doi: 10.3934/dcdsb.2022120
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Confining integro-differential equations originating from evolutionary biology: Ground states and long time dynamics

1. 

Université de Rouen Normandie, CNRS, Laboratoire de Mathématiques Raphaël Salem, Saint-Etienne-du-Rouvray, France

2. 

Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 Avenue des États-Unis, 78035 Versailles cedex, France

* Corresponding author: Matthieu Alfaro

To the memory of Mayan Mimura, a friend and a very inspiring applied mathematician

Received  October 2021 Revised  April 2022 Early access June 2022

We consider nonlinear mutation selection models, known as replicator-mutator equations in evolutionary biology. They involve a nonlocal mutation kernel and a confining fitness potential. We prove that the long time behaviour of the Cauchy problem is determined by the principal eigenelement of the underlying linear operator. The novelties compared to the literature on these models are about the case of symmetric mutations: we propose a new milder sufficient condition for the existence of a principal eigenfunction, and we provide what is to our knowledge the first quantification of the spectral gap. We also recover existing results in the non-symmetric case, through a new approach.

Citation: Matthieu Alfaro, Pierre Gabriel, Otared Kavian. Confining integro-differential equations originating from evolutionary biology: Ground states and long time dynamics. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022120
References:
[1]

M. Alfaro and R. Carles, Explicit solutions for replicator-mutator equations: Extinction versus acceleration, SIAM J. Appl. Math., 74 (2014), 1919-1934.  doi: 10.1137/140979411.

[2]

M. Alfaro and R. Carles, Replicator-mutator equations with quadratic fitness, Proc. Amer. Math. Soc., 145 (2017), 5315-5327.  doi: 10.1090/proc/13669.

[3]

M. Alfaro and M. Veruete, Evolutionary branching via replicator–mutator equations, J. Dynam. Differential Equations, 31 (2019), 2029-2052.  doi: 10.1007/s10884-018-9692-9.

[4]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, volume 1184 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[5]

V. Bansaye, B. Cloez, P. Gabriel and A. Marguet, A non-conservative Harris ergodic theorem, J. Lond. Math. Soc., 2022. To appear.

[6]

O. BonnefonJ. Coville and G. Legendre, Concentration phenomenon in some non-local equation, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 763-781.  doi: 10.3934/dcdsb.2017037.

[7]

R. Bürger, On the maintenance of genetic variation: Global analysis of Kimura's continuum-of-alleles model, J. Math. Biol., 24 (1986), 341-351.  doi: 10.1007/BF00275642.

[8]

R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272.  doi: 10.1007/BF01215194.

[9]

R. Bürger, An integro-differential equation from population genetics and perturbations of differentiable semigroups in Fréchet spaces, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), 63-73.  doi: 10.1017/S0308210500028894.

[10]

R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000.

[11]

R. Bürger and I. M. Bomze, Stationary distributions under mutation-selection balance: Structure and properties, Adv. in Appl. Probab., 28 (1996), 227-251.  doi: 10.2307/1427919.

[12]

J. A. CañizoP. Gabriel and H. Yoldaş, Spectral gap for the growth-fragmentation equation via Harris's Theorem, SIAM J. Math. Anal., 53 (2021), 5185-5214.  doi: 10.1137/20M1338654.

[13]

N. ChampagnatR. Ferrière and S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution, Stochastic Models, 24 (2008), 2-44.  doi: 10.1080/15326340802437710.

[14]

B. Cloez and P. Gabriel, Fast, slow convergence, and concentration in the house of cards replicator-mutator model, arXiv: 2203.07924.

[15]

B. Cloez and P. Gabriel, On an irreducibility type condition for the ergodicity of nonconservative semigroups, C. R. Math. Acad. Sci. Paris, 358 (2020), 733-742.  doi: 10.5802/crmath.92.

[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]

J. Coville, Singular measure as principal eigenfunctions of some nonlocal operators, Appl. Math. Lett., 26 (2013), 831-835.  doi: 10.1016/j.aml.2013.03.005.

[18]

J. Coville and F. Hamel, On generalized principal eigenvalues of nonlocal operators with a drift, Nonlinear Anal., 193 (2020), 111569, 20 pp. doi: 10.1016/j.na.2019.07.002.

[19]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics. Springer-Verlag, New-York, 2000.

[20]

W. H. Fleming, Equilibrium distributions of continuous polygenic traits, SIAM J. Appl. Math., 36 (1979), 148-168.  doi: 10.1137/0136014.

[21]

P. Gabriel and H. Martin., Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation, Annales Henri Lebesgue, 5 (2022), 275-301.  doi: 10.5802/ahl.123.

[22]

M.-E. GilF. HamelG. Martin and L. Roques, Mathematical properties of a class of integro-differential models from population genetics, SIAM J. Appl. Math., 77 (2017), 1536-1561.  doi: 10.1137/16M1108224.

[23]

M.-E. GilF. HamelG. Martin and L. Roques, Dynamics of fitness distributions in the presence of a phenotypic optimum: an integro-differential approach, Nonlinearity, 32 (2019), 3485-3522.  doi: 10.1088/1361-6544/ab1bbe.

[24]

Q. Griette, Singular measure traveling waves in an epidemiological model with continuous phenotypes, Trans. Amer. Math. Soc., 371 (2019), 4411-4458.  doi: 10.1090/tran/7700.

[25]

F. Hamel, F. Lavigne, G. Martin and L. Roques, Dynamics of adaptation in an anisotropic phenotype-fitness landscape, Nonlinear Anal. Real World Appl., 54 (2020), 103107, 33 pp. doi: 10.1016/j.nonrwa.2020.103107.

[26]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci. USA, 54 (1965), 731-736.  doi: 10.1073/pnas.54.3.731.

[27]

R. Lande, The maintenance of genetic variability by mutation in a polygenic character with linked loci, Genetics Research, 26 (1975), 221-235. 

[28]

F. LiJ. Coville and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879-903.  doi: 10.3934/dcds.2017036.

[29]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.

[30]

M. Reed and B. Simon, Methods of Modern Mathematical Physics (vol IV): Analysis of Operators, Academic Press, 1978.

[31]

L. A. Takhtajan, Quantum Mechanics for Mathematicians, Graduate Studies in Mathematics. American Mathematical Society, 2008. doi: 10.1090/gsm/095.

[32]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lectures Notes of the Unione Matematica Italiana, Springer, 2007.

[33]

J. Y. WakanoT. Funaki and S. Yokoyama, Derivation of replicator-mutator equations from a model in population genetics, Jpn. J. Ind. Appl. Math., 34 (2017), 473-488.  doi: 10.1007/s13160-017-0249-9.

[34]

K. Yosida, Functional Analysis, Number 123 in Die Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York, New York, 1974.

show all references

References:
[1]

M. Alfaro and R. Carles, Explicit solutions for replicator-mutator equations: Extinction versus acceleration, SIAM J. Appl. Math., 74 (2014), 1919-1934.  doi: 10.1137/140979411.

[2]

M. Alfaro and R. Carles, Replicator-mutator equations with quadratic fitness, Proc. Amer. Math. Soc., 145 (2017), 5315-5327.  doi: 10.1090/proc/13669.

[3]

M. Alfaro and M. Veruete, Evolutionary branching via replicator–mutator equations, J. Dynam. Differential Equations, 31 (2019), 2029-2052.  doi: 10.1007/s10884-018-9692-9.

[4]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, volume 1184 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[5]

V. Bansaye, B. Cloez, P. Gabriel and A. Marguet, A non-conservative Harris ergodic theorem, J. Lond. Math. Soc., 2022. To appear.

[6]

O. BonnefonJ. Coville and G. Legendre, Concentration phenomenon in some non-local equation, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 763-781.  doi: 10.3934/dcdsb.2017037.

[7]

R. Bürger, On the maintenance of genetic variation: Global analysis of Kimura's continuum-of-alleles model, J. Math. Biol., 24 (1986), 341-351.  doi: 10.1007/BF00275642.

[8]

R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272.  doi: 10.1007/BF01215194.

[9]

R. Bürger, An integro-differential equation from population genetics and perturbations of differentiable semigroups in Fréchet spaces, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), 63-73.  doi: 10.1017/S0308210500028894.

[10]

R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000.

[11]

R. Bürger and I. M. Bomze, Stationary distributions under mutation-selection balance: Structure and properties, Adv. in Appl. Probab., 28 (1996), 227-251.  doi: 10.2307/1427919.

[12]

J. A. CañizoP. Gabriel and H. Yoldaş, Spectral gap for the growth-fragmentation equation via Harris's Theorem, SIAM J. Math. Anal., 53 (2021), 5185-5214.  doi: 10.1137/20M1338654.

[13]

N. ChampagnatR. Ferrière and S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution, Stochastic Models, 24 (2008), 2-44.  doi: 10.1080/15326340802437710.

[14]

B. Cloez and P. Gabriel, Fast, slow convergence, and concentration in the house of cards replicator-mutator model, arXiv: 2203.07924.

[15]

B. Cloez and P. Gabriel, On an irreducibility type condition for the ergodicity of nonconservative semigroups, C. R. Math. Acad. Sci. Paris, 358 (2020), 733-742.  doi: 10.5802/crmath.92.

[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]

J. Coville, Singular measure as principal eigenfunctions of some nonlocal operators, Appl. Math. Lett., 26 (2013), 831-835.  doi: 10.1016/j.aml.2013.03.005.

[18]

J. Coville and F. Hamel, On generalized principal eigenvalues of nonlocal operators with a drift, Nonlinear Anal., 193 (2020), 111569, 20 pp. doi: 10.1016/j.na.2019.07.002.

[19]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics. Springer-Verlag, New-York, 2000.

[20]

W. H. Fleming, Equilibrium distributions of continuous polygenic traits, SIAM J. Appl. Math., 36 (1979), 148-168.  doi: 10.1137/0136014.

[21]

P. Gabriel and H. Martin., Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation, Annales Henri Lebesgue, 5 (2022), 275-301.  doi: 10.5802/ahl.123.

[22]

M.-E. GilF. HamelG. Martin and L. Roques, Mathematical properties of a class of integro-differential models from population genetics, SIAM J. Appl. Math., 77 (2017), 1536-1561.  doi: 10.1137/16M1108224.

[23]

M.-E. GilF. HamelG. Martin and L. Roques, Dynamics of fitness distributions in the presence of a phenotypic optimum: an integro-differential approach, Nonlinearity, 32 (2019), 3485-3522.  doi: 10.1088/1361-6544/ab1bbe.

[24]

Q. Griette, Singular measure traveling waves in an epidemiological model with continuous phenotypes, Trans. Amer. Math. Soc., 371 (2019), 4411-4458.  doi: 10.1090/tran/7700.

[25]

F. Hamel, F. Lavigne, G. Martin and L. Roques, Dynamics of adaptation in an anisotropic phenotype-fitness landscape, Nonlinear Anal. Real World Appl., 54 (2020), 103107, 33 pp. doi: 10.1016/j.nonrwa.2020.103107.

[26]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci. USA, 54 (1965), 731-736.  doi: 10.1073/pnas.54.3.731.

[27]

R. Lande, The maintenance of genetic variability by mutation in a polygenic character with linked loci, Genetics Research, 26 (1975), 221-235. 

[28]

F. LiJ. Coville and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879-903.  doi: 10.3934/dcds.2017036.

[29]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.

[30]

M. Reed and B. Simon, Methods of Modern Mathematical Physics (vol IV): Analysis of Operators, Academic Press, 1978.

[31]

L. A. Takhtajan, Quantum Mechanics for Mathematicians, Graduate Studies in Mathematics. American Mathematical Society, 2008. doi: 10.1090/gsm/095.

[32]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lectures Notes of the Unione Matematica Italiana, Springer, 2007.

[33]

J. Y. WakanoT. Funaki and S. Yokoyama, Derivation of replicator-mutator equations from a model in population genetics, Jpn. J. Ind. Appl. Math., 34 (2017), 473-488.  doi: 10.1007/s13160-017-0249-9.

[34]

K. Yosida, Functional Analysis, Number 123 in Die Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York, New York, 1974.

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