Inside dynamics of solutions of integro-differential equations

Discrete and Continuous Dynamical Systems - Series B

Volumn 19, Issue 10, 2014, Pages 3057-3085

  Bonnefon, Olivier ^a   Coville, Jérôme ^a   Garnier, Jimmy ^a,b,c   Roques, Lionel ^a  

^a INRA   (France)

^b AIX MARSEILLE UNIVERSITY   (France)

^c UNIVERSITÉ DE SAVOIE   (France)

Author keywords

Integro differential equation; Long distance dispersal; Monostable; Pulled and pushed solutions; Thin tailed fat tailed kernel; Traveling waves

Indexed keywords

EID: 84907482173     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2014.19.3057     Document Type: Article

References (50)

1. H. G. Aronson, The asymptotic speed of propagation of a simple epidemic, in Nonlinear Diffusion, (1977).
2. H. G. Aronson and D. G. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, in Partial Differential Equations and Related Topics, 446 (1975), 5-49.
3. D. G Aronson and H. F Weinberger, Multidimensional non-linear diffusion arising in population-genetics, Adv. Math., 30 (1978), 33-76.
4. F. Austerlitz and P. H. Garnier-Géré, Modelling the impact of colonisation on genetic diversity and differentiation of forest trees: Interaction of life cycle, pollen flow and seed long-distance dispersal, Heredity, 90 (2003), 282-290.
5. H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.
6. G. Bohrer, R. Nathan and S. Volis, Effects of long-distance dispersal for metapopulation survival and genetic structure at ecological time and spatial scales, J. Ecol., 93 (2005), 1029-1040.
7. O. Bonnefon, J. Garnier, F. Hamel and L. Roques, Inside dynamics of delayed traveling waves, Math. Model. Nat. Phenom., 8 (2013), 42-59.
8. O. Bonnefon, J. Coville, J. Garnier, F. Hamel and L. Roques, The spatio-temporal dynamics of neutral genetic diversity, Preprint.
9. M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190.
10. M. Cain, B. Milligan and A. Strand, Long distance seed dispersal in plant populations, Am. J. Bot., 87 (2000), 1217-1227.
11. A. Carr and J. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132, (2004), 2433-2439.
12. J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, Am. Nat., 152 (1998), 204-224.
13. J. S. Clark, C. Fastie, G. Hurtt, S. T. Jackson, C. Johnson, G. A. King, M. Lewis, J. Lynch, S. Pacala, C. Prentice, E. W. Schupp, T. Webb III and P. Wyckoff, Reid's paradox of rapid plant migration, BioScience, 48 (1998), 13-24.
14. J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1-29.
15. J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Diff. Equations, 244 (2008), 3080-3118.
16. E. C. M. Crooks, Travelling fronts for monostable reaction-diffusion systems with gradient-dependence, Adv. Diff. Equations, 8 (2003), 279-314.
17. O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Diff. Equations, 33 (1979), 58-73.
18. J. P. Eckmann and C. E. Wayne, The nonlinear stability of front solutions for parabolic differential equations, Comm. Math. Phys., 161 (1994), 323-334.
19. J. Fayard, E. K. Klein and F. Lefèvre, Long distance dispersal and the fate of a gene from the colonization front, J. Evol. Biol., 22 (2009), 2171-2182.
20. P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, 1979.
21. P. C. Fife and J. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.
22. J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.
23. J. Garnier, T. Giletti, F. Hamel and L. Roques, Inside dynamics of pulled and pushed fronts, J. Math. Pures Appl., 11 (2012), 173-188.
24. O. Hallatschek and D. R. Nelson, Gene surfing in expanding populations, Theor. Popul. Biol., 73 (2008), 158-170.
25. Ja. I. Kanel, Certain problem of burning-theory equations, Dokl. Akad. Nauk SSSR, 136 (1961), 277-280.
26. E. K. Klein, C. Lavigne and P. H. Gouyon, Mixing of propagules from discrete sources at long distance : Comparing a dispersal tail to an exponential, BMC Ecology, 6 (2006).
27. N. S. Kolmogorov, N. Petrovsky and I. G. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique, Bull. Univ. Moscou, Sér A, 1 (1937) 1-26.
28. M. Kot, M. Lewis and P. Van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.
29. J. G. Lambrinos, How interactions between ecology and evolution influence contemporary invasion dynamics, Ecol., 85 (2004), 2061-2070.
30. K. S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky and Piscounov, J. Diff. Equations, 59 (1985), 44-70.
31. J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.
32. D. Mollison, Spatial contact models for ecological and epidemic spread, J. R. Stat. Ser. B Stat. Methodol., 39 (1977),283-326.
33. R. Nathan and H. Muller-Landau, Spatial patterns of seed dispersal, their determinants and consequences for recruitment, Trends Ecol. Evol., 15 (2000), 278-285.
34. J. M. Pringle, F. Lutscher and E. Glick, Going against the flow: effects of non-Gaussian dispersal kernels and reproduction over multiple generations, Mar. Ecol. Prog. Ser., 377 (2009), 13-17.
35. C. Reid, The Origin of the British Flora, Dulau & Co, 1899.
36. L. Roques, F. Hamel, J. Fayard, B. Fady and E. K. Klein, Recolonisation by diffusion can generate increasing rates of spread, Theor. Popul. Biol., 77 (2010) 205-21.
37. L. Roques, J. Garnier, F. Hamel and E. Klein, Allee effect promotes diversity in traveling waves of colonization, Proc. Natl. Acad. Sci. USA, 109 (2012), 8828-8833.
38. F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 213-234.
39. F. Rothe, Convergence to pushed fronts, Rocky Mountain J. Math., 11 (1981), 617-634.
40. D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 25 (1976), 312-355.
41. D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Diff. Equations, 25 (1977), 130-144.
42. K. Schumacher, Travelling-front solutions for integro-differential equations. I, J. Reine Angew. Math., 316 (1980), 54-70.
43. J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.
44. A. N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosci., 31 (1976), 307-315.
45. H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.
46. P. Turchin, Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants, Sinauer Associates, 1998.
47. K. Uchiyama, The behaviour of solutions of some non-linear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.
48. M. O. Vlad, L. L. Cavalli-Sforza and J. Ross, Enhanced (hydrodynamic) transport induced by population growth in reaction-diffusion systems with application to population genetics, Proc. Natl. Acad. Sci. USA, 101 (2004), 10249-10253.
49. H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
50. H. F. Weinberger, On spreading speeds and traveling waves for growth and migration in periodic habitat, J. Math. Biol., 45 (2002), 511-548.