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 and 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, (). 

[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), 321-340.

[3]

J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 543-590. doi: 10.1016/j.anihpc.2003.06.001.

[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), 1091-1117. doi: 10.1142/S0218202505000637.

[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), 137-161. doi: 10.1016/j.mbs.2006.09.012.

[6]

N. Champagnat, R. Ferrière and S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution, Stoch. Models, 24 (2008), 2-44. doi: 10.1080/15326340802437710.

[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.

[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), 179-195. doi: 10.1016/j.jde.2011.03.007.

[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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.

[10]

L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On mutation-selection dynamics for continuous structured populations, Commun. Math. Sci., 6 (2008), 729-747.

[11]

O. Diekmann, A beginner's guide to adaptive dynamics, in "Mathematical Modelling of Population Dynamics," Banach Center Publ., 63, Polish Acad. Sci., Warsaw, (2004), 47-86.

[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), 257-271.

[13]

I. Eshel, Evolutionary and continuous stability, Journal of Theoretical Biology, 103 (1983), 99-111. doi: 10.1016/0022-5193(83)90201-1.

[14]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. R. Soc. Edinb. Sec. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631.

[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), 35-57.

[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), 2024-2027.

[17]

P.-E. Jabin and G. Raoul, Selection dynamics with competition,, J. Math. Biol., ().  doi: 10.1007/s00285-010-0370-8.

[18]

S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle, The American Naturalist, 104 (1970), 413-423.

[19]

S. Lion and M. van Baalen, Self-structuring in spatial evolutionary ecology, Ecology Letters, 11 (2008), 277-295.

[20]

A. Lorz, S. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1071-1098. doi: 10.1080/03605302.2010.538784.

[21]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.

[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), 078105.1-078105.4.

[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), 198-202.

[24]

S. Mirrahimi and P. E. Souganidis, A homogenization approach for the motion of motor proteins,, Nonlinear Differential Equations and Applications NoDEA, (). 

[25]

B. Perthame and G. Barles, Dirac concentrations in Lotka-Volterra parabolic {PDEs}, Indiana Univ. Math. J., 57 (2008), 3275-3301. doi: 10.1512/iumj.2008.57.3398.

[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), 2055-2071. doi: 10.1016/j.anihpc.2008.10.003.

[27]

B. Perthame and P. E. Souganidis, Asymmetric potentials and motor effect: A large deviation approach, Arch. Ration. Mech. Anal., 193 (2009), 153-169. doi: 10.1007/s00205-008-0198-1.

[28]

G. Raoul, Long time evolution of populations under selection and vanishing mutations, Acta Applicandae Mathematica, 114 (2011), 1-14. doi: 10.1007/s10440-011-9603-0.

[29]

T. W. Schoener, Resource partitioning in ecological communities, Science, 13 (1974), 27-39.

[30]

A. Szilágyi and G. Meszéna, Two-patch model of spatial niche segregation, Evolutionary Ecology, 23 (2009), 187-205.

show all references

References:
[1]

V. Bansaye and A. Lambert, Past, growth and persistence of source-sink metapopulations,, preprint, (). 

[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), 321-340.

[3]

J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 543-590. doi: 10.1016/j.anihpc.2003.06.001.

[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), 1091-1117. doi: 10.1142/S0218202505000637.

[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), 137-161. doi: 10.1016/j.mbs.2006.09.012.

[6]

N. Champagnat, R. Ferrière and S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution, Stoch. Models, 24 (2008), 2-44. doi: 10.1080/15326340802437710.

[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.

[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), 179-195. doi: 10.1016/j.jde.2011.03.007.

[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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.

[10]

L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On mutation-selection dynamics for continuous structured populations, Commun. Math. Sci., 6 (2008), 729-747.

[11]

O. Diekmann, A beginner's guide to adaptive dynamics, in "Mathematical Modelling of Population Dynamics," Banach Center Publ., 63, Polish Acad. Sci., Warsaw, (2004), 47-86.

[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), 257-271.

[13]

I. Eshel, Evolutionary and continuous stability, Journal of Theoretical Biology, 103 (1983), 99-111. doi: 10.1016/0022-5193(83)90201-1.

[14]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. R. Soc. Edinb. Sec. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631.

[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), 35-57.

[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), 2024-2027.

[17]

P.-E. Jabin and G. Raoul, Selection dynamics with competition,, J. Math. Biol., ().  doi: 10.1007/s00285-010-0370-8.

[18]

S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle, The American Naturalist, 104 (1970), 413-423.

[19]

S. Lion and M. van Baalen, Self-structuring in spatial evolutionary ecology, Ecology Letters, 11 (2008), 277-295.

[20]

A. Lorz, S. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1071-1098. doi: 10.1080/03605302.2010.538784.

[21]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.

[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), 078105.1-078105.4.

[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), 198-202.

[24]

S. Mirrahimi and P. E. Souganidis, A homogenization approach for the motion of motor proteins,, Nonlinear Differential Equations and Applications NoDEA, (). 

[25]

B. Perthame and G. Barles, Dirac concentrations in Lotka-Volterra parabolic {PDEs}, Indiana Univ. Math. J., 57 (2008), 3275-3301. doi: 10.1512/iumj.2008.57.3398.

[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), 2055-2071. doi: 10.1016/j.anihpc.2008.10.003.

[27]

B. Perthame and P. E. Souganidis, Asymmetric potentials and motor effect: A large deviation approach, Arch. Ration. Mech. Anal., 193 (2009), 153-169. doi: 10.1007/s00205-008-0198-1.

[28]

G. Raoul, Long time evolution of populations under selection and vanishing mutations, Acta Applicandae Mathematica, 114 (2011), 1-14. doi: 10.1007/s10440-011-9603-0.

[29]

T. W. Schoener, Resource partitioning in ecological communities, Science, 13 (1974), 27-39.

[30]

A. Szilágyi and G. Meszéna, Two-patch model of spatial niche segregation, Evolutionary Ecology, 23 (2009), 187-205.

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