We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the integro-differential equation
$\partial_t u(t,x)\\=\int_{\Omega }m(x,y)\left(u(t,y)-u(t,x)\right)\,dy+\left(a(x)-\int_{\Omega }k(x,y)u(t,y)\,dy\right)u(t,x),$
supplemented by the initial condition
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Figure 1.
Numerical approximation of the solution to (10) at different times for two configurations, in which only the mutation rate differs. The competition rate is constant and set to
Figure 2.
Numerical approximation of the solution of problem (10)-(11) at different times for two configurations, which differ only in their initial datum. The mutation and competition rates are constant and set respectively to
Figure 3.
Numerical approximation of the solution of problem (10)-(11) at different times. The mutation and competition rates are constant and set respectively to
Figure 4.
Numerical approximation of the solution of problem (10)-(11) at different times. The mutation and competition rates are constant and set respectively to
[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, 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., 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., 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.![]() ![]() ![]() |