Small populations corrections for selection-mutation models

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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

Abstract
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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 and Heterogeneous Media, 2012, 7 (4) : 805-836. doi: 10.3934/nhm.2012.7.805

References:

[1]	, M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations".
[2]	G. Barles, "Solutions de Viscosite et Équations de Hamilton-Jacobi," Collec. SMAI, Springer-Verlag, 2002.
[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), 321-340.
[4]	G. Barles and B. Perthame, Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics, Recent Developments in Nonlinear Partial Differential Equations, 57-68, Contemp. Math., 439, Amer. Math. Soc., Providence, RI, (2007). doi: 10.1090/conm/439/08463.
[5]	R. Bürger and I. M. Bomze, Stationary distributions under mutation-selection balance: structure and properties, Adv. Appl. Prob., 28 (1996), 227-251. doi: 10.2307/1427919.
[6]	A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, J. Math. Biol., 48 (2004), 135-159. doi: 10.1007/s00285-003-0226-6.
[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), 137-161. doi: 10.1016/j.mbs.2006.09.012.
[8]	N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models, Stoch. Proc. Appl., 116 (2006), 1127-1160. doi: 10.1016/j.spa.2006.01.004.
[9]	N. Champagnat, R. Ferrière and G. Ben Arous, The canonical equation of adaptive dynamics: A mathematical view, Selection, 2 (2001), 71-81.
[10]	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.
[11]	N. Champagnat and P.-E. Jabin, The evolutionary limit for models of populations interacting competitively via several resources, J. Differential Equations 251 (2011), 176-195. doi: 10.1016/j.jde.2011.03.007.
[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), 1267-1272. doi: 10.1016/j.crma.2010.11.001.
[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.
[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), 167.
[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), 47-59.
[16]	L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Commun. Math. Sci., 6 (2008), 729-747.
[17]	U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612. doi: 10.1007/s002850050022.
[18]	O. Diekmann, A beginner's guide to adaptive dynamics. In Mathematical modelling of population dynamics, Banach Center Publ., 63, 47-86, Polish Acad. Sci., Warsaw, (2004).
[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), 157-189. doi: 10.1007/s002850170002.
[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), 257-271.
[21]	S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching, Math. Comput. Modelling, 49 (2009), 2109-2115. doi: 10.1016/j.mcm.2008.07.018.
[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), 2024-2027.
[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), 35-57.
[24]	M. Gyllenberg and G. Meszéna, On the impossibility of coexistence of infinitely many strategies, J. Math. Biol., 50 (2005), 133-160. doi: 10.1007/s00285-004-0283-5.
[25]	J. Hofbauer and R. Sigmund, Adaptive dynamics and evolutionary stability, Applied Math. Letters, 3 (1990), 75-79. doi: 10.1016/0893-9659(90)90051-C.
[26]	P. E. Jabin and G. Raoul, Selection dynamics with competition, J. Math. Biol., 63 (2011), 493-517. doi: 10.1007/s00285-010-0370-8.
[27]	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.
[28]	S. Méléard, Introduction to stochastic models for evolution, Markov Process. Related Fields, 15 (2009), 259-264.
[29]	S. Méléard and V. C. Tran, Trait substitution sequence process and canonical equation for age-structured populations, J. Math. Biol., 58 (2009), 881-921. doi: 10.1007/s00285-008-0202-2.
[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), 198-202.
[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 "Stochastic and Spatial Structures of Dynamical Systems" (eds. S. J. van Strien & S. M. Verduyn Lunel), North Holland, Amsterdam, (1996), 183-231.
[32]	S. Mirrahimi, G. Barles, B. Perthame and P. E. Souganidis, Singular Hamilton-Jacobi equation for the tail problem, Submitted.
[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, to appear (published online, 2009). doi: 10.1093/imammb/dqp018.
[34]	B. Perthame and S. Génieys, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit, Math. Model. Nat. Phenom., 2 (2007), 135-151. doi: 10.1051/mmnp:2008029.
[35]	G. Raoul, Long time evolution of populations under selection and rare mutations, Acta Applicandae Mathematica, 114 (2011), 114. doi: 10.1007/s10440-011-9603-0.
[36]	F. Yu, Stationary distributions of a model of sympatric speciation, Ann. Appl. Probab., 17 (2007), 840-874. doi: 10.1214/105051606000000916.

show all references

References:

[1]	, M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations".
[2]	G. Barles, "Solutions de Viscosite et Équations de Hamilton-Jacobi," Collec. SMAI, Springer-Verlag, 2002.
[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), 321-340.
[4]	G. Barles and B. Perthame, Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics, Recent Developments in Nonlinear Partial Differential Equations, 57-68, Contemp. Math., 439, Amer. Math. Soc., Providence, RI, (2007). doi: 10.1090/conm/439/08463.
[5]	R. Bürger and I. M. Bomze, Stationary distributions under mutation-selection balance: structure and properties, Adv. Appl. Prob., 28 (1996), 227-251. doi: 10.2307/1427919.
[6]	A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, J. Math. Biol., 48 (2004), 135-159. doi: 10.1007/s00285-003-0226-6.
[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), 137-161. doi: 10.1016/j.mbs.2006.09.012.
[8]	N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models, Stoch. Proc. Appl., 116 (2006), 1127-1160. doi: 10.1016/j.spa.2006.01.004.
[9]	N. Champagnat, R. Ferrière and G. Ben Arous, The canonical equation of adaptive dynamics: A mathematical view, Selection, 2 (2001), 71-81.
[10]	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.
[11]	N. Champagnat and P.-E. Jabin, The evolutionary limit for models of populations interacting competitively via several resources, J. Differential Equations 251 (2011), 176-195. doi: 10.1016/j.jde.2011.03.007.
[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), 1267-1272. doi: 10.1016/j.crma.2010.11.001.
[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.
[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), 167.
[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), 47-59.
[16]	L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Commun. Math. Sci., 6 (2008), 729-747.
[17]	U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612. doi: 10.1007/s002850050022.
[18]	O. Diekmann, A beginner's guide to adaptive dynamics. In Mathematical modelling of population dynamics, Banach Center Publ., 63, 47-86, Polish Acad. Sci., Warsaw, (2004).
[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), 157-189. doi: 10.1007/s002850170002.
[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), 257-271.
[21]	S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching, Math. Comput. Modelling, 49 (2009), 2109-2115. doi: 10.1016/j.mcm.2008.07.018.
[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), 2024-2027.
[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), 35-57.
[24]	M. Gyllenberg and G. Meszéna, On the impossibility of coexistence of infinitely many strategies, J. Math. Biol., 50 (2005), 133-160. doi: 10.1007/s00285-004-0283-5.
[25]	J. Hofbauer and R. Sigmund, Adaptive dynamics and evolutionary stability, Applied Math. Letters, 3 (1990), 75-79. doi: 10.1016/0893-9659(90)90051-C.
[26]	P. E. Jabin and G. Raoul, Selection dynamics with competition, J. Math. Biol., 63 (2011), 493-517. doi: 10.1007/s00285-010-0370-8.
[27]	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.
[28]	S. Méléard, Introduction to stochastic models for evolution, Markov Process. Related Fields, 15 (2009), 259-264.
[29]	S. Méléard and V. C. Tran, Trait substitution sequence process and canonical equation for age-structured populations, J. Math. Biol., 58 (2009), 881-921. doi: 10.1007/s00285-008-0202-2.
[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), 198-202.
[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 "Stochastic and Spatial Structures of Dynamical Systems" (eds. S. J. van Strien & S. M. Verduyn Lunel), North Holland, Amsterdam, (1996), 183-231.
[32]	S. Mirrahimi, G. Barles, B. Perthame and P. E. Souganidis, Singular Hamilton-Jacobi equation for the tail problem, Submitted.
[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, to appear (published online, 2009). doi: 10.1093/imammb/dqp018.
[34]	B. Perthame and S. Génieys, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit, Math. Model. Nat. Phenom., 2 (2007), 135-151. doi: 10.1051/mmnp:2008029.
[35]	G. Raoul, Long time evolution of populations under selection and rare mutations, Acta Applicandae Mathematica, 114 (2011), 114. doi: 10.1007/s10440-011-9603-0.
[36]	F. Yu, Stationary distributions of a model of sympatric speciation, Ann. Appl. Probab., 17 (2007), 840-874. doi: 10.1214/105051606000000916.

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