# American Institute of Mathematical Sciences

January  2012, 11(1): 83-96. doi: 10.3934/cpaa.2012.11.83

## Existence of nontrivial steady states for populations structured with respect to space and a continuous trait

 1 Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, 1040 Wien 2 ENS Cachan, CMLA, IUF & CNRS, PRES UniverSud, 61, Av. du Pdt Wilson, 94235 Cachan Cedex 3 MAPMO, Université d'Orléans, F-45067 Orléans Cedex, France

Received  January 2010 Revised  July 2010 Published  September 2011

We prove the existence of nontrivial steady states to reaction-diffusion equations with a continuous parameter appearing in selection/mutation/competition/migration models for populations, which are structured both with respect to space and a continuous trait.
Citation: Anton Arnold, Laurent Desvillettes, Céline Prévost. Existence of nontrivial steady states for populations structured with respect to space and a continuous trait. Communications on Pure and Applied Analysis, 2012, 11 (1) : 83-96. doi: 10.3934/cpaa.2012.11.83
##### References:
 [1] H. Brezis, "Analyse Fonctionnelle," Masson, Paris, 1987. [2] F. Brezzi and G. Gilardi, "Fundamentals of P.D.E. for Numerical Analysis," preprint n. 446 of Istituto di Analisi Numerica, Pavia, 1984. [3] A. Calsina and S. Cuadrado, Asymptotic stability of equilibria of selection-mutation equations, J. Math. Biol., 54 (2007), 489-511. doi: doi:10.1007/s00285-006-0056-4. [4] J. Carrillo, L. Desvillettes and K. Fellner, Exponential decay towards equilibrium for the inhomogeneous Aizenman-Bak model, Commun. Math. Phys., 278 (2008), 433-451. doi: doi:10.1007/s00220-007-0404-2. [5] J. Carrillo, L. Desvillettes and K. Fellner, Fast-reaction limit for the inhomogeneous Aizenman-Bak model, Kinetic and Related Models, 1 (2008), 127-137. [6] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," vol 1. Physical Origins and Classical Methods, Springer-Verlag, Berlin-Heidelberg-New York, 1990. [7] L. Desvillettes, R. Ferrières and C. Prévost, "Infinite Dimensional Reaction-Diffusion for Population Dynamics,", preprint n. 2003-04 du CMLA, (): 2003. [8] L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On mutation-selection dynamics, Commun. Math. Sc., 6 (2008), 729-747. [9] O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol., 67 (2005), 257-271. doi: doi:10.1016/j.tpb.2004.12.003. [10] P. Laurençot and S. Mischler, Global existence for the discrete diffusive coagulation-fragmentation equations in L1, Rev. Mat. Iberoamericana, 18 (2002), 731-745. [11] P. Laurençot and S. Mischler, The continuous coagulation-fragmentation equations with diffusion, Arch. Rational Mech. Anal., 162 (2002), 45-99. [12] G. Raoul, Local stability of evolutionary attractors for continuous structured populations, to appear in Monatshefte für Mathematik, 2011. [13] F. Rothe, "Global Solutions of Reaction-Diffusion Systems," Lecture Notes in Mathematics, vol. 1072. Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984. [14] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Second edition, Grundlehren der Mathematischen Wissenschaften, vol. 258. Springer-Verlag, New York, 1994.

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##### References:
 [1] H. Brezis, "Analyse Fonctionnelle," Masson, Paris, 1987. [2] F. Brezzi and G. Gilardi, "Fundamentals of P.D.E. for Numerical Analysis," preprint n. 446 of Istituto di Analisi Numerica, Pavia, 1984. [3] A. Calsina and S. Cuadrado, Asymptotic stability of equilibria of selection-mutation equations, J. Math. Biol., 54 (2007), 489-511. doi: doi:10.1007/s00285-006-0056-4. [4] J. Carrillo, L. Desvillettes and K. Fellner, Exponential decay towards equilibrium for the inhomogeneous Aizenman-Bak model, Commun. Math. Phys., 278 (2008), 433-451. doi: doi:10.1007/s00220-007-0404-2. [5] J. Carrillo, L. Desvillettes and K. Fellner, Fast-reaction limit for the inhomogeneous Aizenman-Bak model, Kinetic and Related Models, 1 (2008), 127-137. [6] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," vol 1. Physical Origins and Classical Methods, Springer-Verlag, Berlin-Heidelberg-New York, 1990. [7] L. Desvillettes, R. Ferrières and C. Prévost, "Infinite Dimensional Reaction-Diffusion for Population Dynamics,", preprint n. 2003-04 du CMLA, (): 2003. [8] L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On mutation-selection dynamics, Commun. Math. Sc., 6 (2008), 729-747. [9] O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol., 67 (2005), 257-271. doi: doi:10.1016/j.tpb.2004.12.003. [10] P. Laurençot and S. Mischler, Global existence for the discrete diffusive coagulation-fragmentation equations in L1, Rev. Mat. Iberoamericana, 18 (2002), 731-745. [11] P. Laurençot and S. Mischler, The continuous coagulation-fragmentation equations with diffusion, Arch. Rational Mech. Anal., 162 (2002), 45-99. [12] G. Raoul, Local stability of evolutionary attractors for continuous structured populations, to appear in Monatshefte für Mathematik, 2011. [13] F. Rothe, "Global Solutions of Reaction-Diffusion Systems," Lecture Notes in Mathematics, vol. 1072. Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984. [14] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Second edition, Grundlehren der Mathematischen Wissenschaften, vol. 258. Springer-Verlag, New York, 1994.
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