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

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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 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.
[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.
[3]	F. E. Browder, On the spectral theory of elliptic differential operators. I, Mathematische Annalen, 142 (1960), 22-130. doi: 10.1007/BF01343363.
[4]	R. Bürger, Perturbations of positive semigroups and applications to population genetics, Mathematische Zeitschrift, 197 (1988), 259-272. doi: 10.1007/BF01215194.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	H. Von Foerster, Some Remarks on Changing Populations, Grune and Stratton, 1959.
[14]	N. Gao, Extensions of Perron-Frobenius theory, Positivity, 56 (2013), 965-977. doi: 10.1007/s11117-012-0215-3.
[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.
[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.
[17]	M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics: Models, Methods and Numerics, Springer, 2017.
[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.
[19]	T. Kato, Perturbation Theory for Linear Operators, Springer Science & Business Media, 2013.
[20]	M. G. Krein and M. A. Rutman, Population-size-dependent and age-dependent branching processes, Uspekhi Matematicheskikh Nauk, 3 (1948), 3-95.
[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.
[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.
[23]	J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, 2014.
[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.
[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.
[26]	B. Perthame, Transport Equations in Biology, Springer Science & Business Media, 2006.
[27]	B. Perthame and S. K. Tumuluri, Nonlinear renewal equations, Selected Topics in Cancer Modeling, (2008), 65-96.
[28]	S. T. Rachev, Probability Metrics and Stability of Stochastic Processes, Wiley, 1991.
[29]	H. Schaefer and M. P. Wolff, Topological Vector Spaces, Graduate Texts in Mathematics, 1971.
[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.
[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).
[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.
[33]	C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2003.

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.
[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.
[3]	F. E. Browder, On the spectral theory of elliptic differential operators. I, Mathematische Annalen, 142 (1960), 22-130. doi: 10.1007/BF01343363.
[4]	R. Bürger, Perturbations of positive semigroups and applications to population genetics, Mathematische Zeitschrift, 197 (1988), 259-272. doi: 10.1007/BF01215194.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	H. Von Foerster, Some Remarks on Changing Populations, Grune and Stratton, 1959.
[14]	N. Gao, Extensions of Perron-Frobenius theory, Positivity, 56 (2013), 965-977. doi: 10.1007/s11117-012-0215-3.
[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.
[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.
[17]	M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics: Models, Methods and Numerics, Springer, 2017.
[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.
[19]	T. Kato, Perturbation Theory for Linear Operators, Springer Science & Business Media, 2013.
[20]	M. G. Krein and M. A. Rutman, Population-size-dependent and age-dependent branching processes, Uspekhi Matematicheskikh Nauk, 3 (1948), 3-95.
[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.
[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.
[23]	J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, 2014.
[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.
[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.
[26]	B. Perthame, Transport Equations in Biology, Springer Science & Business Media, 2006.
[27]	B. Perthame and S. K. Tumuluri, Nonlinear renewal equations, Selected Topics in Cancer Modeling, (2008), 65-96.
[28]	S. T. Rachev, Probability Metrics and Stability of Stochastic Processes, Wiley, 1991.
[29]	H. Schaefer and M. P. Wolff, Topological Vector Spaces, Graduate Texts in Mathematics, 1971.
[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.
[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).
[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.
[33]	C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2003.

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