Advertisement

Journal of Mathematical Biology

, Volume 66, Issue 1–2, pp 225–279 | Cite as

What life cycle graphs can tell about the evolution of life histories

  • Claus Rueffler
  • Johan A. J. Metz
  • Tom J. M. Van Dooren
Article

Abstract

We analyze long-term evolutionary dynamics in a large class of life history models. The model family is characterized by discrete-time population dynamics and a finite number of individual states such that the life cycle can be described in terms of a population projection matrix. We allow an arbitrary number of demographic parameters to be subject to density-dependent population regulation and two or more demographic parameters to be subject to evolutionary change. Our aim is to identify structural features of life cycles and modes of population regulation that correspond to specific evolutionary dynamics. Our derivations are based on a fitness proxy that is an algebraically simple function of loops within the life cycle. This allows us to phrase the results in terms of properties of such loops which are readily interpreted biologically. The following results could be obtained. First, we give sufficient conditions for the existence of optimisation principles in models with an arbitrary number of evolving traits. These models are then classified with respect to their appropriate optimisation principle. Second, under the assumption of just two evolving traits we identify structural features of the life cycle that determine whether equilibria of the monomorphic adaptive dynamics (evolutionarily singular points) correspond to fitness minima or maxima. Third, for one class of frequency-dependent models, where optimisation is not possible, we present sufficient conditions that allow classifying singular points in terms of the curvature of the trade-off curve. Throughout the article we illustrate the utility of our framework with a variety of examples.

Keywords

Adaptive dynamics Density dependence Frequency dependence Life history theory Matrix model Evolutionary optimisation 

Mathematics Subject Classification (2000)

92D15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abrams PA (2005) ‘Adaptive Dynamics’ vs. ‘adaptive dynamics’. J Evol Biol 18: 1162–1165CrossRefGoogle Scholar
  2. Armstrong RA, McGehee R (1980) Competitive exclusion. Am Nat 115: 151–170MathSciNetCrossRefGoogle Scholar
  3. Bowers RG (2010) On the determination of evolutionary outcomes directly from the population dynamics of the resident. J Math Biol 62: 901–924MathSciNetCrossRefGoogle Scholar
  4. Bowers RG, Hoyle A, White A, Boots M (2005) The geometric theory of adaptive evolution: trade-off and invasion plots. J Theor Biol 233: 363–377MathSciNetCrossRefGoogle Scholar
  5. Brault S, Caswell H (1993) Pod-specific demography of killer whales (Orcinus orca). Ecology 74: 1444–1454CrossRefGoogle Scholar
  6. Caswell H (1982) Optimal life histories and the maximization of reproductive value—a general theorem for complex life-cycles. Ecology 63: 1218–1222CrossRefGoogle Scholar
  7. Caswell H (2001) Matrix population models, 2nd edn. Sinauer, SunderlandGoogle Scholar
  8. Charlesworth B (1994) Evolution in age-structured populations, 2nd edn. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  9. Charnov EL (1993) Life history invariants: some explorations of symmetry in evolutionary ecology. Oxford University Press, OxfordGoogle Scholar
  10. de Mazancourt C, Dieckmann U (2004) Trade-off geometries and frequency-dependent selection. American Nat 164: 765–778CrossRefGoogle Scholar
  11. Dercole F, Ferrière R, Rinaldi S (2002) Ecological bistability and evolutionary reversals under asymmetrical competition. Evolution 56: 1081–1090Google Scholar
  12. Dercole F, Rinaldi S (2008) Analysis of evolutionary processes: the adaptive dynamics approach and its applications. Princeton University Press, PrincetonzbMATHGoogle Scholar
  13. Dieckmann U, Law R (1996) The dynamical theory of coevolution: A derivation from stochastic ecological processes. J Math Biol 34: 579–612MathSciNetzbMATHCrossRefGoogle Scholar
  14. Dieckmann U, Metz JAJ (2006) Surprising evolutionary predictions from enhanced ecological realism. Theor Popul Biol 69: 263–381zbMATHCrossRefGoogle Scholar
  15. Diekmann O (2004) A beginners guide to adaptive dynamics. In: Rudnicki R (ed) Mathematical modelling of population dynamics. Banach Center Publications, vol 63. Polish Academy of Sciences, Warszawa, pp 47–86Google Scholar
  16. Diekmann O, Gyllenberg M, Metz JAJ, Thieme HR (1998) On the formulation and analysis of general deterministic structured population models. I. Linear theory. J Math Biol 36: 349–388MathSciNetzbMATHCrossRefGoogle Scholar
  17. Diekmann O, Gyllenberg M, Huang H, Kirkilionis M, Metz JAJ, Thieme HR (2001) On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory. J Math Biol 43: 157–189MathSciNetzbMATHCrossRefGoogle Scholar
  18. Diekmann O, Gyllenberg M, Metz JAJ (2003) Steady state analysis of structured population models. Theor Popul Biol 63: 309–338zbMATHCrossRefGoogle Scholar
  19. Ellner S, Hairston NG (1994) Role of overlapping generations in maintaining genetic variation in a fluctuating environment. Am Nat 143: 403–417CrossRefGoogle Scholar
  20. Geritz SAH (2005) Resident–invader dynamics and the coexistence of similar strategies. J Math Biol 50: 67–82MathSciNetzbMATHCrossRefGoogle Scholar
  21. Geritz SAH, Gyllenberg M, Jacobs FJA, Parvinen K (2002) Invasion dynamics and attractor inheritance. J Math Biol 44: 548–560MathSciNetzbMATHCrossRefGoogle Scholar
  22. Geritz SAH, Kisdi É, Meszéna G, Metz JAJ (1998) Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol Ecol 12: 35–57CrossRefGoogle Scholar
  23. Gyllenberg M, Service R (2011) Necessary and sufficient conditions for the existence of an optimisation principle in evolution. J Math Biol 62: 359–369MathSciNetzbMATHCrossRefGoogle Scholar
  24. Heino M, Metz JAJ, Kaitala V (1998) The enigma of frequency-dependent selection. Trends Ecol Evol 13: 367–370CrossRefGoogle Scholar
  25. Horn R, Johnson C (1985) Matrix analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
  26. Hubbell S, Werner P (1979) Measuring the intrinsic rate of increase of populations with heterogeneous life-histories. Am Nat 113: 277–293MathSciNetCrossRefGoogle Scholar
  27. Leimar O (2009) Multidimensional convergence stability. Evol Ecol Res 11: 191–208Google Scholar
  28. Levins SA (1970) Community equilibria and stability, and an extension of the competitive exclusion principle. Am Nat 104: 413–423CrossRefGoogle Scholar
  29. Levins R (1962) Theory of fitness in a heterogeneous environment. I. The fitness set and the adaptive function. Am Nat 96: 361–373CrossRefGoogle Scholar
  30. Levins R (1968) Evolution in changing environments. Princeton University Press, PrincetonGoogle Scholar
  31. MacArthur RH (1970) Species packing and competitive equilibrium for many species. Theor Popul Biol 1: 1–11CrossRefGoogle Scholar
  32. Meszéna G, Gyllenberg M, Jacobs FJA, Metz JAJ (2005) Dynamics of similar populations: the link between population dynamics and evolution. Phys Rev Lett 95: 078105(4)CrossRefGoogle Scholar
  33. Meszéna G, Gyllenberg M, Pásztor L, Metz JAJ (2006) Competitive exclusion and limiting similarity: a unified theory. Theor Popul Biol 69: 68–87zbMATHCrossRefGoogle Scholar
  34. Metz JAJ (2005) Eight personal rules for doing science. J Evol Biol 18: 1178–1181CrossRefGoogle Scholar
  35. Metz JAJ (2008) Fitness. In: Jørgensen S, Fath B (eds) Evolutionary ecology. Encyclopedia of Ecology, vol [2]. Elsevier, , pp 1599–1612Google Scholar
  36. Metz JAJ (2011) Thoughts on the geometry of meso-evolution: collecting mathematical elements for a post-modern synthesis. In: Chalub FACC, Rodrigues JF (eds) The mathematics of Darwin’s Legacy. Birkhauser, BaselGoogle Scholar
  37. Metz JAJ, Diekmann O (1986) The dynamics of physiologically structured populations. Lecture Notes in Biomathematics, vol 68. Springer, BerlinGoogle Scholar
  38. Metz JAJ, Geritz SAH, Meszéna G, Jacobs FJA, Van Heerwaarden JS (1996a) Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. In: van Strien S, Verduyn Lunel S (eds) Stochastic and spatial structures of dynamical systems, Proceedings of the Royal Dutch Academy of Science. North Holland, Dordrecht, Netherlands, pp 183–231. http://www.iiasa.ac.at/Research/ADN/Series.html
  39. Metz JAJ, Klinkhamer PGL, de Jong TJ (2009) A different model to explain delayed germination. Evol Ecol Res 11: 177–190Google Scholar
  40. Metz JAJ, Leimar O (2011) A simple fitness proxy for ESS calculations in structured populations with continuous traits, with applications to the evolution of haplo-diploids and genetic dimorphisms. J Biol Dyn 5: 163–190MathSciNetCrossRefGoogle Scholar
  41. Metz JAJ, Mylius SD, Diekmann O (1996b) When does evolution optimize? On the relation between types of density dependence and evolutionarily stable life history parameters. IIASA working paper WP-96-04. http://www.iiasa.ac.at/Research/ADN/Series.html
  42. Metz JAJ, Mylius SD, Diekmann O (2008) When does evolution optimise. Evol Ecol Res 10: 629–654Google Scholar
  43. Metz JAJ, Mylius SD, Diekmann O (2008) Even in the odd cases when evolution optimises, unrelated population dynamical details may shine through in the ESS. Evol Ecol Res 10: 655–666Google Scholar
  44. Metz JAJ, Nisbet RM, Geritz SAH (1992) How should we define ‘fitness’ for general ecological scenarios?. Trends Ecol Evol 7: 198–202CrossRefGoogle Scholar
  45. Mylius SD, Diekmann O (1995) On evolutionary stable life histories, optimisation and the need to be specific about density dependence. Oikos 74: 218–224CrossRefGoogle Scholar
  46. Nowak M (1990) An evolutionary stable strategy may be inaccessible. J Theor Biol 142: 237–241CrossRefGoogle Scholar
  47. Otto SP, Day T (2007) A biologist’s guide to mathematical modeling in ecology and evolution. Princeton University Press, PrincetonzbMATHGoogle Scholar
  48. Pásztor L, Meszéna G, Kisdi É (1996) R 0 or r: a matter of taste. J Evol Biol 1996: 511–518CrossRefGoogle Scholar
  49. Powell EO (1958) Criteria for the growth of contaminants and mutants in continuous culture. J Gen Microbiol 18: 259–268CrossRefGoogle Scholar
  50. Ravigné V, Dieckmann U, Olivieri I (2009) Live where you thrive: joint evolution of habitat choice and local adaptation facilitates specialization and promotes diversity. Am Nat 174: E141–E169CrossRefGoogle Scholar
  51. Roff D (2002) Life history evolution. Sinauer, SunderlandGoogle Scholar
  52. Rueffler C (submitted) A new formula for the basic reproduction ratio R 0. Bull Math BiolGoogle Scholar
  53. Rueffler C, Van Dooren TJM, Leimar O, Abrams PA (2006) Disruptive selection and then what?. Trends Ecol Evol 21: 238–245CrossRefGoogle Scholar
  54. Rueffler C, Van Dooren TJM, Metz JAJ (2004) Adaptive walks on changing landscapes: Levins’ approached extended. Theor Popul Biol 65: 165–178zbMATHCrossRefGoogle Scholar
  55. Schneider K (2006) A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition. J Math Biol 52: 483–523MathSciNetzbMATHCrossRefGoogle Scholar
  56. Stearns SC (1992) The evolution of life histories. Oxford University Press, OxfordGoogle Scholar
  57. Takada T, Nakajima H (1992) An analysis of life history evolution in terms of the density-dependent Lefkovitch matrix model. Math Biosci 112: 155–176MathSciNetzbMATHCrossRefGoogle Scholar
  58. Takada T, Nakajima H (1996) The optimal allocation for seed reproduction and vegetative reproduction in perennial plants: an application to the density-dependent transition matrix model. J Theor Biol 182: 179–191CrossRefGoogle Scholar
  59. Takada T, Nakajima H (1998) Theorems on the invasion process in stage-structured populations with density-dependent dynamics. J Math Biol 36: 497–514MathSciNetzbMATHCrossRefGoogle Scholar
  60. Van Dooren TJM (2006) Protected polymorphism and evolutionary stability in pleiotropic models with trait-specific dominance. Evolution 60: 1991–2003CrossRefGoogle Scholar
  61. Van Dooren TJM (2012) Adaptive dynamics for mendelian genetics. In: Metz JAJ, Dieckmann U (eds) Elements of adaptive dynamics. Cambridge University Press, Cambridge (in press)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Claus Rueffler
    • 1
  • Johan A. J. Metz
    • 2
    • 3
    • 4
  • Tom J. M. Van Dooren
    • 4
    • 5
  1. 1.Mathematics and Biosciences Group, Department of MathematicsUniversity of ViennaViennaAustria
  2. 2.Mathematical Institute and Institute of BiologyLeiden UniversityLeidenThe Netherlands
  3. 3.Evolution and Ecology ProgramInternational Institute of Applied Systems AnalysisLaxenburgAustria
  4. 4.Netherlands Centre for Biodiversity, NaturalisLeidenThe Netherlands
  5. 5.UMR 7625 Ecology and Evolution, Eco-Evolutionary Mathematics, Ecole Normale SupérieureParis Cedex 05France

Personalised recommendations