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

Inside dynamics of solutions of integro-differential equations

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

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.
Citation: Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. 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. doi: 10.1017/S0308210504000721.

[15]

J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Diff. Equations, 244 (2008), 3080. 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.

[17]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic,, J. Diff. Equations, 33 (1979), 58. 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. 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. 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.

[22]

J. Garnier, Accelerating solutions in integro-differential equations,, SIAM J. Math. Anal., 43 (2011), 1955. 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. 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. 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.

[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, 1 (1937), 1.

[28]

M. Kot, M. Lewis and P. Van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027. doi: 10.2307/2265698.

[29]

J. G. Lambrinos, How interactions between ecology and evolution influence contemporary invasion dynamics,, Ecol., 85 (2004), 2061. 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. 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. 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.

[33]

R. Nathan and H. Muller-Landau, Spatial patterns of seed dispersal, their determinants and consequences for recruitment,, Trends Ecol. Evol., 15 (2000), 278. 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. 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. 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. 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. doi: 10.1017/S0308210500010258.

[39]

F. Rothe, Convergence to pushed fronts,, Rocky Mountain J. Math., 11 (1981), 617. 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. 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. 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. doi: 10.1515/crll.1980.316.54.

[43]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. 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. 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. 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.

[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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. 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. doi: 10.1017/S0308210504000721.

[15]

J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Diff. Equations, 244 (2008), 3080. 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.

[17]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic,, J. Diff. Equations, 33 (1979), 58. 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. 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. 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.

[22]

J. Garnier, Accelerating solutions in integro-differential equations,, SIAM J. Math. Anal., 43 (2011), 1955. 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. 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. 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.

[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, 1 (1937), 1.

[28]

M. Kot, M. Lewis and P. Van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027. doi: 10.2307/2265698.

[29]

J. G. Lambrinos, How interactions between ecology and evolution influence contemporary invasion dynamics,, Ecol., 85 (2004), 2061. 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. 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. 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.

[33]

R. Nathan and H. Muller-Landau, Spatial patterns of seed dispersal, their determinants and consequences for recruitment,, Trends Ecol. Evol., 15 (2000), 278. 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. 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. 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. 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. doi: 10.1017/S0308210500010258.

[39]

F. Rothe, Convergence to pushed fronts,, Rocky Mountain J. Math., 11 (1981), 617. 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. 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. 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. doi: 10.1515/crll.1980.316.54.

[43]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. 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. 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. 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.

[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. 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. 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. doi: 10.1007/s00285-002-0169-3.

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