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June  2019, 24(6): 2551-2576. doi: 10.3934/dcdsb.2018265

On the long-time behaviour of age and trait structured population dynamics

Tristan Roget ,

CMAP-École Polytechnique, Route de Saclay, Palaiseau

* Corresponding author

Received  November 2017 Revised  April 2018 Published  June 2019 Early access  October 2018

We study the long-time behaviour of a population structured by age and a phenotypic trait under a selection-mutation dynamics. By analysing spectral properties of a family of positive operators on measure spaces, we show the existence of eventually singular stationary solutions. When the stationary measures are absolutely continuous with a continuous density, we show the convergence of the dynamics to the unique equilibrium.

Keywords: Partial differential equation, age-structured population dynamics, selection-mutation equation, Long-time behaviour, non-local operators.
Mathematics Subject Classification: Primary: 35B40, 92D25; Secondary: 35P05, 60J80.
Citation: Tristan Roget. On the long-time behaviour of age and trait structured population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2551-2576. doi: 10.3934/dcdsb.2018265
References:
[1]

A. S. Ackleh, J. Cleveland and H. R. Thieme, Population dynamics under selection and mutation: Long-time behavior for differential equations in measure spaces, Journal of Differential Equations, 261 (2016), 1472-1505.  doi: 10.1016/j.jde.2016.04.008.  Google Scholar

[2]

O. Bonnefon, J. 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.  Google Scholar

[3]

F. E. Browder, On the spectral theory of elliptic differential operators. I, Mathematische Annalen, 142 (1960), 22-130.  doi: 10.1007/BF01343363.  Google Scholar

[4]

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

[5]

L. Burlando, Monotonicity of spectral radius for positive operators on ordered Banach spaces, Archiv der Mathematik, 56 (1991), 49-57.  doi: 10.1007/BF01190081.  Google Scholar

[6]

A. Calsina and J. M. Palmada, Steady states of a selection-mutation model for an age structured population, Journal of Mathematical Analysis and Applications, 400 (2013), 386-395.  doi: 10.1016/j.jmaa.2012.11.042.  Google Scholar

[7]

J. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.  Google Scholar

[8]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, Journal of Differential Equations, 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[9]

J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[10]

J. Coville, Singular measure as principal eigenfunction of some nonlocal operators, Applied Mathematics Letters, 26 (2013), 831-835.  doi: 10.1016/j.aml.2013.03.005.  Google Scholar

[11]

D. Dawson, Measure-valued Markov processes, École d'été de Probabilités de Saint-Flour XXI-1991, 1541 (1993), 1-260.  doi: 10.1007/BFb0084190.  Google Scholar

[12]

L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Communications in Mathematical Sciences, 6 (2008), 729-747.  doi: 10.4310/CMS.2008.v6.n3.a10.  Google Scholar

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H. Von Foerster, Some Remarks on Changing Populations, Grune and Stratton, 1959. Google Scholar

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N. Gao, Extensions of Perron-Frobenius theory, Positivity, 56 (2013), 965-977.  doi: 10.1007/s11117-012-0215-3.  Google Scholar

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E. M. Gurtin and R. MacCamy, Non-linear age-dependent population dynamics, Archive for Rational Mechanics and Analysis, 54 (1974), 281-300.  doi: 10.1007/BF00250793.  Google Scholar

[16]

P. Gwiazda and E. Wiedemann, Generalized entropy method for the renewal equation with measure data, Commun. Math. Sci., 15 (2017), 577-586.  doi: 10.4310/CMS.2017.v15.n2.a13.  Google Scholar

[17]

M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics: Models, Methods and Numerics, Springer, 2017. Google Scholar

[18]

P. Jagers and F. Klebaner, Population-size-dependent and age-dependent branching processes, Stochastic Processes and their Applications, 87 (2000), 235-254.  doi: 10.1016/S0304-4149(99)00111-8.  Google Scholar

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T. Kato, Perturbation Theory for Linear Operators, Springer Science & Business Media, 2013. Google Scholar

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M. G. Krein and M. A. Rutman, Population-size-dependent and age-dependent branching processes, Uspekhi Matematicheskikh Nauk, 3 (1948), 3-95.   Google Scholar

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J. Kristensen and F. Rindler, Relaxation of signed integral functionals in BV, Calculus of Variations and Partial Differential Equations, 37 (2010), 29-62.  doi: 10.1007/s00526-009-0250-5.  Google Scholar

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H. Leman, S. Meleard and S. Mirrahimi, Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 469-493.  doi: 10.3934/dcdsb.2015.20.469.  Google Scholar

[23]

J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, 2014. Google Scholar

[24]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, Journal de Mathématiques Pures Et Appliquées, 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001.  Google Scholar

[25]

S. Nordmann, B. Perthame and C. Taing, Dynamics of concentration in a population model structured by age and a phenotypical trait, Acta Applicandae Mathematicae, 155 (2018), 197-225.  doi: 10.1007/s10440-017-0151-0.  Google Scholar

[26]

B. Perthame, Transport Equations in Biology, Springer Science & Business Media, 2006. Google Scholar

[27]

B. Perthame and S. K. Tumuluri, Nonlinear renewal equations, Selected Topics in Cancer Modeling, (2008), 65-96.   Google Scholar

[28]

S. T. Rachev, Probability Metrics and Stability of Stochastic Processes, Wiley, 1991. Google Scholar

[29]

H. Schaefer and M. P. Wolff, Topological Vector Spaces, Graduate Texts in Mathematics, 1971. Google Scholar

[30]

D. Spector, Simple proofs of some results of Reshetnyak, Proceedings of the American Mathematical Society, 139 (2011), 1681-1690.  doi: 10.1090/S0002-9939-2010-10593-2.  Google Scholar

[31]

C. V. Tran, Modèles Particulaires Stochastiques Pour des Problèmes d'évolution Adaptative et Pour L'approximation de Solutions Statistiques, Université de Nanterre-Paris X, PhD, (2006). Google Scholar

[32]

C. V. Tran, Large population limit and time behaviour of a stochastic particle model describing an age-structured population, ESAIM: Probability and Statistics, 12 (2008), 345-386.  doi: 10.1051/ps:2007052.  Google Scholar

[33]

C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2003. Google Scholar

show all references

References:
[1]

A. S. Ackleh, J. Cleveland and H. R. Thieme, Population dynamics under selection and mutation: Long-time behavior for differential equations in measure spaces, Journal of Differential Equations, 261 (2016), 1472-1505.  doi: 10.1016/j.jde.2016.04.008.  Google Scholar

[2]

O. Bonnefon, J. 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.  Google Scholar

[3]

F. E. Browder, On the spectral theory of elliptic differential operators. I, Mathematische Annalen, 142 (1960), 22-130.  doi: 10.1007/BF01343363.  Google Scholar

[4]

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

[5]

L. Burlando, Monotonicity of spectral radius for positive operators on ordered Banach spaces, Archiv der Mathematik, 56 (1991), 49-57.  doi: 10.1007/BF01190081.  Google Scholar

[6]

A. Calsina and J. M. Palmada, Steady states of a selection-mutation model for an age structured population, Journal of Mathematical Analysis and Applications, 400 (2013), 386-395.  doi: 10.1016/j.jmaa.2012.11.042.  Google Scholar

[7]

J. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.  Google Scholar

[8]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, Journal of Differential Equations, 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[9]

J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 30 (2013), 179-223.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[10]

J. Coville, Singular measure as principal eigenfunction of some nonlocal operators, Applied Mathematics Letters, 26 (2013), 831-835.  doi: 10.1016/j.aml.2013.03.005.  Google Scholar

[11]

D. Dawson, Measure-valued Markov processes, École d'été de Probabilités de Saint-Flour XXI-1991, 1541 (1993), 1-260.  doi: 10.1007/BFb0084190.  Google Scholar

[12]

L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Communications in Mathematical Sciences, 6 (2008), 729-747.  doi: 10.4310/CMS.2008.v6.n3.a10.  Google Scholar

[13]

H. Von Foerster, Some Remarks on Changing Populations, Grune and Stratton, 1959. Google Scholar

[14]

N. Gao, Extensions of Perron-Frobenius theory, Positivity, 56 (2013), 965-977.  doi: 10.1007/s11117-012-0215-3.  Google Scholar

[15]

E. M. Gurtin and R. MacCamy, Non-linear age-dependent population dynamics, Archive for Rational Mechanics and Analysis, 54 (1974), 281-300.  doi: 10.1007/BF00250793.  Google Scholar

[16]

P. Gwiazda and E. Wiedemann, Generalized entropy method for the renewal equation with measure data, Commun. Math. Sci., 15 (2017), 577-586.  doi: 10.4310/CMS.2017.v15.n2.a13.  Google Scholar

[17]

M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics: Models, Methods and Numerics, Springer, 2017. Google Scholar

[18]

P. Jagers and F. Klebaner, Population-size-dependent and age-dependent branching processes, Stochastic Processes and their Applications, 87 (2000), 235-254.  doi: 10.1016/S0304-4149(99)00111-8.  Google Scholar

[19]

T. Kato, Perturbation Theory for Linear Operators, Springer Science & Business Media, 2013. Google Scholar

[20]

M. G. Krein and M. A. Rutman, Population-size-dependent and age-dependent branching processes, Uspekhi Matematicheskikh Nauk, 3 (1948), 3-95.   Google Scholar

[21]

J. Kristensen and F. Rindler, Relaxation of signed integral functionals in BV, Calculus of Variations and Partial Differential Equations, 37 (2010), 29-62.  doi: 10.1007/s00526-009-0250-5.  Google Scholar

[22]

H. Leman, S. Meleard and S. Mirrahimi, Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 469-493.  doi: 10.3934/dcdsb.2015.20.469.  Google Scholar

[23]

J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, 2014. Google Scholar

[24]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, Journal de Mathématiques Pures Et Appliquées, 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001.  Google Scholar

[25]

S. Nordmann, B. Perthame and C. Taing, Dynamics of concentration in a population model structured by age and a phenotypical trait, Acta Applicandae Mathematicae, 155 (2018), 197-225.  doi: 10.1007/s10440-017-0151-0.  Google Scholar

[26]

B. Perthame, Transport Equations in Biology, Springer Science & Business Media, 2006. Google Scholar

[27]

B. Perthame and S. K. Tumuluri, Nonlinear renewal equations, Selected Topics in Cancer Modeling, (2008), 65-96.   Google Scholar

[28]

S. T. Rachev, Probability Metrics and Stability of Stochastic Processes, Wiley, 1991. Google Scholar

[29]

H. Schaefer and M. P. Wolff, Topological Vector Spaces, Graduate Texts in Mathematics, 1971. Google Scholar

[30]

D. Spector, Simple proofs of some results of Reshetnyak, Proceedings of the American Mathematical Society, 139 (2011), 1681-1690.  doi: 10.1090/S0002-9939-2010-10593-2.  Google Scholar

[31]

C. V. Tran, Modèles Particulaires Stochastiques Pour des Problèmes d'évolution Adaptative et Pour L'approximation de Solutions Statistiques, Université de Nanterre-Paris X, PhD, (2006). Google Scholar

[32]

C. V. Tran, Large population limit and time behaviour of a stochastic particle model describing an age-structured population, ESAIM: Probability and Statistics, 12 (2008), 345-386.  doi: 10.1051/ps:2007052.  Google Scholar

[33]

C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2003. Google Scholar

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