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Inside dynamics of solutions of integro-differential equations
1. | INRA, UR 546 Biostatistique et Processus Spatiaux (BioSP), F-84914 Avignon, France, France, France, France |
References:
[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.
doi: 10.1007/BFb0070595. |
[3] |
D. G Aronson and H. F Weinberger, Multidimensional non-linear diffusion arising in population-genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[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.
doi: 10.1038/sj.hdy.6800243. |
[5] |
H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.
doi: 10.1002/cpa.21389. |
[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.
doi: 10.1111/j.1365-2745.2005.01048.x. |
[7] |
O. Bonnefon, J. Garnier, F. Hamel and L. Roques, Inside dynamics of delayed traveling waves, Math. Model. Nat. Phenom., 8 (2013), 42-59.
doi: 10.1051/mmnp/20138305. |
[8] |
O. Bonnefon, J. Coville, J. Garnier, F. Hamel and L. Roques, The spatio-temporal dynamics of neutral genetic diversity,, Preprint., ().
doi: 10.1016/j.ecocom.2014.05.003. |
[9] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190.
doi: 10.1090/memo/0285. |
[10] |
M. Cain, B. Milligan and A. Strand, Long distance seed dispersal in plant populations, Am. J. Bot., 87 (2000), 1217-1227.
doi: 10.2307/2656714. |
[11] |
A. Carr and J. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132, (2004), 2433-2439.
doi: 10.1090/S0002-9939-04-07432-5. |
[12] |
J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, Am. Nat., 152 (1998), 204-224.
doi: 10.1086/286162. |
[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.
doi: 10.2307/1313224. |
[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.
doi: 10.1017/S0308210504000721. |
[15] |
J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Diff. Equations, 244 (2008), 3080-3118.
doi: 10.1016/j.jde.2007.11.002. |
[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.
doi: 10.1016/0022-0396(79)90080-9. |
[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.
doi: 10.1007/BF02099781. |
[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.
doi: 10.1111/j.1420-9101.2009.01832.x. |
[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.
doi: 10.1137/10080693X. |
[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.
doi: 10.3934/cpaa.2012.11.173. |
[24] |
O. Hallatschek and D. R. Nelson, Gene surfing in expanding populations, Theor. Popul. Biol., 73 (2008), 158-170.
doi: 10.1016/j.tpb.2007.08.008. |
[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.
doi: 10.2307/2265698. |
[29] |
J. G. Lambrinos, How interactions between ecology and evolution influence contemporary invasion dynamics, Ecol., 85 (2004), 2061-2070.
doi: 10.1890/03-8013. |
[30] |
K. S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky and Piscounov, J. Diff. Equations, 59 (1985), 44-70.
doi: 10.1016/0022-0396(85)90137-8. |
[31] |
J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.
doi: 10.1016/S0025-5564(03)00041-5. |
[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.
doi: 10.1016/S0169-5347(00)01874-7. |
[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.
doi: 10.3354/meps07836. |
[35] |
C. Reid, The Origin of the British Flora, Dulau & Co, 1899.
doi: 10.5962/bhl.title.7595. |
[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), 205-212.
doi: 10.1016/j.tpb.2010.02.002. |
[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.
doi: 10.1073/pnas.1201695109. |
[38] |
F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 213-234.
doi: 10.1017/S0308210500010258. |
[39] |
F. Rothe, Convergence to pushed fronts, Rocky Mountain J. Math., 11 (1981), 617-634.
doi: 10.1216/RMJ-1981-11-4-617. |
[40] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[41] |
D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Diff. Equations, 25 (1977), 130-144.
doi: 10.1016/0022-0396(77)90185-1. |
[42] |
K. Schumacher, Travelling-front solutions for integro-differential equations. I, J. Reine Angew. Math., 316 (1980), 54-70.
doi: 10.1515/crll.1980.316.54. |
[43] |
J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.
doi: 10.1093/biomet/38.1-2.196. |
[44] |
A. N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosci., 31 (1976), 307-315.
doi: 10.1016/0025-5564(76)90087-0. |
[45] |
H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.
doi: 10.1007/BF00279720. |
[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.
doi: 10.1073/pnas.0403419101. |
[49] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[50] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration in periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
show all references
References:
[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.
doi: 10.1007/BFb0070595. |
[3] |
D. G Aronson and H. F Weinberger, Multidimensional non-linear diffusion arising in population-genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[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.
doi: 10.1038/sj.hdy.6800243. |
[5] |
H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.
doi: 10.1002/cpa.21389. |
[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.
doi: 10.1111/j.1365-2745.2005.01048.x. |
[7] |
O. Bonnefon, J. Garnier, F. Hamel and L. Roques, Inside dynamics of delayed traveling waves, Math. Model. Nat. Phenom., 8 (2013), 42-59.
doi: 10.1051/mmnp/20138305. |
[8] |
O. Bonnefon, J. Coville, J. Garnier, F. Hamel and L. Roques, The spatio-temporal dynamics of neutral genetic diversity,, Preprint., ().
doi: 10.1016/j.ecocom.2014.05.003. |
[9] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190.
doi: 10.1090/memo/0285. |
[10] |
M. Cain, B. Milligan and A. Strand, Long distance seed dispersal in plant populations, Am. J. Bot., 87 (2000), 1217-1227.
doi: 10.2307/2656714. |
[11] |
A. Carr and J. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132, (2004), 2433-2439.
doi: 10.1090/S0002-9939-04-07432-5. |
[12] |
J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, Am. Nat., 152 (1998), 204-224.
doi: 10.1086/286162. |
[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.
doi: 10.2307/1313224. |
[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.
doi: 10.1017/S0308210504000721. |
[15] |
J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Diff. Equations, 244 (2008), 3080-3118.
doi: 10.1016/j.jde.2007.11.002. |
[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.
doi: 10.1016/0022-0396(79)90080-9. |
[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.
doi: 10.1007/BF02099781. |
[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.
doi: 10.1111/j.1420-9101.2009.01832.x. |
[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.
doi: 10.1137/10080693X. |
[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.
doi: 10.3934/cpaa.2012.11.173. |
[24] |
O. Hallatschek and D. R. Nelson, Gene surfing in expanding populations, Theor. Popul. Biol., 73 (2008), 158-170.
doi: 10.1016/j.tpb.2007.08.008. |
[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.
doi: 10.2307/2265698. |
[29] |
J. G. Lambrinos, How interactions between ecology and evolution influence contemporary invasion dynamics, Ecol., 85 (2004), 2061-2070.
doi: 10.1890/03-8013. |
[30] |
K. S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky and Piscounov, J. Diff. Equations, 59 (1985), 44-70.
doi: 10.1016/0022-0396(85)90137-8. |
[31] |
J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.
doi: 10.1016/S0025-5564(03)00041-5. |
[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.
doi: 10.1016/S0169-5347(00)01874-7. |
[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.
doi: 10.3354/meps07836. |
[35] |
C. Reid, The Origin of the British Flora, Dulau & Co, 1899.
doi: 10.5962/bhl.title.7595. |
[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), 205-212.
doi: 10.1016/j.tpb.2010.02.002. |
[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.
doi: 10.1073/pnas.1201695109. |
[38] |
F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 213-234.
doi: 10.1017/S0308210500010258. |
[39] |
F. Rothe, Convergence to pushed fronts, Rocky Mountain J. Math., 11 (1981), 617-634.
doi: 10.1216/RMJ-1981-11-4-617. |
[40] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[41] |
D. H. Sattinger, Weighted norms for the stability of traveling waves, J. Diff. Equations, 25 (1977), 130-144.
doi: 10.1016/0022-0396(77)90185-1. |
[42] |
K. Schumacher, Travelling-front solutions for integro-differential equations. I, J. Reine Angew. Math., 316 (1980), 54-70.
doi: 10.1515/crll.1980.316.54. |
[43] |
J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.
doi: 10.1093/biomet/38.1-2.196. |
[44] |
A. N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosci., 31 (1976), 307-315.
doi: 10.1016/0025-5564(76)90087-0. |
[45] |
H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.
doi: 10.1007/BF00279720. |
[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.
doi: 10.1073/pnas.0403419101. |
[49] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[50] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration in periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
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