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Stability and bifurcation in an age-structured model with stocking rate and time delays
On the long-time behaviour of age and trait structured population dynamics
CMAP-École Polytechnique, Route de Saclay, Palaiseau |
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.
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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