# American Institute of Mathematical Sciences

February  2011, 4(1): 1-13. doi: 10.3934/dcdss.2011.4.1

## The periodic patch model for population dynamics with fractional diffusion

 1 Ecole des Hautes Etudes en Sciences Sociales, CAMS, 54, bd Raspail F-75270 Paris, France 2 Institut de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 4, France 3 Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63 - 35121 Padova, Italy

Received  May 2010 Published  October 2010

Fractional diffusions arise in the study of models from population dynamics. In this paper, we derive a class of integro-differential reaction-diffusion equations from simple principles. We then prove an approximation result for the first eigenvalue of linear integro-differential operators of the fractional diffusion type, and we study from that the dynamics of a population in a fragmented environment with fractional diffusion.
Citation: Henri Berestycki, Jean-Michel Roquejoffre, Luca Rossi. The periodic patch model for population dynamics with fractional diffusion. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 1-13. doi: 10.3934/dcdss.2011.4.1
##### References:
 [1] H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Funct. Anal, 40 (1981), 1-29. doi: doi:10.1016/0022-1236(81)90069-0. [2] H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480. doi: doi:10.1007/s00220-004-1201-9. [3] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol, 51 (2005), 75-113. doi: doi:10.1007/s00285-004-0313-3. [4] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl, 84 (2005), 1101-1146. doi: doi:10.1016/j.matpur.2004.10.006. [5] 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, 18 (1968), 369-521. [6] X. Cabré and J.-M. Roquejoffre, Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire, C. R. Math. Acad. Sci. Paris, 347 (2009), 1361-1366. [7] Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift, Proc. Royal Soc. Edinburgh, 132 (2002), 567-594. doi: doi:10.1017/S0308210500001785. [8] P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatos, Diffusion and mixing in fluid flow, Annals of Math., 168 (2008), 643-674. doi: doi:10.4007/annals.2008.168.643. [9] J. Coville, PhD thesis, 2003. [10] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer-Verlag, 1979. [11] 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 (1937), 1-26. [12] J. D. Murray, "Mathematical Biology," 2nd edition, Biomathematics, 19, Springer-Verlag, Berlin, 1993.

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##### References:
 [1] H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Funct. Anal, 40 (1981), 1-29. doi: doi:10.1016/0022-1236(81)90069-0. [2] H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480. doi: doi:10.1007/s00220-004-1201-9. [3] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol, 51 (2005), 75-113. doi: doi:10.1007/s00285-004-0313-3. [4] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl, 84 (2005), 1101-1146. doi: doi:10.1016/j.matpur.2004.10.006. [5] 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, 18 (1968), 369-521. [6] X. Cabré and J.-M. Roquejoffre, Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire, C. R. Math. Acad. Sci. Paris, 347 (2009), 1361-1366. [7] Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift, Proc. Royal Soc. Edinburgh, 132 (2002), 567-594. doi: doi:10.1017/S0308210500001785. [8] P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatos, Diffusion and mixing in fluid flow, Annals of Math., 168 (2008), 643-674. doi: doi:10.4007/annals.2008.168.643. [9] J. Coville, PhD thesis, 2003. [10] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer-Verlag, 1979. [11] 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 (1937), 1-26. [12] J. D. Murray, "Mathematical Biology," 2nd edition, Biomathematics, 19, Springer-Verlag, Berlin, 1993.
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