Advertisement

Thoughts on the Geometry of Meso-evolution: Collecting Mathematical Elements for a Postmodern Synthesis

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

Abstract

Adaptive dynamics (AD) is a recently developed framework geared towards making the transition from micro-evolution to long-term evolution based on a time scale separation approximation. This assumption allows defining the fitness of a mutant as the rate constant of initial exponential growth of the mutant population in the environment created by the resident community dynamics. This definition makes that all resident types have fitness zero. If in addition it is assumed that mutational steps are small, evolution can be visualized as an uphill walk in a fitness landscape that keeps changing as a result of the evolution it engenders. The chapter summarises the main tools for analysing special eco-evolutionary models based on these simplifications. In addition it describes a number of general predictions that directly derive from the AD perspective a such, without making any further assumptions.

Keywords

Adaptive dynamics Fitness landscapes Evo-devo Meso-evolution Macro-evolution Internal selection Speciation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    W. Ewens, What changes has mathematics made to the Darwinian theory? In F.A.C.C. Chalub and J.F. Rodrigues (eds.), The Mathematics of Darwin’s Legacy, 7–26, Birkh¨auser, Basel, 2011, This issue.Google Scholar
  2. 2.
    R. B¨urger, Some mathematical models in evolutionary genetics. In F.A.C.C. Chalub and J.F. Rodrigues (eds.), The Mathematics of Darwin’s Legacy, 67–89, Birkh¨auser, Basel, 2011, This issue.Google Scholar
  3. 3.
    G.L. Jepsen, G.G. Simpson, and E. Mayr (eds.), Genetics, Paleontology and Evolution. Princeton University Press, USA, 1949.Google Scholar
  4. 4.
    E. Mayr, The Growth of Biological Thought: Diversity, Evolution, and Inheritance. Belknap Press, USA, 1982.Google Scholar
  5. 5.
    E. Mayr and W.B. Provine, The Evolutionary Synthesis: Perspectives on the Unification of Biology. Harvard University Press, USA, 1980.Google Scholar
  6. 6.
    R. Amundson, The Changing Role of the Embryo in Evolutionary Thought: Roots of Evo-Devo. Cambridge: Cambridge University Press, UK, 2005.Google Scholar
  7. 7.
    W.D. Hamilton, Extraordinary sex ratios. Science 156 (1967), 477.Google Scholar
  8. 8.
    J. Maynard Smith and G.R. Price, The logic of animal conflict. Nature 246 (1973), 15–18.Google Scholar
  9. 9.
    R.A. Fisher, The Genetical Theory of Natural Selection. Clarendon Press, Oxford, 1930.Google Scholar
  10. 10.
    I. Eshel, M.W. Feldman, and A. Bergman, Long-term evolution, short-term evolution, and population genetic theory. J. Theor. Biol. 191 (1998), 391–396.CrossRefGoogle Scholar
  11. 11.
    I. Eshel, Short-term and long-term evolution. In U. Dieckmann and J.A.J. Metz (eds.), Elements of Adaptive Dynamics, Cambridge University Press, UK, In press.Google Scholar
  12. 12.
    T.L. Vincent and J.S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics. Cambridge University Press, UK, 2005.Google Scholar
  13. 13.
    P.A. Abrams, Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: an assessment of three methods. Ecol. Lett. 4 (2001), 166–175.CrossRefGoogle Scholar
  14. 14.
    R.A. Fisher, On the dominance ratio. Proc. Roy. Soc. Edin. 42 (1922), 321–341.Google Scholar
  15. 15.
    J.A.J. Metz, Fitness. In S.E. J¨orgensen and B.D. Fath (eds.), Evolutionary Ecology, volume 2 of Encyclopedia of Ecology, 1599–1612, Elsevier, UK, 2008.Google Scholar
  16. 16.
    J.A.J. Metz and O. Diekmann (eds.), The dynamics of physiologically structured populations, volume 68 of Lecture Notes in Biomathematics. Springer-Verlag, Berlin, 1986.Google Scholar
  17. 17.
    J.A.J. Metz and A.M. de Roos, The role of physiologically structured population models within a general individual-based modeling perspective. In D. DeAngelis and L. Gross (eds.), Individual-based models and approaches in ecology: Concepts and Models, 88–111, Chapman & Hall, USA, 1992.Google Scholar
  18. 18.
    J.A.J. Metz, R.M. Nisbet, and S.A.H. Geritz, How should we define “fitness” for general ecological scenarios? Trends Ecol. Evol. 7 (1992), 198–202.CrossRefGoogle Scholar
  19. 19.
    R. Ferri`ere and M. Gatto, Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations. Theor. Popul. Biol. 48 (1995), 126–171.Google Scholar
  20. 20.
    P. Jagers, Branching Processes with Biological Applications. Wiley Series in Probability and Mathematical Statistics-Applied, London, UK, 1975.Google Scholar
  21. 21.
    K.B. Athreya and S. Karlin, Branching processes with random environments I – extinction probabilities. Ann. Math. Stat. 42 (1971), 1499.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    K.B. Athreya and S. Karlin, Branching processes with random environments II – limit theorems. Ann. Math. Stat. 42 (1971), 1843.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    P. Haccou, P. Jagers, and V.A. Vatutin, Branching Processes. Variation, Growth, and Extinction of Populations, volume 5 of Cambridge Studies in Adaptive Dynamics. Cambridge University Press, UK, 2005.Google Scholar
  24. 24.
    F.J.A. Jacobs and J.A.J. Metz, On the concept of attractor for community-dynamical processes I: the case of unstructured populations. J. Math. Biol. 47 (2003), 222–234.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    M. Gyllenberg, F.J.A. Jacobs, and J.A.J. Metz, On the concept of attractor for community-dynamical processes II: the case of structured populations. J. Math. Biol. 47 (2003), 235–248.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    O. Diekmann, M. Gyllenberg, and J.A.J. Metz, Steady-state analysis of structured population models. Theor. Popul. Biol. 63 (2003), 309–338.zbMATHCrossRefGoogle Scholar
  27. 27.
    M. Gyllenberg, J.A.J. Metz, and R. Service, When do optimisation arguments make evolutionary sense? In F.A.C.C. Chalub and J.F. Rodrigues (eds.), The Mathematics of Darwin’s Legacy, 233–268, Birkh¨auser, Basel, 2011, This issue.Google Scholar
  28. 28.
    I. Eshel, Evolutionary and continuous stability. J. Theor. Biol. 103 (1983), 99–111.CrossRefMathSciNetGoogle Scholar
  29. 29.
    J.A.J. Metz, S.D. Mylius, and O. Diekmann, When does evolution optimize? Evol. Ecol. Res. 10 (2008), 629–654.Google Scholar
  30. 30.
    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
  31. 31.
    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
  32. 32.
    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
  33. 33.
    N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch. Process. Their Appl. 116 (2006), 1127–1160.zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    S.A.H. Geritz, M. Gyllenberg, F.J.A. Jacobs, and K. Parvinen, Invasion dynamics and attractor inheritance. J. Math. Biol. 44 (2002), 548–560.zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    S.A.H. Geritz, Resident-invader dynamics and the coexistence of similar strategies. J. Math. Biol. 50 (2005), 67–82.zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes: the Adaptive Dynamics Approach and its Applications. Princeton University Press, USA, 2008.Google Scholar
  37. 37.
    N. Champagnat, Convergence of adaptive dynamics n-morphic jump processes to the canonical equation and degenerate diffusion approximation. Preprint of the University of Nanterre (Paris 10) No. 03/7. (2003).Google Scholar
  38. 38.
    M. Durinx, J.A.J. Metz, and G. Meszena, Adaptive dynamics for physiologically structured population models. J. Math. Biol. 56 (2008), 673–742.zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    S. M´el´eard and V.C. Tran, Trait substitution sequence process and canonical equation for age-structured populations. J. Math. Biol. 58 (2009), 881–921.Google Scholar
  40. 40.
    J.A.J. Metz, Invasion fitness, canonical equations, and global invasion criteria for Mendelian populations. In U. Dieckmann and J.A.J. Metz (eds.), Elements of Adaptive Dynamics, Cambridge University Press, UK, In press.Google Scholar
  41. 41.
    J.A.J. Metz and C.C. de Kovel, The canonical equation of adaptive dynamics for diploid and haplo-diploid mendelian populations (in preperation).Google Scholar
  42. 42.
    T.J.M. Van Dooren, Adaptive dynamics with Mendelian genetics. In U. Dieckmann and J.A.J. Metz (eds.), Elements of Adaptive Dynamics, Cambridge University Press, UK, In press.Google Scholar
  43. 43.
    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
  44. 44.
    I. Salazar-Ciudad, On the origins of morphological disparity and its diverse developmental bases. Bioessays 28 (2006), 1112–1122.CrossRefGoogle Scholar
  45. 45.
    I. Salazar-Ciudad, Developmental constraints vs. variational properties: How pattern formation can help to understand evolution and development. J. Exp. Zool. Part B 306B (2006), 107–125.Google Scholar
  46. 46.
    O. Leimar, Evolutionary change and Darwinian demons. Selection 2 (2001), 65–72.Google Scholar
  47. 47.
    O. Leimar, The evolution of phenotypic polymorphism: Randomized strategies versus evolutionary branching. Am. Nat. 165 (2005), 669–681.CrossRefGoogle Scholar
  48. 48.
    O. Leimar, Multidimensional convergence stability. Evol. Ecol. Res. 11 (2009), 191– 208.Google Scholar
  49. 49.
    U. Dieckmann and M. Doebeli, On the origin of species by sympatric speciation. Nature 400 (1999), 354–357.CrossRefGoogle Scholar
  50. 50.
    S.A.H. Geritz and ´E. Kisdi, Adaptive dynamics in diploid, sexual populations and the evolution of reproductive isolation. Proc. R. Soc. Lond. Ser. B-Biol. Sci. 267 (2000), 1671–1678.Google Scholar
  51. 51.
    M. Doebeli and U. Dieckmann, Speciation along environmental gradients. Nature 421 (2003), 259–264.Google Scholar
  52. 52.
    P.S. Pennings, M. Kopp, G. Meszena, U. Dieckmann, and J. Hermisson, An analytically tractable model for competitive speciation. Am. Nat. 171 (2008), E44–E71.CrossRefGoogle Scholar
  53. 53.
    J. Ripa, When is sympatric speciation truly adaptive? An analysis of the joint evolution of resource utilization and assortative mating. Evol. Ecol. 23 (2009), 31–52.Google Scholar
  54. 54.
    M. Egas, M.W. Sabelis, F. Vala, and I. Lesna, Adaptive speciation in agricultural pests. In U. Dieckmann, M. Doebeli, J.A.J. Metz, and D. Tautz (eds.), Adaptive Speciation, 249–263, Cambridge University Press, UK, 2004.Google Scholar
  55. 55.
    J. Maynard Smith, Sympatric speciation. Am. Nat. 100 (1966), 637.Google Scholar
  56. 56.
    R.F. Hoekstra, R. Bijlsma, and A.J. Dolman, Polymorphism from environmental heterogeneity – models are only robust if the heterozygote is close in fitness to the favored homozygote in each environment. Genet. Res. 45 (1985), 299–314.CrossRefGoogle Scholar
  57. 57.
    ´ E. Kisdi and S.A.H. Geritz, Evolutionary branching and speciation: Insights from few-locus models. In U. Dieckmann and J.A.J. Metz (eds.), Elements of Adaptive Dynamics, Cambridge University Press, UK, In press.Google Scholar
  58. 58.
    N. Eldredge and S.J. Gould, Punctuated equilibria: an alternative to phyletic gradualism. In T. Schopf (ed.), Models in Paleobiology, 82–115, Freeman, Cooper and Company,, EUA, 1972.Google Scholar
  59. 59.
    S.J. Gould and N. Eldredge, Punctuated equilibria: the tempo and mode of evolution reconsidered. Paleobiology 3 (1977), 115.151.Google Scholar
  60. 60.
    J.H. Gillespie, The Causes of Molecular Evolution. Oxford University Press, UK, 1991.Google Scholar
  61. 61.
    A.G. Wouters, Explanation without a cause (verklaren zonder oorzaken te geven). Questiones Infinitae, Publications of the Zeno Institute of Philosophy, XXIX, Utrecht (1999).Google Scholar
  62. 62.
    F. Galis and J.A.J. Metz, Testing the vulnerability of the phylotypic stage: On modularity and evolutionary conservation. J. Exp. Zool. 291 (2001), 195–204.CrossRefGoogle Scholar
  63. 63.
    F. Galis, J.J.M. van Alphen, and J.A.J. Metz, Why do we have five fingers? The evolutionary constraint on digit numbers. Trends Ecol. Evol. 16 (2001), 637–646.Google Scholar
  64. 64.
    F. Galis, T.J.M. van Dooren, and J.A.J. Metz, Conservation of the segmented germband stage: modularity and robustness or pleiotropy and stabilizing selection? TIG 18 (2002), 504–509.CrossRefGoogle Scholar
  65. 65.
    F. Galis, T.J.M. Van Dooren, J.D. Feuth, J.A.J. Metz, A. Witkam, S. Ruinard, M.J. Steigenga, and L.C.D. Wijnaendts, Extreme selection in humans against homeotic transformations of cervical vertebrae. Evolution 60 (2006), 2643–2654.Google Scholar
  66. 66.
    F. Rodr´ıguez-Trelles, R. Tarrio, and F.J. Ayala, Molecular clocks: whence and whither? In P.C.J. Donoghue and M.P. Smith (eds.), Telling the Evolutionary Time: Molecular Clocks and the Fossil Record, 5–26, CRC Press, Boca Raton, USA, 2004.Google Scholar
  67. 67.
    S.H. Rice, A geometric model for the evolution of development. J. Theor. Biol. 143 (1990), 319–342.CrossRefMathSciNetGoogle Scholar
  68. 68.
    D. Waxman and J.J. Welch, Fisher’s microscope and Haldane’s ellipse. Am. Nat. 166 (2005), 447–457.CrossRefGoogle Scholar
  69. 69.
    D. Waxman, Fisher’s geometrical model of evolutionary adaptation – beyond spherical geometry. J. Theor. Biol. 241 (2006), 887–895.MathSciNetGoogle Scholar
  70. 70.
    D. Waxman, Mean curvature versus normality: A comparison of two approximations of fisher’s geometrical model. Theor. Popul. Biol. 71 (2007), 30–36.zbMATHCrossRefGoogle Scholar
  71. 71.
    F. Galis, I. Van Der Sluijs, T.J. Van Dooren, J.A. Metz, and M. Nussbaumer, Do large dogs die young? J. Exp. Zool. Part B 308B (2007), 119–126.CrossRefGoogle Scholar
  72. 72.
    M. Kopp and J. Hermisson, The evolution of genetic architecture under frequencydependent disruptive selection. Evolution 60 (2006), 1537–1550.Google Scholar
  73. 73.
    G.S. Van Doorn and U. Dieckmann, The long-term evolution of multilocus traits under frequency-dependent disruptive selection. Evolution 60 (2006), 2226–2238.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute of Biology and Mathematical InstituteLeidenNetherlands
  2. 2.Evolution and Ecology ProgramIIASALaxenburgAustria

Personalised recommendations