May  2017, 22(3): 763-781. doi: 10.3934/dcdsb.2017037

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

* Corresponding author: Jérôme Coville

Dedicated to Professor Stephen Cantrell, with all our admiration

Received  September 2015 Revised  April 2016 Published  January 2017

Fund Project: The research leading to these results has received funding from the french ANR program under the "ANR JCJC" project MODEVOL: ANR-13-JS01-0009 held by Gael Raoul and the "ANR DEFI" project NONLOCAL: ANR-14-CE25-0013 held by Fran¸cois Hamel. J. Coville wants to thank G. Raoul for interesting discussions on this topic.

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
$u(0,\cdot)=u_0$
in
$\Omega $
, where the domain
$\Omega $
is a, the functions
$k$
and
$m$
are non-negative kernels satisfying integrability conditions and the function
$a$
is continuous. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function
$u$
represents the density of individuals characterized by the trait, the domain of trait values
$\Omega $
is a bounded subset of
$\mathbb{R}^N$
, the kernels
$k$
and
$m$
respectively account for the competition between individuals and the mutations occurring in every generation, and the function
$a$
represents a growth rate. When the competition is independent of the trait, that is, the kernel
$k$
is independent of
$x$
, (
$k(x,y)=k(y)$
), we construct a positive stationary solution which belongs to
$d\mu$
inthe space of Radon measures on
$\Omega $
.
$\mathbb{M}(\Omega )$
.Moreover, in the case where this measure
$d\mu$
is regular and bounded, we prove its uniqueness and show that, for any non-negative initial datum in
$L^1(\Omega )\cap L^{\infty}(\Omega )$
, the solution of the Cauchy problem converges to this limit measure in
$L^2(\Omega )$
. We also exhibit an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. The numerical simulations seem to reveal a dependence of the limit measure with respect to the initial datum.
Citation: Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037
References:
[1]

U. M. AsherS. 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.  Google Scholar

[2]

U. M. AsherS. 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.  Google Scholar

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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.  Google Scholar

[4]

H. BerestyckiJ. 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.  Google Scholar

[5]

H. BerestyckiJ. 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.  Google Scholar

[6]

H. BerestyckiF. 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.  Google Scholar

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Reaction-diffusion equations for population dynamics with forced speed. Ⅰ. The case of the whole space, Discrete Contin. Dyn. Syst. , 21 (2008), 41–67. Google Scholar

[10]

R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation Wiley series in mathematical and computational biology, John Wiley, 2000. Google Scholar

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R. Bürger and J. Hofbauer, Mutation load and mutation-selection-balance in quantitative genetic traits, J. Math. Biol., 32 (1994), 193-218.   Google Scholar

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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.  Google Scholar

[13]

Asymptotic stability of equilibria of selection-mutation equations, J. Math. Biol. , 54 (2007), 489–511. Google Scholar

[14]

A. CalsinaS. CuadradoL. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

Convergence to equilibrium for positive solutions of some mutation-selection model, preprint 2013. arXiv: 1308.6471. Google Scholar

[18]

Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett. , 26 (2013), 831–835. Google Scholar

[19]

Nonlocal refuge model with a partial control, Discrete Contin. Dynam. Systems, 35 (2015), 1421–1446. Google Scholar

[20]

J. CovilleJ. 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.  Google Scholar

[21]

L. DesvillettesP. E. JabinS. 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.  Google Scholar

[22]

O. Diekmann, A beginner's guide to adaptive dynamics, Banach Center Publ., 63 (2003), 47-86.   Google Scholar

[23]

O. DiekmannP. E. JabinS. 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.  Google Scholar

[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.  Google Scholar

[25]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.   Google Scholar

[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.  Google Scholar

[27]

A. LorzS. 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.  Google Scholar

[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.  Google Scholar

[29]

P. MichelS. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[33]

Local stability of evolutionary attractors for continuous structured populations, Monatsh. Math. , 165 (2012), 117–144. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

U. M. AsherS. 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.  Google Scholar

[2]

U. M. AsherS. 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.  Google Scholar

[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.  Google Scholar

[4]

H. BerestyckiJ. 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.  Google Scholar

[5]

H. BerestyckiJ. 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.  Google Scholar

[6]

H. BerestyckiF. 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.  Google Scholar

[7]

H. BerestyckiL. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation Wiley series in mathematical and computational biology, John Wiley, 2000. Google Scholar

[11]

R. Bürger and J. Hofbauer, Mutation load and mutation-selection-balance in quantitative genetic traits, J. Math. Biol., 32 (1994), 193-218.   Google Scholar

[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.  Google Scholar

[13]

Asymptotic stability of equilibria of selection-mutation equations, J. Math. Biol. , 54 (2007), 489–511. Google Scholar

[14]

A. CalsinaS. CuadradoL. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

Convergence to equilibrium for positive solutions of some mutation-selection model, preprint 2013. arXiv: 1308.6471. Google Scholar

[18]

Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett. , 26 (2013), 831–835. Google Scholar

[19]

Nonlocal refuge model with a partial control, Discrete Contin. Dynam. Systems, 35 (2015), 1421–1446. Google Scholar

[20]

J. CovilleJ. 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.  Google Scholar

[21]

L. DesvillettesP. E. JabinS. 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.  Google Scholar

[22]

O. Diekmann, A beginner's guide to adaptive dynamics, Banach Center Publ., 63 (2003), 47-86.   Google Scholar

[23]

O. DiekmannP. E. JabinS. 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.  Google Scholar

[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.  Google Scholar

[25]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.   Google Scholar

[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.  Google Scholar

[27]

A. LorzS. 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.  Google Scholar

[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.  Google Scholar

[29]

P. MichelS. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[33]

Local stability of evolutionary attractors for continuous structured populations, Monatsh. Math. , 165 (2012), 117–144. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $1$ and the growth rate function achieves its maximum only at the origin while the initial datum $u_0$ is uniform with value 1. We have set $\rho=1$ for the first simulation (subfigures (A) to (D)), and $\rho=0.1$ for the second one (subfigures (E) to (H)). In both situations, we observe the convergence to a stationary solution, either to a regular measure (see subfigure (D)) or to a singular measure with one Dirac mass at the origin (see subfigure (H)), the latter being characteristic of a concentration phenomenon. In the regular case (subfigures (A) to (D)), the stationarity being attained numerically around $t=590$
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 $2$ and $1$, the growth rate function achieves its maximum at four points and the initial datum $u_0$ is such that it vanishes on three (subfigures (A) to (D)) or two (subfigures (E) to (H)) of these points. In both cases, rapid convergence of the approximate solution towards an identical regular stationary state is observed, the numerical stationarity being attained around $t=85$
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 $0.01$ and $1$, the growth rate function achieves its maximum at four points and the initial datum $u_0$ is such that it vanishes on three of these four points. We observe a slow convergence of the numerical solution towards the approximation of a singular stationary measure containing a single Dirac mass. The approximate solution continues to take large increasing values in a single element at $t=1000$)
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 $0.01$ and $1$, the growth rate function achieves its maximum at four points and the initial datum $u_0$ is such that it vanishes on two of these four points. We observe a slow convergence of the numerical solution towards the approximation of a singular stationary measure containing two Dirac masses
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