Advertisement

Inside Dynamics of Integrodifference Equations with Mutations

  • Nathan G. MarculisEmail author
  • Mark A. Lewis
Original Article
  • 16 Downloads

Abstract

The method of inside dynamics provides a theory that can track the dynamics of neutral gene fractions in spreading populations. However, the role of mutations has so far been absent in the study of the gene flow of neutral fractions via inside dynamics. Using integrodifference equations, we develop a neutral genetic mutation model by extending a previously established scalar inside dynamics model. To classify the mutation dynamics, we define a mutation class as the set of neutral fractions that can mutate into one another. We show that the spread of neutral genetic fractions is dependent on the leading edge of population as well as the structure of the mutation matrix. Specifically, we show that the neutral fractions that contribute to the spread of the population must belong to the same mutation class as the neutral fraction found in the leading edge of the population. We prove that the asymptotic proportion of individuals at the leading edge of the population spread is given by the dominant right eigenvector of the associated mutation matrix, independent of growth and dispersal parameters. In addition, we provide numerical simulations to demonstrate our mathematical results, to extend their generality and to develop new conjectures about our model.

Keywords

Integrodifference equations Mutations Neutral genetic diversity Range expansion Spreading speed 

Notes

Acknowledgements

This research was supported by a grant to MAL from the Natural Science and Engineering Research Council of Canada (Grant No. NET GP 434810-12) to the TRIA Network, with contributions from Alberta Agriculture and Forestry, Foothills Research Institute, Manitoba Conservation and Water Stewardship, Natural Resources Canada-Canadian Forest Service, Northwest Territories Environment and Natural Resources, Ontario Ministry of Natural Resources and Forestry, Saskatchewan Ministry of Environment, West Fraser and Weyerhaeuser. MAL is also grateful for support through NSERC and the Canada Research Chair Program. NGM acknowledges support from NSERC TRIA-Net Collaborative Research Grant and would like to express his thanks to the Lewis Research Group for the many discussions and constructive feedback throughout this work. We are also grateful for the helpful and detailed comments from anonymous reviewers.

Supplementary material

References

  1. Arenas M (2015) Trends in substitution models of molecular evolution. Front Genet 6:319CrossRefGoogle Scholar
  2. Bateman AW, Buttenschön A, Erickson KD, Marculis NG (2017) Barnacles vs bullies: modelling biocontrol of the invasive european green crab using a castrating barnacle parasite. Theor Ecol 10(3):305–318CrossRefGoogle Scholar
  3. Bonnefon O, Garnier J, Hamel F, Roques L (2013) Inside dynamics of delayed travelling waves. Math Mod Nat Phen 8:44–61zbMATHCrossRefGoogle Scholar
  4. Bonnefon O, Coville J, Garnier J, Roques L (2014) Inside dynamics of solutions of integro-differential equations. Discret Contin Dyn Syst Ser B 19(10):3057–3085MathSciNetzbMATHGoogle Scholar
  5. Bromham L, Penny D (2003) The modern molecular clock. Nat Rev Genet 4(3):216CrossRefGoogle Scholar
  6. Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19(90):297–301MathSciNetzbMATHCrossRefGoogle Scholar
  7. Duret L (2008) Neutral theory: the null hypothesis of molecular evolution. Nat Educ 1:803–806Google Scholar
  8. Edmonds CA, Lillie AS, Cavalli-Sforza LL (2004) Mutations arising in the wave front of an expanding population. Proc Natl Acad Sci 101(4):975–979CrossRefGoogle Scholar
  9. Excoffier L, Ray N (2008) Surfing during population expansions promotes genetic revolutions and structuration. Trends Ecol Evol 23(7):347–351CrossRefGoogle Scholar
  10. Excoffier L, Foll M, Petit RJ (2009) Genetic consequences of range expansions. Annu Rev of Ecol Evol Syst 40:481–501CrossRefGoogle Scholar
  11. Felsenstein J (1981) Evolutionary trees from DNA sequences: a maximum likelihood approach. J Mol Evol 17(6):368–376CrossRefGoogle Scholar
  12. Garnier J, Lewis MA (2016) Expansion under climate change: the genetic consequences. Bull Math Biol 78(11):2165–2185MathSciNetzbMATHCrossRefGoogle Scholar
  13. Garnier J, Giletti T, Hamel F, Roques L (2012) Inside dynamics of pulled and pushed fronts. J Math Pures Appl 98(4):428–449MathSciNetzbMATHCrossRefGoogle Scholar
  14. Hallatschek O, Nelson DR (2008) Gene surfing in expanding populations. Theor Popul Biol 73(1):158–170zbMATHCrossRefGoogle Scholar
  15. Hallatschek O, Hersen P, Ramanathan S, Nelson DR (2007) Genetic drift at expanding frontiers promotes gene segregation. Proc Natl Acad Sci 104(50):19926–19930CrossRefGoogle Scholar
  16. Ho S (2008) The molecular clock and estimating species divergence. Nat Educ 1(1):1–2Google Scholar
  17. Jukes T, Cantor C (1969) Evolution of protein molecules. In: ‘mammalian protein metabolism’. (ed. hn munro.) pp 21–132CrossRefGoogle Scholar
  18. Kimura M (1980) A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. J Mol Evol 16(2):111–120CrossRefGoogle Scholar
  19. Klopfstein S, Currat M, Excoffier L (2006) The fate of mutations surfing on the wave of a range expansion. Mol Biol Evol 23(3):482–490CrossRefGoogle Scholar
  20. Kot M, Lewis MA, van den Driessche P (1996) Dispersal data and the spread of invading organisms. Ecology 77(7):2027–2042CrossRefGoogle Scholar
  21. Krkošek M, Lauzon-Guay JS, Lewis MA (2007) Relating dispersal and range expansion of California sea otters. Theor Popul Biol 71(4):401–407zbMATHCrossRefGoogle Scholar
  22. Lande R (1992) Neutral theory of quantitative genetic variance in an island model with local extinction and colonization. Evolution 46(2):381–389CrossRefGoogle Scholar
  23. Lehe R, Hallatschek O, Peliti L (2012) The rate of beneficial mutations surfing on the wave of a range expansion. PLoS Comput Biol 8(3):e1002447MathSciNetCrossRefGoogle Scholar
  24. Lewis MA, Petrovskii SV, Potts JR (2016) The mathematics behind biological invasions, vol 44. Springer, Cham zbMATHCrossRefGoogle Scholar
  25. Lewis MA, Marculis NG, Shen Z (2018) Integrodifference equations in the presence of climate change: persistence criterion, travelling waves and inside dynamics. J Math Biol 77(6–7):1649–1687MathSciNetzbMATHCrossRefGoogle Scholar
  26. Li B, Weinberger HF, Lewis MA (2005) Spreading speeds as slowest wave speeds for cooperative systems. Math Biosci 196(1):82–98MathSciNetzbMATHCrossRefGoogle Scholar
  27. Lui R (1983) Existence and stability of travelling wave solutions of a nonlinear integral operator. J Math Biol 16(3):199–220MathSciNetzbMATHCrossRefGoogle Scholar
  28. Lutscher F, Pachepsky E, Lewis MA (2005) The effect of dispersal patterns on stream populations. SIAM Rev 47(4):749–772MathSciNetzbMATHCrossRefGoogle Scholar
  29. Lynch M (1988) The divergence of neutral quantitative characters among partially isolated populations. Evolution 42(3):455–466CrossRefGoogle Scholar
  30. Marculis NG, Lui R, Lewis MA (2017) Neutral genetic patterns for expanding populations with nonoverlapping generations. Bull Math Biol 79(4):828–852MathSciNetzbMATHCrossRefGoogle Scholar
  31. Marculis NG, Garnier J, Lui R, Lewis MA (2019) Inside dynamics for stage-structured integrodifference equations. In Press, J Math BiolGoogle Scholar
  32. Mayr E (1940) Speciation phenomena in birds. Am Nat 74(752):249–278CrossRefGoogle Scholar
  33. Morin PA, Luikart G, Wayne RK (2004) Snps in ecology, evolution and conservation. Trends Ecol Evol 19(4):208–216CrossRefGoogle Scholar
  34. Pannell JR, Charlesworth B (1999) Neutral genetic diversity in a metapopulation with recurrent local extinction and recolonization. Evolution 53(3):664–676CrossRefGoogle Scholar
  35. Pannell JR, Charlesworth B (2000) Effects of metapopulation processes on measures of genetic diversity. Philos Trans R Soc Lond B Biol Sci 355(1404):1851–1864CrossRefGoogle Scholar
  36. Reimer JR, Bonsall MB, Maini PK (2016) Approximating the critical domain size of integrodifference equations. Bull Math Biol 78(1):72–109MathSciNetzbMATHCrossRefGoogle Scholar
  37. Roques L, Garnier J, Hamel F, Klein EK (2012) Allee effect promotes diversity in traveling waves of colonization. Proc Natl Acad Sci 109(23):8828–8833MathSciNetCrossRefGoogle Scholar
  38. Roques L, Hosono Y, Bonnefon O, Boivin T (2015) The effect of competition on the neutral intraspecific diversity of invasive species. J Math Biol 71(2):465–489MathSciNetzbMATHCrossRefGoogle Scholar
  39. Selkoe KA, Toonen RJ (2006) Microsatellites for ecologists: a practical guide to using and evaluating microsatellite markers. Ecol Lett 9(5):615–629CrossRefGoogle Scholar
  40. Slatkin M (1985) Gene flow in natural populations. Annu Rev Ecol Syst 16(1):393–430CrossRefGoogle Scholar
  41. Slatkin M, Excoffier L (2012) Serial founder effects during range expansion: a spatial analog of genetic drift. Genetics 191(1):171–181CrossRefGoogle Scholar
  42. Stokes A (1976) On two types of moving front in quasilinear diffusion. Math Biosci 31(3–4):307–315MathSciNetzbMATHCrossRefGoogle Scholar
  43. Van Kirk RW, Lewis MA (1997) Integrodifference models for persistence in fragmented habitats. Bull Math Biol 59(1):107zbMATHCrossRefGoogle Scholar
  44. Weinberger HF (1982) Long-time behavior of a class of biological models. SIAM J Math Anal 13(3):353–396MathSciNetzbMATHCrossRefGoogle Scholar
  45. Zhou Y, Kot M (2011) Discrete-time growth-dispersal models with shifting species ranges. Theor Ecol 4(1):13–25CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2020

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Department of Mathematical and Statistical Sciences & Department of Biological SciencesUniversity of AlbertaEdmontonCanada

Personalised recommendations