Advertisement

Population Formulation of Adaptative Meso-evolution: Theory and Numerics

Chapter
Part of the Mathematics and Biosciences in Interaction book series (MBI)

Abstract

The population formalism of “adaptive evolution” has been developed in the last twenty years along ideas presented in other chapters in this volume. This mathematical formalism addresses the question of explaining how selection of a favorable phenotypical trait in a population occurs. In the language of Metz’s Chapter, it refers to meso-evolution. It uses models based, usually, on integro-differential equations for the population structured by a phenotypical trait. A self-contained mathematical formulation of adaptive evolution also contains the description of mutations and leads to partial differential equations. Then the complete evolution picture follows from the model ingredients mostly driven by the changing adaptive landscape.

Keywords

Adaptive evolution Lotka-Volterra equation Darwinian evolution Hamilton-Jacobi equation Viscosity solutions Dirac concentrations MonteCarlo simulations Finite differences 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.A.J. Metz, Thoughts on the geometry of meso-evolution: Collecting mathematical elements for a post-modern synthesis. In F.A.C.C. Chalub and J.F. Rodrigues (eds.), The Mathematics of Darwin’s Legacy, 193–231, Birkh¨auser, Basel, 2011, This issue.Google Scholar
  2. 2.
    J. Maynard Smith, Theory of games and evolution of animal conflicts. J. Theor. Biol. 47 (1974), 209–221.CrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics. Cambridge Univ. Press, Cambridge, UK, 1998.Google Scholar
  4. 4.
    J. Hofbauer and K. Sigmund, Evolutionary game dynamics. Bull. Amer. Math. Soc. (N.S.) 40 (2003), 479–519.Google Scholar
  5. 5.
    J. Hofbauer and K. Sigmund, Adaptive dynamics and evolutionary stability. Appl. Math. Lett. 3 (1990), 75–79.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J.A.J. Metz, S.A.H. Geritz, G. Mesz´ena, F.J.A. 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 (Amsterdam,1995), Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, 45, 183– 231, North-Holland, Amsterdam, 1996.Google Scholar
  7. 7.
    S.A.H. Geritz, ´E. Kisdi, G. Meszena, and J.A.J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12 (1998), 35–57.Google Scholar
  8. 8.
    O. Diekmann, A beginner’s guide to adaptive dynamics. In Mathematical Modelling of Population Dynamics, volume 63 of Banach Center Publ., 47–86, Polish Acad. Sci., Warsaw, 2004.Google Scholar
  9. 9.
    A. Sasaki and S. Ellner, The evolutionarily stable phenotype distribution in a random environment. Evolution 49 (1995), 337–350.CrossRefGoogle Scholar
  10. 10.
    A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics. J. Math. Biol. 48 (2004), 135–159.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. Calsina and C. Perello, Equations for biological evolution. Proc. R. Soc. Edinb. Sect. A-Math. 125 (1995), 939–958.zbMATHMathSciNetGoogle Scholar
  12. 12.
    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.zbMATHCrossRefGoogle Scholar
  13. 13.
    G. Barles and B. Perthame, Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics. In D. Danielli (ed.), Recent Developments in Nonlinear Partial Differential Equations, volume 439 of Contemporary Mathematics Series, 57–68, 2007.Google Scholar
  14. 14.
    B. Perthame and G. Barles, Dirac concentrations in Lotka-Volterra parabolic PDEs. Indiana Univ. Math. J. 57 (2008), 3275–3301.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    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.zbMATHMathSciNetGoogle Scholar
  16. 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.zbMATHMathSciNetGoogle Scholar
  17. 17.
    P.-E. Jabin and G. Raoul, On selection dynamics for competitive interactions. Preprint CMLA-ENS Cachan 17 (2009).Google Scholar
  18. 18.
    N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch. Process. Their Appl. 116 (2006), 1127–1160.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    N. Champagnat, R. Ferri`ere, and S. M´el´eard, Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models. Theor. Popul. Biol. 69 (2006), 297–321.Google Scholar
  20. 20.
    N. Champagnat, R. Ferri`ere, and S. M´el´eard, Individual-based probabilistic models of adaptive evolution and various scaling approximations. In R.C. Dalang, M. Dozzi, and F. Russo (eds.), Seminar On Stochastic Analysis, Random Fields And Applications V, volume 59 of Progress In Probability, 75–113, 2008, 5th Seminar on Stochastic Analysis, Random Fields and Applications, Ascona, Switzerland, May 30–Jun 03, 2005.Google Scholar
  21. 21.
    S. M´el´eard, Random modeling of adaptive dynamics and evolutionary branching. In F.A.C.C. Chalub and J. Rodrigues (eds.), The Mathematics of Darwin’s Legacy, 175–192, Birkh¨auser, Basel, 2011, This issue.Google Scholar
  22. 22.
    A. Lorz, S. Mirrahimi, and B. Perthame, Dirac concentration in a multidimensional nonlocal parabolic equation. In preperation.Google Scholar
  23. 23.
    U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes. J. Math. Biol. 34 (1996), 579–612.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    L.C. Evans and P.E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations. Indiana Univ. Math. J. 38 (1989), 141–172.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    G. Barles, L.C. Evans, and P.E. Souganidis, Wavefront propagation for reactiondiffusion systems of PDE. Duke Math. J. 61 (1990), 835–858.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    P.E. Souganidis, Front propagation: theory and applications. In Viscosity solutions and applications (Montecatini Terme, 1995), volume 1660 of Lecture Notes in Math., 186–242, Springer, Berlin, 1997.Google Scholar
  27. 27.
    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. 27 (1992), 1–67.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    G. Mesz´ena, 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.Google Scholar
  30. 30.
    S. Genieys, V. Volpert, and P. Auger, Adaptive dynamics: modelling darwin’s divergence principle. C. R. Biol. 329 (2006), 876–879.CrossRefGoogle Scholar
  31. 31.
    B. Perthame and S. G´enieys, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit. Math. Model. Nat. Phenom. 2 (2007), 135–151.Google Scholar
  32. 32.
    H. Berestycki, G. Nadin, B. Perthame, and L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states. Nonlinearity 22 (2009), 2813–2844.zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    E. Brigatti, V. Schwammle, and M.A. Neto, Individual-based model with global competition interaction: fluctuation effects in pattern formation. Phys. Rev. E 77 (2008).Google Scholar
  34. 34.
    V. Schw¨ammle and E. Brigatti, Speciational view of macroevolution: Are micro and macroevolution decoupled? Europhys. Lett. 75 (2006), 342–348.Google Scholar
  35. 35.
    B. Bolker and S.W. Pacala, Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol. 52 (1997), 179–197.zbMATHCrossRefGoogle Scholar
  36. 36.
    S.A. Gourley, Travelling front solutions of a nonlocal Fisher equation. J. Math. Biol. 41 (2000), 272–284.zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Z.C. Wang, W.T. Li, and S.G. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays. J. Differ. Equ. 222 (2006), 185–232.zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    N. Fournier and S. M´el´eard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 (2004), 1880–1919.Google Scholar
  39. 39.
    M. Gauduchon and B. Perthame, Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations. Mathematical Medicine and Biology 27 (2010), 195–210.zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    D. Claessen, J. Andersson, L. Persson, and A.M. de Roos, Delayed evolutionary branching in small populations. Evol. Ecol. Res. 9 (2007), 51–69.Google Scholar
  41. 41.
    S.R. Proulx and T. Day, What can invasion analyses tell us about evolution under stochasticity in finite populations? Selection 2 (2002), 2–15.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUPMC Univ Paris 06, UMR 7598ParisFrance
  2. 2.Laboratoire Jacques-Louis LionsInstitut Universitaire de France & UPMC Univ Paris 06, UMR 7598ParisFrance

Personalised recommendations