Inside dynamics of solutions of integro-differential equations

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December 2014, 19(10): 3057-3085. doi: 10.3934/dcdsb.2014.19.3057

Inside dynamics of solutions of integro-differential equations

Olivier Bonnefon ^1, , Jérôme Coville ^1, , Jimmy Garnier ^1, and Lionel Roques ^1,

1.	INRA, UR 546 Biostatistique et Processus Spatiaux (BioSP), F-84914 Avignon, France, France, France, France

Received July 2013 Revised January 2014 Published October 2014

Abstract
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In this paper, we investigate the inside dynamics of the positive solutions of integro-differential equations \begin{equation*} \partial_t u(t,x)= (J\star u)(t,x) -u(t,x) + f(u(t,x)), \ t>0 \hbox{ and } x\in\mathbb{R}, \end{equation*} with both thin-tailed and fat-tailed dispersal kernels $J$ and a monostable reaction term $f.$ The notion of inside dynamics has been introduced to characterize traveling waves of some reaction-diffusion equations [23]. Assuming that the solution is made of several fractions $\upsilon^i\ge 0$ ($i\in I \subset \mathbb{N}$), its inside dynamics is given by the spatio-temporal evolution of $\upsilon^i$. According to this dynamics, the traveling waves can be classified in two categories: pushed and pulled waves. For thin-tailed kernels, we observe no qualitative differences between the traveling waves of the above integro-differential equations and the traveling waves of the classical reaction-diffusion equations. In particular, in the KPP case ($f(u)\leq f'(0)u$ for all $u\in(0,1)$) we prove that all the traveling waves are pulled. On the other hand for fat-tailed kernels, the integro-differential equations do not admit any traveling waves. Therefore, to analyse the inside dynamics of a solution in this case, we introduce new notions of pulled and pushed solutions. Within this new framework, we provide analytical and numerical results showing that the solutions of integro-differential equations involving a fat-tailed dispersal kernel are pushed. Our results have applications in population genetics. They show that the existence of long distance dispersal events during a colonization tend to preserve the genetic diversity.

Keywords: long distance dispersal, integro-differential equation, monostable, pulled and pushed solutions, thin-tailed/fat-tailed kernel., Traveling waves.

Mathematics Subject Classification: Primary: 35R09, 45K05; Secondary: 35B06, 35K5.

Citation: Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057

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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Olivier Bonnefon Jérôme Coville Jimmy Garnier Lionel Roques
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Olivier Bonnefon Jérôme Coville Jimmy Garnier Lionel Roques

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