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December  2012, 7(4): 805-836. doi: 10.3934/nhm.2012.7.805

Small populations corrections for selection-mutation models

Pierre-Emmanuel Jabin 1,

1. 

CSCAMM and Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States

Received  March 2012 Revised  August 2012 Published  December 2012

We consider integro-differential models describing the evolution of a population structured by a quantitative trait. Individuals interact competitively, creating a strong selection pressure on the population. On the other hand, mutations are assumed to be small. Following the formalism of [20], this creates concentration phenomena, typically consisting in a sum of Dirac masses slowly evolving in time. We propose a modification to those classical models that takes the effect of small populations into accounts and corrects some abnormal behaviours.
Keywords: Hamilton-Jacobi equation with constraints, Adaptive dynamics, small populations., Dirac concentration.
Mathematics Subject Classification: Primary: 35B25, 35K55, 92D1.
Citation: Pierre-Emmanuel Jabin. Small populations corrections for selection-mutation models. Networks & Heterogeneous Media, 2012, 7 (4) : 805-836. doi: 10.3934/nhm.2012.7.805
References:
[1]

, M. Bardi and I. Capuzzo Dolcetta,, M, ().   Google Scholar

[2]

G. Barles, "Solutions de Viscosite et Équations de Hamilton-Jacobi,", Collec. SMAI, (2002).   Google Scholar

[3]

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

[4]

G. Barles and B. Perthame, Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics,, Recent Developments in Nonlinear Partial Differential Equations, 439 (2007), 57.  doi: 10.1090/conm/439/08463.  Google Scholar

[5]

R. Bürger and I. M. Bomze, Stationary distributions under mutation-selection balance: structure and properties,, Adv. Appl. Prob., 28 (1996), 227.  doi: 10.2307/1427919.  Google Scholar

[6]

A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics,, J. Math. Biol., 48 (2004), 135.  doi: 10.1007/s00285-003-0226-6.  Google Scholar

[7]

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

[8]

N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models,, Stoch. Proc. Appl., 116 (2006), 1127.  doi: 10.1016/j.spa.2006.01.004.  Google Scholar

[9]

N. Champagnat, R. Ferrière and G. Ben Arous, The canonical equation of adaptive dynamics: A mathematical view,, Selection, 2 (2001), 71.   Google Scholar

[10]

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

[11]

N. Champagnat and P.-E. Jabin, The evolutionary limit for models of populations interacting competitively via several resources,, J. Differential Equations 251 (2011), 251 (2011), 176.  doi: 10.1016/j.jde.2011.03.007.  Google Scholar

[12]

N. Champagnat, P.-E. Jabin and G. Raoul, Convergence to equilibrium in competitive Lotka-Volterra and chemostat systems,, C. R. Math. Acad. Sci. Paris, 348 (2010), 1267.  doi: 10.1016/j.crma.2010.11.001.  Google Scholar

[13]

N. Champagnat and S. Méléard, Polymorphic evolution sequence and evolutionary branching,, To appear in Probab. Theory Relat. Fields (published online, (2010).  doi: 10.1007/s00440-010-0292-9.  Google Scholar

[14]

M. G. Crandall and P.-L. Lions, Users guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc., 27 (1992).   Google Scholar

[15]

R. Cressman and J. Hofbauer, Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics,, Theor. Pop. Biol., 67 (2005), 47.   Google Scholar

[16]

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

[17]

U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes,, J. Math. Biol., 34 (1996), 579.  doi: 10.1007/s002850050022.  Google Scholar

[18]

O. Diekmann, A beginner's guide to adaptive dynamics. In Mathematical modelling of population dynamics,, Banach Center Publ., 63 (2004), 47.   Google Scholar

[19]

O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory,, J. Math. Biol., 43 (2001), 157.  doi: 10.1007/s002850170002.  Google Scholar

[20]

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.   Google Scholar

[21]

S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching,, Math. Comput. Modelling, 49 (2009), 2109.  doi: 10.1016/j.mcm.2008.07.018.  Google Scholar

[22]

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

[23]

S. A. H. Geritz, E. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree,, Evol. Ecol., 12 (1998), 35.   Google Scholar

[24]

M. Gyllenberg and G. Meszéna, On the impossibility of coexistence of infinitely many strategies,, J. Math. Biol., 50 (2005), 133.  doi: 10.1007/s00285-004-0283-5.  Google Scholar

[25]

J. Hofbauer and R. Sigmund, Adaptive dynamics and evolutionary stability,, Applied Math. Letters, 3 (1990), 75.  doi: 10.1016/0893-9659(90)90051-C.  Google Scholar

[26]

P. E. Jabin and G. Raoul, Selection dynamics with competition,, J. Math. Biol., 63 (2011), 493.  doi: 10.1007/s00285-010-0370-8.  Google Scholar

[27]

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

[28]

S. Méléard, Introduction to stochastic models for evolution,, Markov Process. Related Fields, 15 (2009), 259.   Google Scholar

[29]

S. Méléard and V. C. Tran, Trait substitution sequence process and canonical equation for age-structured populations,, J. Math. Biol., 58 (2009), 881.  doi: 10.1007/s00285-008-0202-2.  Google Scholar

[30]

J. A. J. Metz, R. M. Nisbet and S. A. H. Geritz, How should we define 'fitness' for general ecological scenarios?,, Trends in Ecology and Evolution, 7 (1992), 198.   Google Scholar

[31]

J. A. J. Metz, S. A. H. Geritz, G. Meszéna, F. A. J. Jacobs and J. S. van Heerwaarden, Adaptive Dynamics, a geometrical study of the consequences of nearly faithful reproduction,, in, (1996), 183.   Google Scholar

[32]

S. Mirrahimi, G. Barles, B. Perthame and P. E. Souganidis, Singular Hamilton-Jacobi equation for the tail problem,, Submitted., ().   Google Scholar

[33]

B. Perthame and M. Gauduchon, Survival thresholds and mortality rates in adaptive dynamics: Conciliating deterministic and stochastic simulations,, IMA Journal of Mathematical Medicine and Biology, (2009).  doi: 10.1093/imammb/dqp018.  Google Scholar

[34]

B. Perthame and S. Génieys, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit,, Math. Model. Nat. Phenom., 2 (2007), 135.  doi: 10.1051/mmnp:2008029.  Google Scholar

[35]

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

[36]

F. Yu, Stationary distributions of a model of sympatric speciation,, Ann. Appl. Probab., 17 (2007), 840.  doi: 10.1214/105051606000000916.  Google Scholar

show all references

References:
[1]

, M. Bardi and I. Capuzzo Dolcetta,, M, ().   Google Scholar

[2]

G. Barles, "Solutions de Viscosite et Équations de Hamilton-Jacobi,", Collec. SMAI, (2002).   Google Scholar

[3]

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

[4]

G. Barles and B. Perthame, Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics,, Recent Developments in Nonlinear Partial Differential Equations, 439 (2007), 57.  doi: 10.1090/conm/439/08463.  Google Scholar

[5]

R. Bürger and I. M. Bomze, Stationary distributions under mutation-selection balance: structure and properties,, Adv. Appl. Prob., 28 (1996), 227.  doi: 10.2307/1427919.  Google Scholar

[6]

A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics,, J. Math. Biol., 48 (2004), 135.  doi: 10.1007/s00285-003-0226-6.  Google Scholar

[7]

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

[8]

N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models,, Stoch. Proc. Appl., 116 (2006), 1127.  doi: 10.1016/j.spa.2006.01.004.  Google Scholar

[9]

N. Champagnat, R. Ferrière and G. Ben Arous, The canonical equation of adaptive dynamics: A mathematical view,, Selection, 2 (2001), 71.   Google Scholar

[10]

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

[11]

N. Champagnat and P.-E. Jabin, The evolutionary limit for models of populations interacting competitively via several resources,, J. Differential Equations 251 (2011), 251 (2011), 176.  doi: 10.1016/j.jde.2011.03.007.  Google Scholar

[12]

N. Champagnat, P.-E. Jabin and G. Raoul, Convergence to equilibrium in competitive Lotka-Volterra and chemostat systems,, C. R. Math. Acad. Sci. Paris, 348 (2010), 1267.  doi: 10.1016/j.crma.2010.11.001.  Google Scholar

[13]

N. Champagnat and S. Méléard, Polymorphic evolution sequence and evolutionary branching,, To appear in Probab. Theory Relat. Fields (published online, (2010).  doi: 10.1007/s00440-010-0292-9.  Google Scholar

[14]

M. G. Crandall and P.-L. Lions, Users guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc., 27 (1992).   Google Scholar

[15]

R. Cressman and J. Hofbauer, Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics,, Theor. Pop. Biol., 67 (2005), 47.   Google Scholar

[16]

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

[17]

U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes,, J. Math. Biol., 34 (1996), 579.  doi: 10.1007/s002850050022.  Google Scholar

[18]

O. Diekmann, A beginner's guide to adaptive dynamics. In Mathematical modelling of population dynamics,, Banach Center Publ., 63 (2004), 47.   Google Scholar

[19]

O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory,, J. Math. Biol., 43 (2001), 157.  doi: 10.1007/s002850170002.  Google Scholar

[20]

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.   Google Scholar

[21]

S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching,, Math. Comput. Modelling, 49 (2009), 2109.  doi: 10.1016/j.mcm.2008.07.018.  Google Scholar

[22]

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

[23]

S. A. H. Geritz, E. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree,, Evol. Ecol., 12 (1998), 35.   Google Scholar

[24]

M. Gyllenberg and G. Meszéna, On the impossibility of coexistence of infinitely many strategies,, J. Math. Biol., 50 (2005), 133.  doi: 10.1007/s00285-004-0283-5.  Google Scholar

[25]

J. Hofbauer and R. Sigmund, Adaptive dynamics and evolutionary stability,, Applied Math. Letters, 3 (1990), 75.  doi: 10.1016/0893-9659(90)90051-C.  Google Scholar

[26]

P. E. Jabin and G. Raoul, Selection dynamics with competition,, J. Math. Biol., 63 (2011), 493.  doi: 10.1007/s00285-010-0370-8.  Google Scholar

[27]

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

[28]

S. Méléard, Introduction to stochastic models for evolution,, Markov Process. Related Fields, 15 (2009), 259.   Google Scholar

[29]

S. Méléard and V. C. Tran, Trait substitution sequence process and canonical equation for age-structured populations,, J. Math. Biol., 58 (2009), 881.  doi: 10.1007/s00285-008-0202-2.  Google Scholar

[30]

J. A. J. Metz, R. M. Nisbet and S. A. H. Geritz, How should we define 'fitness' for general ecological scenarios?,, Trends in Ecology and Evolution, 7 (1992), 198.   Google Scholar

[31]

J. A. J. Metz, S. A. H. Geritz, G. Meszéna, F. A. J. Jacobs and J. S. van Heerwaarden, Adaptive Dynamics, a geometrical study of the consequences of nearly faithful reproduction,, in, (1996), 183.   Google Scholar

[32]

S. Mirrahimi, G. Barles, B. Perthame and P. E. Souganidis, Singular Hamilton-Jacobi equation for the tail problem,, Submitted., ().   Google Scholar

[33]

B. Perthame and M. Gauduchon, Survival thresholds and mortality rates in adaptive dynamics: Conciliating deterministic and stochastic simulations,, IMA Journal of Mathematical Medicine and Biology, (2009).  doi: 10.1093/imammb/dqp018.  Google Scholar

[34]

B. Perthame and S. Génieys, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit,, Math. Model. Nat. Phenom., 2 (2007), 135.  doi: 10.1051/mmnp:2008029.  Google Scholar

[35]

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

[36]

F. Yu, Stationary distributions of a model of sympatric speciation,, Ann. Appl. Probab., 17 (2007), 840.  doi: 10.1214/105051606000000916.  Google Scholar

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