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Inside dynamics of solutions of integro-differential equations

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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.
    Mathematics Subject Classification: Primary: 35R09, 45K05; Secondary: 35B06, 35K57.

    Citation:

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  • [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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