From adaptive dynamics to adaptive walks

Abstract

We consider an asexually reproducing population on a finite type space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event. On the individual-based level this population can be modelled as a measure-valued Markov process. Multiple variations of this system have been studied in the simultaneous limit of large populations and rare mutations, where the regime is chosen such that mutations are separated. We consider the deterministic system, resulting from the large population limit, and then let the mutation probability tend to zero. This corresponds to a much higher frequency of mutations, where multiple microscopic types are present at the same time. The limiting process resembles an adaptive walk or flight and jumps between different equilibria of coexisting types. The graph structure on the type space, determined by the possibilities to mutate, plays an important role in defining this jump process. In a variation of the above model, where the radius in which mutants can be spread is limited, we study the possibility of crossing valleys in the fitness landscape and derive different kinds of limiting walks.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Baar M, Bovier A (2018) The polymorphic evolution sequence for populations with phenotypic plasticity. Electron J Probab 23(72):1–27. https://doi.org/10.1214/18-EJP194

    MathSciNet  Article  MATH  Google Scholar 

  2. Baar M, Bovier A, Champagnat N (2017) From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step. Ann Appl Probab 27(2):1093–1170. https://doi.org/10.1214/16-AAP1227

    MathSciNet  Article  MATH  Google Scholar 

  3. Barton NH, Polechová J (2005) The limitations of adaptive dynamics as a model of evolution. J Evol Biol 18(5):1186–1190. https://doi.org/10.1111/j.1420-9101.2005.00943.x

    Article  Google Scholar 

  4. Berestycki J, Brunet E, Shi Z (2016) The number of accessible paths in the hypercube. Bernoulli 22(2):653–680. https://doi.org/10.3150/14-BEJ641

    MathSciNet  Article  MATH  Google Scholar 

  5. Berestycki J, Brunet E, Shi Z (2017) Accessibility percolation with backsteps. ALEA Lat Am J Probab Math Stat 14(1):45–62

    MathSciNet  Article  Google Scholar 

  6. Bolker B, Pacala SW (1997) Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor Popul Biol 52(3):179–197. https://doi.org/10.1006/tpbi.1997.1331

    Article  MATH  Google Scholar 

  7. Bolker BM, Pacala SW (1999) Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Am Nat 153(6):575–602. https://doi.org/10.1086/303199

    Article  Google Scholar 

  8. Bovier A, Wang SD (2013) Trait substitution trees on two time scales analysis. Markov Process Relat Fields 19(4):607–642

    MathSciNet  MATH  Google Scholar 

  9. Bovier A, Coquille L, Neukirch R (2018) The recovery of a recessive allele in a Mendelian diploid model. J Math Biol 77(4):971–1033. https://doi.org/10.1007/s00285-018-1240-z

    MathSciNet  Article  MATH  Google Scholar 

  10. Bovier A, Coquille L, Smadi C (2019) Crossing a fitness valley as a metastable transition in a stochastic population model. Ann Appl Probab (online first)

  11. Champagnat N (2006) A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch Process Appl 116(8):1127–1160. https://doi.org/10.1016/j.spa.2006.01.004

    MathSciNet  Article  MATH  Google Scholar 

  12. Champagnat N, Méléard S (2007) Invasion and adaptive evolution for individual-based spatially structured populations. J Math Biol 55(2):147–188. https://doi.org/10.1007/s00285-007-0072-z

    MathSciNet  Article  MATH  Google Scholar 

  13. Champagnat N, Méléard S (2011) Polymorphic evolution sequence and evolutionary branching. Probab Theory Relat Fields 151(1–2):45–94. https://doi.org/10.1007/s00440-010-0292-9

    MathSciNet  Article  MATH  Google Scholar 

  14. Champagnat N, Ferrière R, Méléard S (2008) From individual stochastic processes to macroscopic models in adaptive evolution. Stoch Models 24(suppl. 1):2–44. https://doi.org/10.1080/15326340802437710

    MathSciNet  Article  MATH  Google Scholar 

  15. Champagnat N, Jabin PE, Raoul G (2010) Convergence to equilibrium in competitive Lotka–Volterra and chemostat systems. C R Math Acad Sci Paris 348(23–24):1267–1272. https://doi.org/10.1016/j.crma.2010.11.001

    MathSciNet  Article  MATH  Google Scholar 

  16. Collet P, Méléard S, Metz JAJ (2013) A rigorous model study of the adaptive dynamics of Mendelian diploids. J Math Biol 67(3):569–607. https://doi.org/10.1007/s00285-012-0562-5

    MathSciNet  Article  MATH  Google Scholar 

  17. Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34(5–6):579–612. https://doi.org/10.1007/s002850050022

    MathSciNet  Article  MATH  Google Scholar 

  18. Dieckmann U, Law R (2000) Moment approximations of individual-based models. In: The geometry of ecological interactions: simplifying spatial complexity, Camb. Univ. Press, pp 252–270

  19. Durrett R, Mayberry J (2011) Traveling waves of selective sweeps. Ann Appl Probab 21(2):699–744. https://doi.org/10.1214/10-AAP721

    MathSciNet  Article  MATH  Google Scholar 

  20. Ethier SN, Kurtz TG (1986) Markov processes. Wiley Ser. in Probab. and Math. Stat. Wiley, New York. https://doi.org/10.1002/9780470316658

    Google Scholar 

  21. Fournier N, Méléard S (2004) A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann Appl Probab 14(4):1880–1919. https://doi.org/10.1214/105051604000000882

    MathSciNet  Article  MATH  Google Scholar 

  22. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Camb. Univ. Press, Cambridge. https://doi.org/10.1017/CBO9781139173179

    Google Scholar 

  23. Jain K (2007) Evolutionary dynamics of the most populated genotype on rugged fitness landscapes. Phys Rev E Stat Nonlinear Soft Matter Phys 76(3 Pt 1):031922

    Article  Google Scholar 

  24. Jain K, Krug J (2005) Evolutionary trajectories in rugged fitness landscapes. J Stat Mech: Theory Exp 2005(04):P04,008. https://doi.org/10.1088/1742-5468/2005/04/p04008

    Article  MATH  Google Scholar 

  25. Jain K, Krug J (2007) Deterministic and stochastic regimes of asexual evolution on rugged fitness landscapes. Genetics 175:1275–88. https://doi.org/10.1534/genetics.106.067165

    Article  Google Scholar 

  26. Kauffman S, Levin S (1987) Towards a general theory of adaptive walks on rugged landscapes. J Theor Biol 128(1):11–45. https://doi.org/10.1016/S0022-5193(87)80029-2

    MathSciNet  Article  Google Scholar 

  27. Kauffman SA (1992) The origins of order: Self-organization and selection in evolution. In: Spin glasses and biology, World Scientific, pp 61–100

  28. Krug J, Karl C (2003) Punctuated evolution for the quasispecies model. Physica A Stat Mech Appl 318(1):137–143. https://doi.org/10.1016/S0378-4371(02)01417-6 sTATPHYS - Kolkata IV

    Article  MATH  Google Scholar 

  29. Leman H (2016) Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence. Stoch Partial Differ Equ Anal Comput 4(4):791–826. https://doi.org/10.1007/s40072-016-0077-y

    MathSciNet  Article  MATH  Google Scholar 

  30. Maynard Smith J (1962) The scientist speculates: an anthology of partly-baked ideas. Basic Books, New York

    Google Scholar 

  31. Maynard Smith J (1970) Natural selection and the concept of a protein space. Nature 225:563–564. https://doi.org/10.1038/225563a0

    Article  Google Scholar 

  32. Metz JAJ, Geritz SAH, Meszéna G, Jacobs FJA, van Heerwaarden JS (1996) 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, vol 45, North-Holland, Amsterdam, pp 183–231

  33. Neidhart J, Krug J (2011) Adaptive walks and extreme value theory. Phys Rev Lett 107(178):102. https://doi.org/10.1103/PhysRevLett.107.178102

    Article  MATH  Google Scholar 

  34. Neukirch R, Bovier A (2017) Survival of a recessive allele in a Mendelian diploid model. J Math Biol 75(1):145–198. https://doi.org/10.1007/s00285-016-1081-6

    MathSciNet  Article  MATH  Google Scholar 

  35. Nowak S, Krug J (2015) Analysis of adaptive walks on NK fitness landscapes with different interaction schemes. J Stat Mech Theory Exp (6):P06,014, 27, https://doi.org/10.1088/1742-5468/2015/06/p06014

    MathSciNet  Article  Google Scholar 

  36. Orr HA (2003) A minimum on the mean number of steps taken in adaptive walks. J Theor Biol 220(2):241–247. https://doi.org/10.1006/jtbi.2003.3161

    MathSciNet  Article  Google Scholar 

  37. Schmiegelt B, Krug J (2014) Evolutionary accessibility of modular fitness landscapes. J Stat Phys 154(1–2):334–355. https://doi.org/10.1007/s10955-013-0868-8

    MathSciNet  Article  MATH  Google Scholar 

  38. Tran VC (2008) Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM Probab Stat 12:345–386. https://doi.org/10.1051/ps:2007052

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anna Kraut.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy GZ 2047/1, Projekt-ID 390685813 and GZ 2151, Project-ID 390873048 and through the Priority Programme 1590 “Probabilistic Structures in Evolution”

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kraut, A., Bovier, A. From adaptive dynamics to adaptive walks. J. Math. Biol. 79, 1699–1747 (2019). https://doi.org/10.1007/s00285-019-01408-6

Download citation

Keywords

  • Adaptive dynamics
  • Adaptive walks
  • Individual-based models
  • Competitive Lotka–Volterra systems with mutation

Mathematics Subject Classification

  • 37N25
  • 60J27
  • 92D15
  • 92D25