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

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  • 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.

    Mathematics Subject Classification: Primary: 35B40, 92D25; Secondary: 35P05, 60J80.

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  •   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.
      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.
      F. E. Browder , On the spectral theory of elliptic differential operators. I, Mathematische Annalen, 142 (1960) , 22-130.  doi: 10.1007/BF01343363.
      R. Bürger , Perturbations of positive semigroups and applications to population genetics, Mathematische Zeitschrift, 197 (1988) , 259-272.  doi: 10.1007/BF01215194.
      L. Burlando , Monotonicity of spectral radius for positive operators on ordered Banach spaces, Archiv der Mathematik, 56 (1991) , 49-57.  doi: 10.1007/BF01190081.
      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.
      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.
      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.
      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.
      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.
      D. Dawson , Measure-valued Markov processes, École d'été de Probabilités de Saint-Flour XXI-1991, 1541 (1993) , 1-260.  doi: 10.1007/BFb0084190.
      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.
      H. Von Foerster, Some Remarks on Changing Populations, Grune and Stratton, 1959.
      N. Gao , Extensions of Perron-Frobenius theory, Positivity, 56 (2013) , 965-977.  doi: 10.1007/s11117-012-0215-3.
      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.
      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.
      M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics: Models, Methods and Numerics, Springer, 2017.
      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.
      T. Kato, Perturbation Theory for Linear Operators, Springer Science & Business Media, 2013.
      M. G. Krein  and  M. A. Rutman , Population-size-dependent and age-dependent branching processes, Uspekhi Matematicheskikh Nauk, 3 (1948) , 3-95. 
      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.
      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.
      J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, 2014.
      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.
      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.
      B. Perthame, Transport Equations in Biology, Springer Science & Business Media, 2006.
      B. Perthame  and  S. K. Tumuluri , Nonlinear renewal equations, Selected Topics in Cancer Modeling, (2008) , 65-96. 
      S. T. Rachev, Probability Metrics and Stability of Stochastic Processes, Wiley, 1991.
      H. Schaefer and M. P. Wolff, Topological Vector Spaces, Graduate Texts in Mathematics, 1971.
      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.
      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).
      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.
      C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2003.
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