Acta Biotheoretica

, Volume 63, Issue 4, pp 341–361 | Cite as

How Complex, Probable, and Predictable is Genetically Driven Red Queen Chaos?

Regular Article

Abstract

Coevolution between two antagonistic species has been widely studied theoretically for both ecologically- and genetically-driven Red Queen dynamics. A typical outcome of these systems is an oscillatory behavior causing an endless series of one species adaptation and others counter-adaptation. More recently, a mathematical model combining a three-species food chain system with an adaptive dynamics approach revealed genetically driven chaotic Red Queen coevolution. In the present article, we analyze this mathematical model mainly focusing on the impact of species rates of evolution (mutation rates) in the dynamics. Firstly, we analytically proof the boundedness of the trajectories of the chaotic attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. By using symbolic dynamics theory, we quantify the complexity of genetically driven Red Queen chaos computing the topological entropy of existing one-dimensional iterated maps using Markov partitions. Co-dimensional two bifurcation diagrams are also built from the period ordering of the orbits of the maps. Then, we study the predictability of the Red Queen chaos, found in narrow regions of mutation rates. To extend the previous analyses, we also computed the likeliness of finding chaos in a given region of the parameter space varying other model parameters simultaneously. Such analyses allowed us to compute a mean predictability measure for the system in the explored region of the parameter space. We found that genetically driven Red Queen chaos, although being restricted to small regions of the analyzed parameter space, might be highly unpredictable.

Keywords

Adaptive dynamics Chaos Coevolution Ecology  Predator-prey Predictability Red Queen 

References

  1. Agrawal AF, Lively CM (2001) Parasites and the evolution of self-fertilization. Evolution 55:869–879CrossRefGoogle Scholar
  2. Billingsley P (1965) Ergodic theory and information. Wiley, New YorkGoogle Scholar
  3. Blanchard F (2009) Topological chaos: What may this mean? J Diff Eq Appl 15:23–46CrossRefGoogle Scholar
  4. Blount ZD, Borland CZ, Lenski RE (2008) Historical contingency and the evolution of a key innovation in an experimental population of Escherichia coli. Proc Natl Acad Sci USA 105:7899–7906CrossRefGoogle Scholar
  5. Carrasco P, de la Iglesia F, Elena SF (2007) Distribution of fitness and virulence effects caused by single-nucleotide substitutions in Tobacco etch virus. J Virol 81:12979–12984CrossRefGoogle Scholar
  6. Changpin L, Guanrong C (2004) Estimating the Lyapunov exponents of discrete systems. Chaos 14:343–346CrossRefGoogle Scholar
  7. Cooper TF, Rozen DE, Lenski RE (2003) Parallel changes in gene expression after 20,000 generations of evolution in Escherichia coli. Proc Natl Acad Sci USA 100:1072–1077CrossRefGoogle Scholar
  8. Crutchfield JP, Packard NH (1982) Symbolic dynamics of one-dimensional maps:entropies. Finite Precis Noise Int J Theor Phys 21:433–466CrossRefGoogle Scholar
  9. Day T (2012) Computability, Gödel’s incompleteness theorem, and an inherent limit on the predictability of evolution. J R Soc Interface 9:624–639CrossRefGoogle Scholar
  10. Decaestecker E, Gaba S, Raeymaekers JAM, Stoks R, Kerckhoven Van, Ebert D, Meester LD (2007) Host-parasite ’Red Queen’ dynamics archived in pond sediment. Nature 450:870–873CrossRefGoogle Scholar
  11. Deng B (2001) Food chain chaos due to junction-fold point. Chaos 11:514–525CrossRefGoogle Scholar
  12. Dercole F, Ferriere R, Rinaldi S (2013) Chaotic Red Queen coevolution in three-species food chains. Proc R Soc Lond B 277:2321–2330. doi:10.1098/rspb.2010.0209 CrossRefGoogle Scholar
  13. Dercole F, Rinaldi S (2008) Analysis of evolutionary processes: the adaptive dynamics approach and its applications. In: Levin Simon A (ed) Princeton series in theoretical and computational biology. Princeton University Press, PrincetonGoogle Scholar
  14. Dercole F, Rinaldi S (2010) Evolutionary dynamics can be chaotic: a first example. Int J Bifurcat Chaos 20:3473CrossRefGoogle Scholar
  15. Dieckmann U, Marrow P, Law R (1995) Evolutionary cycling in predator-prey interactions:population dynamics and the Red Queen. J Theor Biol 176:91–92CrossRefGoogle Scholar
  16. Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34:579–612CrossRefGoogle Scholar
  17. Ebert D (2008) Host-parasite coevolution: insights from the Daphnia-parasite model system. Curr Opin Microbiol 11:290–301CrossRefGoogle Scholar
  18. Eckmann J-P, Ruelle D (1985) Ergodic theory of chaos and strange attractors. Rev Mod Phys 57:617–656CrossRefGoogle Scholar
  19. Elena SF, Cooper VS, Lenski RE (1996) Punctuated evolution caused by selection of rare beneficial mutations. Science 272:1802–1804CrossRefGoogle Scholar
  20. Ellner SP, Turchin P (2005) When can noise induce chaos and why does it matter: a critique. Oikos 111:620–631CrossRefGoogle Scholar
  21. Fraedrich K (1987) Estimating weather and climate predictability on attractors. J Atmosph Sci 44:722–728CrossRefGoogle Scholar
  22. Gaba S, Ebert D (2009) Time-shift experiments as a tool to study antagonistic coevolution. Trends Ecol Evol 24:226–232CrossRefGoogle Scholar
  23. Gandon S (2002) Local adaptation and the geometry of host-parasite coevolution. Ecol Lett 5:246–256CrossRefGoogle Scholar
  24. Hamilton WD (1980) Sex versus non-sex versus parasite. Oikos 35:282–290CrossRefGoogle Scholar
  25. Hamilton WD, Axelrod A, Tanese R (1990) Sexual reproduction as an adaptation to resist parasites (a review). Proc Natl Acad Sci USA 87:3566–3573CrossRefGoogle Scholar
  26. Hastings A, Powell T (1991) Chaos in a three-species food chain. Ecology 72:896–903CrossRefGoogle Scholar
  27. Hoffman A (1991) Testing the Red Queen hypothesis. J Evol Biol 4:1–7CrossRefGoogle Scholar
  28. Jaenike J (1978) An hypothesis to account for the maintenance of sex in populations. Evol Theory 3:191–194Google Scholar
  29. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511809187 CrossRefGoogle Scholar
  30. Khibnik AI, Kondrashov AS (1997) Three mechanisms of Red Queen dynamics. Proc Roy Soc Lond B 264:1049–1056CrossRefGoogle Scholar
  31. King KC, Delph LF, Jokela J, Lively CM (2009) The geographic mosaic of sex and the Red Queen. Curr Biol 19:1438–1441CrossRefGoogle Scholar
  32. Kolmogorov AN (1958) New metric invariant of transitive dynamical systems and auto-morphism of Lebesgue spaces. Dokl Akad Nauk SSSR 119:861–864Google Scholar
  33. Lai YC, Liu Z, Billings L (2003) Noise-induced unstable dimension variability and transition to chaos in random dynamical systems. Phys Rev E 67:026210CrossRefGoogle Scholar
  34. Letellier C, Aguire LA, Maquet J (2005) Relation between observability and differential embeddings for nonlinear dynamics. Phys Rev E 71:066213. doi:10.1103/PhysRevE.71.066213 CrossRefGoogle Scholar
  35. Letellier C, Denis F, Aguire LA (2013) What we can learn from a chaotic cancer model. J Theor Biol 322:7–16. doi:10.1016/j.jtbi.2013.01.003 CrossRefGoogle Scholar
  36. Letellier C, Aguire LA (2002) Investigating nonlinear dynamics from time series: the influence of symmetries and the choice of observables. Chaos 12:549–558. doi:10.1063/1.1487570 CrossRefGoogle Scholar
  37. Lively CM (1987) Evidence from a New Zealand snail for the maintenance of sex by parasitism. Nature 328:519–521CrossRefGoogle Scholar
  38. Lovkovksy AE, Wolf YI, Koonin EV (2011) predictability of evolutionary trajectories in fitness landscapes. PLoS Comp Biol 7:e1002302CrossRefGoogle Scholar
  39. Lozovsky ER, Chookajorn T, Brown KM, Imwong M, Shaw PJ, Kamchonwongpaisan S, Neafsey DE, Weinreich DM, Hartl DL (2009) Stepwise acquisition of pyrimethamine resistance in the malaria parasite. Proc Natl Acad Sci USA 106:12025–12030CrossRefGoogle Scholar
  40. Milnor J, Thurston W (1988) On iterated maps of the interval I and II. Lect. Notes in Math., 1342, Springer, pp 465–563 doi:10.1007/BFb0082847
  41. Misiurewicz M, Szlenk W (1980) Entropy of piecewise monotone mappings. Studia Math 67:45–63Google Scholar
  42. Morran LT, Schmidt OG, Gelarden IA, Parrish RC II, Lively CM (2011) Running with the Red Queen: host-parasite coevolution selects for biparental sex. Science 333:216–218CrossRefGoogle Scholar
  43. Morris SC (2009) The predictability of evolution: glimpses into a post-Darwinian world. Naturwissenschaften 96:1313–1337CrossRefGoogle Scholar
  44. Morris SC (2010) Evolution: like any other science it is predictable. Philos Trans R Soc B 365:133–145CrossRefGoogle Scholar
  45. Parker T, Chua LO (1989) Practical numerical algorithms for chaotic systems. Springer, BerlinCrossRefGoogle Scholar
  46. Pesin YB (1976) Lyapunov characteristic exponent and ergodic properties of smooth dynamical systems with an invariant measure. Sov Math Dokl 17:196–199Google Scholar
  47. Rand DA, Wilson HB (1981) Chaotic stochasticity—a ubiquitous source of unpredictability in epidemics. Proc R Soc Lond B 246:179–184CrossRefGoogle Scholar
  48. Russell J, Cohn R (2013) Gronwalls inequality. Bookvika publishing, CimmeriaGoogle Scholar
  49. Salathe M, Kouyos RD, Bonhoeffer S (2008) The state of affairs in the Kingdom of the Red Queen. Trends Ecol Evol 23:439–445CrossRefGoogle Scholar
  50. Salverda MLM et al (2011) Initial mutations direct alternative pathways of protein evolution. PLoS Genet 7:e1001321CrossRefGoogle Scholar
  51. Sanjuán R, Moya A, Elena SF (2004) The distribution of fitness effects caused by single-nucleotide substitutions in an RNA virus. Proc Natl Acad Sci USA 101:8396–8401CrossRefGoogle Scholar
  52. Saxer G, Doebeli M, Travisano M (2010) The repeatability of adaptive radiation during long-term experimental evolution of Escherichia coli in a multiple nutrient environment. PLoS One 5:e14184CrossRefGoogle Scholar
  53. Schenk MF, Szendro IG, Krug J, de Visser JAGM (2012) Quantifying the adaptive potential of an antibiotic resistance enzyme. PLoS Genet 8:e1002783CrossRefGoogle Scholar
  54. Sinai V (1959) On the concept of entropy for a dynamical system. Dokl Akad Nauk SSSR 124:768–771Google Scholar
  55. Solé RV, Sardanyés J (2013) Red Queen coevolution on fitness landscapes, in Recent Advances in the theory and application of fitness landscapes. In: Richter H, Engelbrecht AP (eds) Emergence, complexity and computation EEC series. Springer, BerlinGoogle Scholar
  56. Stenseth NC, Maynard Smith J (1984) Coevolution in ecosystems: Red Queen evolution or stasis? Evolution 38:870–880CrossRefGoogle Scholar
  57. Thompson JN (1994) The coevolutionary process. Chicago University Press, ChicagoCrossRefGoogle Scholar
  58. Toprak E et al (2012) Evolutionary paths to antibiotic resistance under dynamically sustained drug selection. Nat Genet 44:101–105CrossRefGoogle Scholar
  59. Van Valen L (1973) A new evolutionary law. Evol Theory 1:1–30Google Scholar
  60. Van Valen L (1976) Energy and evolution. Evol Theory 1:179–229Google Scholar
  61. Van Valen L (1980) Evolution as a zero-sum game for energy. Evol Theory 4:129–142Google Scholar
  62. Venturino E (2011) Simple metaecoepidemic models. Bull Math Biol 73:917–950. doi:10.1007/s11538-010-9542-3 CrossRefGoogle Scholar
  63. Vermeij GJ (1994) The evolutionary interaction among species—selection, escalation and coevolution. Annu Rev Ecol Syst 25:219–236CrossRefGoogle Scholar
  64. Weinreich DM, Delaney NF, Depristo MA, Hartl DL (2006) Darwinian evolution can follow only very few mutational paths to fitter proteins. Science 312:111–114CrossRefGoogle Scholar
  65. Wichman HA, Brown CJ (2010) Experimental evolution of viruses: microviridae as a model system. Philos Trans R Soc Lond B Biol Sci 365:2495–2501CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsISEL - Engineering Superior Institute of LisbonLisbonPortugal
  2. 2.Mathematics Department, Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  3. 3.Department of MathematicsESTS - Technology Superior School of SetubalSetubalPortugal
  4. 4.ICREA-Complex Systems LabUniversitat Pompeu Fabra (UPF), Parc de Recerca Biomèdica de Barcelona (PRBB)BarcelonaSpain
  5. 5.Institut de Biologia Evolutiva (UPF-CSIC-PRBB)BarcelonaSpain

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