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Macroalgal allelopathy in the emergence of coral diseases
Concentration phenomenon in some non-local equation
1. | BioSP, INRA Centre de Recherche PACA, 228 route de l'Aérodrome, Domaine Saint Paul -Site Agroparc, 84914 AVIGNON Cedex 9, France |
2. | CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, Place du Maréchal De Lattre De Tassigny, 75775 Paris cedex 16, France |
$\partial_t u(t,x)\\=\int_{\Omega }m(x,y)\left(u(t,y)-u(t,x)\right)\,dy+\left(a(x)-\int_{\Omega }k(x,y)u(t,y)\,dy\right)u(t,x),$ |
$u(0,\cdot)=u_0$ |
$\Omega $ |
$\Omega $ |
$k$ |
$m$ |
$a$ |
$u$ |
$\Omega $ |
$\mathbb{R}^N$ |
$k$ |
$m$ |
$a$ |
$k$ |
$x$ |
$k(x,y)=k(y)$ |
$d\mu$ |
$\Omega $ |
$\mathbb{M}(\Omega )$ |
$d\mu$ |
$L^1(\Omega )\cap L^{\infty}(\Omega )$ |
$L^2(\Omega )$ |
References:
[1] |
U. M. Asher, S. J. Ruuth and B. T. R. Wetton,
Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.
doi: 10.1137/0732037. |
[2] |
U. M. Asher, S. J. Ruuth and R. J. Spiteri,
Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25 (1997), 151-167.
doi: 10.1016/S0168-9274(97)00056-1. |
[3] |
G. Barles and B. Perthame,
Dirac concentrations in Lotka-Volterra parabolic PDEs, Indiana Univ. Math. J., 57 (2008), 3275-3301.
doi: 10.1512/iumj.2008.57.3398. |
[4] |
H. Berestycki, J. Coville and H. Vo,
Persistence criteria for populations with non-local dispersion, Journal of Mathematical Biology(7), 72 (2016), 1693-1745.
doi: 10.1007/s00285-015-0911-2. |
[5] |
H. Berestycki, J. Coville and H. Vo,
On the definition and the properties of the principal eigenvalue of some nonlocal operators, Journal of Functional Analysis, 271 (2016), 2701-2751.
doi: 10.1016/j.jfa.2016.05.017. |
[6] |
H. Berestycki, F. Hamel and L. Rossi,
Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl.(4), 186 (2007), 469-507.
doi: 10.1007/s10231-006-0015-0. |
[7] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[8] |
H. Berestycki and L. Rossi,
On the principal eigenvalue of elliptic operators in $\mathbb{R}^N$ and applications, J. Eur. Math. Soc. (JEMS), 8 (2006), 195-215.
doi: 10.4171/JEMS/47. |
[9] |
Reaction-diffusion equations for population dynamics with forced speed. Ⅰ. The case
of the whole space, Discrete Contin. Dyn. Syst. , 21 (2008), 41–67. |
[10] |
R. Bürger,
The Mathematical Theory of Selection, Recombination, and Mutation Wiley series in mathematical and computational biology, John Wiley, 2000. |
[11] |
R. Bürger and J. Hofbauer,
Mutation load and mutation-selection-balance in quantitative genetic traits, J. Math. Biol., 32 (1994), 193-218.
|
[12] |
A. Calsina and S. Cuadrado,
Stationary solutions of a selection mutation model: The pure mutation case, Math. Models Methods Appl. Sci., 15 (2005), 1091-1117.
doi: 10.1142/S0218202505000637. |
[13] |
Asymptotic stability of equilibria of selection-mutation equations, J. Math. Biol. , 54
(2007), 489–511. |
[14] |
A. Calsina, S. Cuadrado, L. Desvillettes and G. Raoul,
Asymptotics of steady states of a selection-mutation equation for small mutation rate, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1123-1146.
doi: 10.1017/S0308210510001629. |
[15] |
N. Champagnat, R. Ferrière and S. Méléard, Individual-based probabilistic models of adaptive evolution and various scaling approximations, Seminar on Stochastic Analysis, Random
Fields and Applications V (R. C. Dalang, F. Russo, and M. Dozzi, eds. ), Progress in Probability, vol. 59, Birkha¨user Basel, 2008, 75–113. |
[16] |
J. Coville,
On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.
doi: 10.1016/j.jde.2010.07.003. |
[17] |
Convergence to equilibrium for positive solutions of some mutation-selection model,
preprint 2013. arXiv: 1308.6471. |
[18] |
Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math.
Lett. , 26 (2013), 831–835. |
[19] |
Nonlocal refuge model with a partial control, Discrete Contin. Dynam. Systems, 35
(2015), 1421–1446. |
[20] |
J. Coville, J. Dávila and S. Martínez,
Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223.
doi: 10.1016/j.anihpc.2012.07.005. |
[21] |
L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul,
On selection dynamics for continuous structured populations, Comm. Math. Sci., 6 (2008), 729-747.
doi: 10.4310/CMS.2008.v6.n3.a10. |
[22] |
O. Diekmann,
A beginner's guide to adaptive dynamics, Banach Center Publ., 63 (2003), 47-86.
|
[23] |
O. Diekmann, P. E. Jabin, S. Mischler and B. Perthame,
The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theoret. Population Biol., 67 (2005), 257-271.
doi: 10.1016/j.tpb.2004.12.003. |
[24] |
N. Fournier and S. Méléard,
A microscopic probabilistic description of a locally regulated population and macroscopic approximations, Ann. Appl. Probab., 14 (2004), 1880-1919.
doi: 10.1214/105051604000000882. |
[25] |
F. Hecht,
New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.
|
[26] |
P. E. Jabin and G. Raoul,
On selection dynamics for competitive interactions, J. Math. Biol., 63 (2011), 493-517.
doi: 10.1007/s00285-010-0370-8. |
[27] |
A. Lorz, S. Mirrahimi and B. Perthame,
Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1071-1098.
doi: 10.1080/03605302.2010.538784. |
[28] |
S. Méléard and S. Mirrahimi,
Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity, Comm. Partial Differential Equations, 40 (2015), 957-993.
doi: 10.1080/03605302.2014.963606. |
[29] |
P. Michel, S. Mischler and B. Perthame,
General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl.(9), 84 (2005), 1235-1260.
doi: 10.1016/j.matpur.2005.04.001. |
[30] |
S. Mirrahimi and G. Raoul,
Dynamics of sexual populations structured by a space variable and a phenotypical trait, Theoret. Population Biol., 84 (2013), 87-103.
doi: 10.1016/j.tpb.2012.12.003. |
[31] |
B. Perthame, From differential equations to structured population dynamics, Transport Equations in Biology, Frontiers in Mathematics, vol. 12, Birkhaüser Basel, 2007, pp. 1–26. |
[32] |
G. Raoul,
Long time evolution of populations under selection and vanishing mutations, Acta Appl. Math., 114 (2011), 1-14.
doi: 10.1007/s10440-011-9603-0. |
[33] |
Local stability of evolutionary attractors for continuous structured populations,
Monatsh. Math. , 165 (2012), 117–144. |
[34] |
W. Shen and X. Xie,
On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665-1696.
doi: 10.3934/dcds.2015.35.1665. |
[35] |
W. Shen and A. Zhang,
Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.
doi: 10.1016/j.jde.2010.04.012. |
show all references
References:
[1] |
U. M. Asher, S. J. Ruuth and B. T. R. Wetton,
Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.
doi: 10.1137/0732037. |
[2] |
U. M. Asher, S. J. Ruuth and R. J. Spiteri,
Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25 (1997), 151-167.
doi: 10.1016/S0168-9274(97)00056-1. |
[3] |
G. Barles and B. Perthame,
Dirac concentrations in Lotka-Volterra parabolic PDEs, Indiana Univ. Math. J., 57 (2008), 3275-3301.
doi: 10.1512/iumj.2008.57.3398. |
[4] |
H. Berestycki, J. Coville and H. Vo,
Persistence criteria for populations with non-local dispersion, Journal of Mathematical Biology(7), 72 (2016), 1693-1745.
doi: 10.1007/s00285-015-0911-2. |
[5] |
H. Berestycki, J. Coville and H. Vo,
On the definition and the properties of the principal eigenvalue of some nonlocal operators, Journal of Functional Analysis, 271 (2016), 2701-2751.
doi: 10.1016/j.jfa.2016.05.017. |
[6] |
H. Berestycki, F. Hamel and L. Rossi,
Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl.(4), 186 (2007), 469-507.
doi: 10.1007/s10231-006-0015-0. |
[7] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[8] |
H. Berestycki and L. Rossi,
On the principal eigenvalue of elliptic operators in $\mathbb{R}^N$ and applications, J. Eur. Math. Soc. (JEMS), 8 (2006), 195-215.
doi: 10.4171/JEMS/47. |
[9] |
Reaction-diffusion equations for population dynamics with forced speed. Ⅰ. The case
of the whole space, Discrete Contin. Dyn. Syst. , 21 (2008), 41–67. |
[10] |
R. Bürger,
The Mathematical Theory of Selection, Recombination, and Mutation Wiley series in mathematical and computational biology, John Wiley, 2000. |
[11] |
R. Bürger and J. Hofbauer,
Mutation load and mutation-selection-balance in quantitative genetic traits, J. Math. Biol., 32 (1994), 193-218.
|
[12] |
A. Calsina and S. Cuadrado,
Stationary solutions of a selection mutation model: The pure mutation case, Math. Models Methods Appl. Sci., 15 (2005), 1091-1117.
doi: 10.1142/S0218202505000637. |
[13] |
Asymptotic stability of equilibria of selection-mutation equations, J. Math. Biol. , 54
(2007), 489–511. |
[14] |
A. Calsina, S. Cuadrado, L. Desvillettes and G. Raoul,
Asymptotics of steady states of a selection-mutation equation for small mutation rate, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1123-1146.
doi: 10.1017/S0308210510001629. |
[15] |
N. Champagnat, R. Ferrière and S. Méléard, Individual-based probabilistic models of adaptive evolution and various scaling approximations, Seminar on Stochastic Analysis, Random
Fields and Applications V (R. C. Dalang, F. Russo, and M. Dozzi, eds. ), Progress in Probability, vol. 59, Birkha¨user Basel, 2008, 75–113. |
[16] |
J. Coville,
On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.
doi: 10.1016/j.jde.2010.07.003. |
[17] |
Convergence to equilibrium for positive solutions of some mutation-selection model,
preprint 2013. arXiv: 1308.6471. |
[18] |
Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math.
Lett. , 26 (2013), 831–835. |
[19] |
Nonlocal refuge model with a partial control, Discrete Contin. Dynam. Systems, 35
(2015), 1421–1446. |
[20] |
J. Coville, J. Dávila and S. Martínez,
Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223.
doi: 10.1016/j.anihpc.2012.07.005. |
[21] |
L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul,
On selection dynamics for continuous structured populations, Comm. Math. Sci., 6 (2008), 729-747.
doi: 10.4310/CMS.2008.v6.n3.a10. |
[22] |
O. Diekmann,
A beginner's guide to adaptive dynamics, Banach Center Publ., 63 (2003), 47-86.
|
[23] |
O. Diekmann, P. E. Jabin, S. Mischler and B. Perthame,
The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theoret. Population Biol., 67 (2005), 257-271.
doi: 10.1016/j.tpb.2004.12.003. |
[24] |
N. Fournier and S. Méléard,
A microscopic probabilistic description of a locally regulated population and macroscopic approximations, Ann. Appl. Probab., 14 (2004), 1880-1919.
doi: 10.1214/105051604000000882. |
[25] |
F. Hecht,
New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.
|
[26] |
P. E. Jabin and G. Raoul,
On selection dynamics for competitive interactions, J. Math. Biol., 63 (2011), 493-517.
doi: 10.1007/s00285-010-0370-8. |
[27] |
A. Lorz, S. Mirrahimi and B. Perthame,
Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1071-1098.
doi: 10.1080/03605302.2010.538784. |
[28] |
S. Méléard and S. Mirrahimi,
Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity, Comm. Partial Differential Equations, 40 (2015), 957-993.
doi: 10.1080/03605302.2014.963606. |
[29] |
P. Michel, S. Mischler and B. Perthame,
General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl.(9), 84 (2005), 1235-1260.
doi: 10.1016/j.matpur.2005.04.001. |
[30] |
S. Mirrahimi and G. Raoul,
Dynamics of sexual populations structured by a space variable and a phenotypical trait, Theoret. Population Biol., 84 (2013), 87-103.
doi: 10.1016/j.tpb.2012.12.003. |
[31] |
B. Perthame, From differential equations to structured population dynamics, Transport Equations in Biology, Frontiers in Mathematics, vol. 12, Birkhaüser Basel, 2007, pp. 1–26. |
[32] |
G. Raoul,
Long time evolution of populations under selection and vanishing mutations, Acta Appl. Math., 114 (2011), 1-14.
doi: 10.1007/s10440-011-9603-0. |
[33] |
Local stability of evolutionary attractors for continuous structured populations,
Monatsh. Math. , 165 (2012), 117–144. |
[34] |
W. Shen and X. Xie,
On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665-1696.
doi: 10.3934/dcds.2015.35.1665. |
[35] |
W. Shen and A. Zhang,
Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.
doi: 10.1016/j.jde.2010.04.012. |




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