# American Institute of Mathematical Sciences

May  2013, 18(3): 753-768. doi: 10.3934/dcdsb.2013.18.753

## Adaptation and migration of a population between patches

 1 CMAP, Ecole Polytechnique, CNRS, INRIA, Route de Saclay, 91128 Palaiseau Cedex, France

Received  April 2012 Revised  September 2012 Published  December 2012

A Hamilton-Jacobi formulation has been established previously for phenotypically structured population models where the solution concentrates as Dirac masses in the limit of small diffusion. Is it possible to extend this approach to spatial models? Are the limiting solutions still in the form of sums of Dirac masses? Does the presence of several habitats lead to polymorphic situations? We study the stationary solutions of a structured population model, while the population is structured by continuous phenotypical traits and discrete positions in space. The growth term varies from one habitable zone to another, for instance because of a change in the temperature. The individuals can migrate from one zone to another with a constant rate. The mathematical modeling of this problem, considering mutations between phenotypical traits and competitive interaction of individuals within each zone via a single resource, leads to a system of coupled parabolic integro-differential equations. We study the asymptotic behavior of the stationary solutions to this model in the limit of small mutations. The limit, which is a sum of Dirac masses, can be described with the help of an effective Hamiltonian. The presence of migration can modify the dominant traits and lead to polymorphic situations.
Citation: Sepideh Mirrahimi. Adaptation and migration of a population between patches. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 753-768. doi: 10.3934/dcdsb.2013.18.753
##### References:
 [1] V. Bansaye and A. Lambert, Past, growth and persistence of source-sink metapopulations,, preprint, ().   Google Scholar [2] G. Barles, S. Mirrahimi and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: A general convergence result,, Methods Appl. Anal., 16 (2009), 321.   Google Scholar [3] J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 543.  doi: 10.1016/j.anihpc.2003.06.001.  Google Scholar [4] A. Calsina and S. Cuadrado, Stationary solutions of a selection mutation model: The pure mutation case,, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1091.  doi: 10.1142/S0218202505000637.  Google Scholar [5] J. A. Carrillo, S. Cuadrado and B. Perthame, Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model,, Math. Biosci., 205 (2007), 137.  doi: 10.1016/j.mbs.2006.09.012.  Google Scholar [6] N. Champagnat, R. Ferrière and S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution,, Stoch. Models, 24 (2008), 2.  doi: 10.1080/15326340802437710.  Google Scholar [7] N. Champagnat, R. Ferrière and S. Méléard, "Individual-Based Probabilistic Models of Adaptive Evolution and Various Scaling Approximations," Progress in Probability, 59,, Birkhäuser, (2008).  doi: 10.1007/978-3-7643-8458-6_6.  Google Scholar [8] N. Champagnat and P.-E. Jabin, The evolutionary limit for models of populations interacting competitively via several resources,, Journal of Differential Equations, 261 (2011), 179.  doi: 10.1016/j.jde.2011.03.007.  Google Scholar [9] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [10] L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On mutation-selection dynamics for continuous structured populations,, Commun. Math. Sci., 6 (2008), 729.   Google Scholar [11] O. Diekmann, A beginner's guide to adaptive dynamics,, in, 63 (2004), 47.   Google Scholar [12] O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach,, Th. Pop. Biol., 67 (2005), 257.   Google Scholar [13] I. Eshel, Evolutionary and continuous stability,, Journal of Theoretical Biology, 103 (1983), 99.  doi: 10.1016/0022-5193(83)90201-1.  Google Scholar [14] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, Proc. R. Soc. Edinb. Sec. A, 111 (1989), 359.  doi: 10.1017/S0308210500018631.  Google Scholar [15] S. A. H. Geritz, E. Kisdi, G. Mészena and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree,, Evol. Ecol, 12 (1998), 35.   Google Scholar [16] S. A. H. Geritz, J. A. J. Metz, E. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching,, Phys. Rev. Lett., 78 (1997), 2024.   Google Scholar [17] P.-E. Jabin and G. Raoul, Selection dynamics with competition,, J. Math. Biol., ().  doi: 10.1007/s00285-010-0370-8.  Google Scholar [18] S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle,, The American Naturalist, 104 (1970), 413.   Google Scholar [19] S. Lion and M. van Baalen, Self-structuring in spatial evolutionary ecology,, Ecology Letters, 11 (2008), 277.   Google Scholar [20] A. Lorz, S. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations,, Comm. Partial Differential Equations, 36 (2011), 1071.  doi: 10.1080/03605302.2010.538784.  Google Scholar [21] J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15.   Google Scholar [22] G. Meszéna, M. Gyllenberg, F. J. Jacobs and J. A. J. Metz, Link between population dynamics and dynamics of Darwinian evolution,, Phys. Rev. Lett., 95 (2005), 1.   Google Scholar [23] J. A. J. Metz, R. M. Nisbet and S. A. H. Geritz, How should we define "fitness" for general ecological scenarios?,, TREE, 7 (1992), 198.   Google Scholar [24] S. Mirrahimi and P. E. Souganidis, A homogenization approach for the motion of motor proteins,, Nonlinear Differential Equations and Applications NoDEA, ().   Google Scholar [25] B. Perthame and G. Barles, Dirac concentrations in Lotka-Volterra parabolic {PDEs},, Indiana Univ. Math. J., 57 (2008), 3275.  doi: 10.1512/iumj.2008.57.3398.  Google Scholar [26] B. Perthame and P. E. Souganidis, Asymmetric potentials and motor effect: a homogenization approach,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 26 (2009), 2055.  doi: 10.1016/j.anihpc.2008.10.003.  Google Scholar [27] B. Perthame and P. E. Souganidis, Asymmetric potentials and motor effect: A large deviation approach,, Arch. Ration. Mech. Anal., 193 (2009), 153.  doi: 10.1007/s00205-008-0198-1.  Google Scholar [28] G. Raoul, Long time evolution of populations under selection and vanishing mutations,, Acta Applicandae Mathematica, 114 (2011), 1.  doi: 10.1007/s10440-011-9603-0.  Google Scholar [29] T. W. Schoener, Resource partitioning in ecological communities,, Science, 13 (1974), 27.   Google Scholar [30] A. Szilágyi and G. Meszéna, Two-patch model of spatial niche segregation,, Evolutionary Ecology, 23 (2009), 187.   Google Scholar

show all references

##### References:
 [1] V. Bansaye and A. Lambert, Past, growth and persistence of source-sink metapopulations,, preprint, ().   Google Scholar [2] G. Barles, S. Mirrahimi and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: A general convergence result,, Methods Appl. Anal., 16 (2009), 321.   Google Scholar [3] J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 543.  doi: 10.1016/j.anihpc.2003.06.001.  Google Scholar [4] A. Calsina and S. Cuadrado, Stationary solutions of a selection mutation model: The pure mutation case,, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1091.  doi: 10.1142/S0218202505000637.  Google Scholar [5] J. A. Carrillo, S. Cuadrado and B. Perthame, Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model,, Math. Biosci., 205 (2007), 137.  doi: 10.1016/j.mbs.2006.09.012.  Google Scholar [6] N. Champagnat, R. Ferrière and S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution,, Stoch. Models, 24 (2008), 2.  doi: 10.1080/15326340802437710.  Google Scholar [7] N. Champagnat, R. Ferrière and S. Méléard, "Individual-Based Probabilistic Models of Adaptive Evolution and Various Scaling Approximations," Progress in Probability, 59,, Birkhäuser, (2008).  doi: 10.1007/978-3-7643-8458-6_6.  Google Scholar [8] N. Champagnat and P.-E. Jabin, The evolutionary limit for models of populations interacting competitively via several resources,, Journal of Differential Equations, 261 (2011), 179.  doi: 10.1016/j.jde.2011.03.007.  Google Scholar [9] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [10] L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On mutation-selection dynamics for continuous structured populations,, Commun. Math. Sci., 6 (2008), 729.   Google Scholar [11] O. Diekmann, A beginner's guide to adaptive dynamics,, in, 63 (2004), 47.   Google Scholar [12] O. Diekmann, P.-E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach,, Th. Pop. Biol., 67 (2005), 257.   Google Scholar [13] I. Eshel, Evolutionary and continuous stability,, Journal of Theoretical Biology, 103 (1983), 99.  doi: 10.1016/0022-5193(83)90201-1.  Google Scholar [14] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, Proc. R. Soc. Edinb. Sec. A, 111 (1989), 359.  doi: 10.1017/S0308210500018631.  Google Scholar [15] S. A. H. Geritz, E. Kisdi, G. Mészena and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree,, Evol. Ecol, 12 (1998), 35.   Google Scholar [16] S. A. H. Geritz, J. A. J. Metz, E. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching,, Phys. Rev. Lett., 78 (1997), 2024.   Google Scholar [17] P.-E. Jabin and G. Raoul, Selection dynamics with competition,, J. Math. Biol., ().  doi: 10.1007/s00285-010-0370-8.  Google Scholar [18] S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle,, The American Naturalist, 104 (1970), 413.   Google Scholar [19] S. Lion and M. van Baalen, Self-structuring in spatial evolutionary ecology,, Ecology Letters, 11 (2008), 277.   Google Scholar [20] A. Lorz, S. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations,, Comm. Partial Differential Equations, 36 (2011), 1071.  doi: 10.1080/03605302.2010.538784.  Google Scholar [21] J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15.   Google Scholar [22] G. Meszéna, M. Gyllenberg, F. J. Jacobs and J. A. J. Metz, Link between population dynamics and dynamics of Darwinian evolution,, Phys. Rev. Lett., 95 (2005), 1.   Google Scholar [23] J. A. J. Metz, R. M. Nisbet and S. A. H. Geritz, How should we define "fitness" for general ecological scenarios?,, TREE, 7 (1992), 198.   Google Scholar [24] S. Mirrahimi and P. E. Souganidis, A homogenization approach for the motion of motor proteins,, Nonlinear Differential Equations and Applications NoDEA, ().   Google Scholar [25] B. Perthame and G. Barles, Dirac concentrations in Lotka-Volterra parabolic {PDEs},, Indiana Univ. Math. J., 57 (2008), 3275.  doi: 10.1512/iumj.2008.57.3398.  Google Scholar [26] B. Perthame and P. E. Souganidis, Asymmetric potentials and motor effect: a homogenization approach,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 26 (2009), 2055.  doi: 10.1016/j.anihpc.2008.10.003.  Google Scholar [27] B. Perthame and P. E. Souganidis, Asymmetric potentials and motor effect: A large deviation approach,, Arch. Ration. Mech. Anal., 193 (2009), 153.  doi: 10.1007/s00205-008-0198-1.  Google Scholar [28] G. Raoul, Long time evolution of populations under selection and vanishing mutations,, Acta Applicandae Mathematica, 114 (2011), 1.  doi: 10.1007/s10440-011-9603-0.  Google Scholar [29] T. W. Schoener, Resource partitioning in ecological communities,, Science, 13 (1974), 27.   Google Scholar [30] A. Szilágyi and G. Meszéna, Two-patch model of spatial niche segregation,, Evolutionary Ecology, 23 (2009), 187.   Google Scholar
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