American Institute of Mathematical Sciences

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

The periodic patch model for population dynamics with fractional diffusion

Henri Berestycki 1, , Jean-Michel Roquejoffre 2, and  Luca Rossi 3,

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.
Keywords: Fractional diffusion, reaction-diffusion equation, KPP nonlinearity, persistence..
Mathematics Subject Classification: Primary: 35R11; Secondary: 35B40, 35K57, 92D2.
Citation: Henri Berestycki, Jean-Michel Roquejoffre, Luca Rossi. The periodic patch model for population dynamics with fractional diffusion. Discrete & 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.  doi: doi:10.1016/0022-1236(81)90069-0.  Google Scholar

[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.  doi: doi:10.1007/s00220-004-1201-9.  Google Scholar

[3]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol, 51 (2005), 75.  doi: doi:10.1007/s00285-004-0313-3.  Google Scholar

[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.  doi: doi:10.1016/j.matpur.2004.10.006.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[7]

Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift,, Proc. Royal Soc. Edinburgh, 132 (2002), 567.  doi: doi:10.1017/S0308210500001785.  Google Scholar

[8]

P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatos, Diffusion and mixing in fluid flow,, Annals of Math., 168 (2008), 643.  doi: doi:10.4007/annals.2008.168.643.  Google Scholar

[9]

J. Coville, PhD thesis,, PhD thesis, (2003).   Google Scholar

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Springer-Verlag, (1979).   Google Scholar

[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.   Google Scholar

[12]

J. D. Murray, "Mathematical Biology," 2nd edition,, Biomathematics, 19 (1993).   Google Scholar

show all references

References:
[1]

H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques,, J. Funct. Anal, 40 (1981), 1.  doi: doi:10.1016/0022-1236(81)90069-0.  Google Scholar

[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.  doi: doi:10.1007/s00220-004-1201-9.  Google Scholar

[3]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol, 51 (2005), 75.  doi: doi:10.1007/s00285-004-0313-3.  Google Scholar

[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.  doi: doi:10.1016/j.matpur.2004.10.006.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[7]

Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift,, Proc. Royal Soc. Edinburgh, 132 (2002), 567.  doi: doi:10.1017/S0308210500001785.  Google Scholar

[8]

P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatos, Diffusion and mixing in fluid flow,, Annals of Math., 168 (2008), 643.  doi: doi:10.4007/annals.2008.168.643.  Google Scholar

[9]

J. Coville, PhD thesis,, PhD thesis, (2003).   Google Scholar

[10]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Springer-Verlag, (1979).   Google Scholar

[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.   Google Scholar

[12]

J. D. Murray, "Mathematical Biology," 2nd edition,, Biomathematics, 19 (1993).   Google Scholar

[1]

Michio Urano, Kimie Nakashima, Yoshio Yamada. Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity. Conference Publications, 2005, 2005 (Special) : 868-877. doi: 10.3934/proc.2005.2005.868
[2]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033
[3]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079
[4]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[5]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085
[6]

Costică Moroşanu. Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1567-1587. doi: 10.3934/dcdss.2020089
[7]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246
[8]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581
[9]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267
[10]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
[11]

Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279
[12]

Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079
[13]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029
[14]

Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407
[15]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875
[16]

Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems & Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285
[17]

Takanori Ide, Kazuhiro Kurata, Kazunaga Tanaka. Multiple stable patterns for some reaction-diffusion equation in disrupted environments. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 93-116. doi: 10.3934/dcds.2006.14.93
[18]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39
[19]

Tarik Mohammed Touaoula, Mohammed Nor Frioui, Nikolay Bessonov, Vitaly Volpert. Dynamics of solutions of a reaction-diffusion equation with delayed inhibition. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020193
[20]

Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences