLAN-0313). H.B. was also supported by an NSF FRG grant DMS-1065979. L.R. was partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”.

## References

- Aronson DG, Weinberger HF (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv Math 30:33–76MathSciNetzbMATHCrossRefGoogle Scholar
- Barles G, Evans LC, Souganidis PE (1990) Wavefront propagation for reaction-diffusion systems of PDE. Duke Math J 61:835–858MathSciNetzbMATHCrossRefGoogle Scholar
- Barroux R (2012) Redouté, le moustique “tigre” est arrivé aux portes de Paris, le Monde, July 18Google Scholar
- Berestycki H, Hamel F (2012) Generalized transition waves and their properties. Comm Pure Appl Math 65:592–648MathSciNetzbMATHCrossRefGoogle Scholar
- Berestycki H, Hamel F (2012) Reaction-diffusion equations and propagation phenomena. In: Applied mathematical sciences. Springer, Berlin, to appearGoogle Scholar
- Berestycki H, Hamel F, Roques L (2005) Analysis of the periodically fragmented environment model I. Species persistence. J Math Biol 51:75–113MathSciNetzbMATHCrossRefGoogle Scholar
- Evans LC, Souganidis PE (1989) A PDE approach to certain large deviations problem for systems of parabolic equations. Ann Inst H Poincaré Anal Non Linéaire 6(suppl.):229–258Google Scholar
- Hillen T (2012) Conference given at the Banff research station. http://temple.birs.ca/11w5106/
- Hillen T, Painter KJ (2012) Transport and anisotropic diffusion models for movement in oriented habitats, preprintGoogle Scholar
- Hirsch MV (1988) Stability and convergence in strongly monotone dynamical systems. J Reine Angew Math 383:1–53MathSciNetzbMATHGoogle Scholar
- Jung T, Blaschke M (2004) Phytophthora root and collar rot of alders in Bavaria: distribution, modes of spread and possible management strategies. Plant Pathol 53:197–208CrossRefGoogle Scholar
- Kolmogorov AN, Petrovskii IG, Piskunov NS (1937) 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 Intern A 1:1–26Google Scholar
- Ladyzhenskaya OA, Ural’tseva NN, Solonnikov VA (1968) Linear and quasilinear parabolic equations of parabolic type. American Math Soc, ProvidenceGoogle Scholar
- McKenzie HW, Merrill EH, Spiteri RJ, Lewis MA (2012) How linear features alter predator movement and the functional response. Interface focus, to appear http://rsfs.royalsocietypublishing.org/content/2/2/205.full
- Siegfried A (1960) Itinéraires des contagions, épidémies et idéologies. A. Colin, Paris, 164Google Scholar
- Weinberger HF (2002) On spreading speeds and traveling waves for growth and migration in periodic habitat. J Math Biol 45:511–548MathSciNetzbMATHCrossRefGoogle Scholar