Abstract
Although long-period population size cycles and chaotic fluctuations in abundance are common in ecological models, such dynamics are uncommon in simple population-genetic models where convergence to a fixed equilibrium is most typical. When genotype-frequency cycling does occur, it is most often due to frequency-dependent selection that results from individual or species interactions. In this paper, we demonstrate that fertility selection and genomic imprinting are sufficient to generate a Hopf bifurcation and complex genotype-frequency cycling in a single-locus population-genetic model. Previous studies have shown that on its own, fertility selection can yield stable two-cycles but not long-period cycling characteristic of a Hopf bifurcation. Genomic imprinting, a molecular mechanism by which the expression of an allele depends on the sex of the donating parent, allows fitness matrices to be nonsymmetric, and this additional flexibility is crucial to the complex dynamics we observe in this fertility selection model. Additionally, we find under certain conditions that stable oscillations and a stable equilibrium point can coexist. These dynamics are characteristic of a Chenciner (generalized Hopf) bifurcation. We believe this model to be the simplest population-genetic model with such dynamics.
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References
Akin E. (1982). Cycling in simple genetic systems. J. Math. Biol. 13(3): 305–324
Altenberg L. (1991). Chaos from linear frequency-dependent selection. Am. Nat. 138(1): 51–68
Andreasen V. and Christiansen F.B. (1995). Slow coevolution of a viral pathogen and its diploid host. Philos. Trans. R. Soc. Lond. B Biol. Sci. 348(1325): 341–354
Arrowsmith D.K. and Place C.M. (1990). An Introduction to Dynamical Systems. Cambridge University Press, Cambridge
Asmussen M.A. (1979). Regular and chaotic cycling in models of ecological genetics. Theor. Popul. Biol. 16(2): 172–190
Asmussen M.A. and Feldman M.W. (1977). Density dependent selection 1: A stable feasible equilibrium may not be attainable. J. Theor. Biol. 64(4): 603–618
Bodmer W.F. (1965). Differential fertility in population genetics models. Genetics 51(3): 411–424
Costantino R.F., Desharnais R.A., Cushing J.M. and Dennis B. (1997). Chaotic dynamics in an insect population. Science 275(5298): 389–391
Cressman R. (1988). Frequency-dependent viability selection (a single-locus, multi-phenotype model). J. Theor. Biol. 130(2): 147–165
Dennis B., Desharnais R.A., Cushing J.M., Henson S.M. and Costantino R.F. (2001). Estimating chaos and complex dynamics in an insect population. Ecol. Monogr. 71(2): 277–303
Doebeli M. and de Jong G. (1998). A simple genetic model with non-equilibrium dynamics. J. Math. Biol. 36(6): 550–556
Ellner S. and Turchin P. (1995). Chaos in a noisy world: New methods and evidence from time-series analysis. Am. Nat. 145(3): 343–375
Ellner S.P. and Turchin P. (2005). When can noise induce chaos and why does it matter: a critique. Oikos 111(3): 620–631
Feldman M.W., Christiansen F.B. and Liberman U. (1983). On some models of fertility selection. Genetics 105(4): 1003–1010
Gavrilets S. (1998). One-locus two-allele models with maternal (parental) selection. Genetics 149(2): 1147–1152
Gavrilets S. and Hastings A. (1995). Intermittency and transient chaos from simple frequency-dependent selection. Proc. R. Soc. Lond. B Biol. Sci. 261(1361): 233–238
Govaerts W., Kuznetsov Y.A. and Dhooge A. (2005). Numerical continuation of bifurcations of limit cycles in MATLAB. SIAM J. Sci. Comput. 27(1): 231–252
Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci., vol 42. Springer, New York (1983)
Hadeler K.P. and Glas D. (1983). Quasimonotone systems and convergence to equilibrium in a population genetic model. J. Math. Anal. Appl. 95(2): 297–303
Hadeler K.P. and Liberman U. (1975). Selection model with fertility differences. J. Math. Biol. 2(1): 19–32
Hastings A. (1981). Stable cycling in discrete-time genetic models. Proc. Natl. Acad. Sci. U. S. A. 78(11): 7224–7225
Hofbauer J. and Iooss G. (1984). A Hopf-bifurcation theorem for difference-equations approximating a differential-equation. Monatsh. Math. 98(2): 99–113
Iwasa Y. and Pomiankowski A. (1995). Continual change in mate preferences. Nature 377(6548): 420–422
Josić K. (1997). Local bifurcations in the symmetric model of selection with fertility differences. J. Theor. Biol. 189(3): 291–295
Karlin S. and Lessard S. (1986). Theoretical Studies on Sex Ratio Evolution. Princeton University Press, Princeton
Kingman J.F.C. (1961). A matrix inequality. Quart. J. Math. 12: 78–80
Koth M. and Kemler F. (1986). A one locus-two allele selection model admitting stable limit cycles. J. Theor. Biol. 122(3): 263–268
Kuznetsov Y.A.: Elements Of Applied Bifurcation Theory, Appl. Math. Sci., vol. 112, 3rd edn. Springer, New York (2004)
Li C. and Chen G. (2003). An improved version of the marotto theorem. Chaos Solitons Fractals 18: 69–77
Li T.Y. and Yorke J.A. (1975). Period three implies chaos. Am. Math. Monthly 82(10): 985–992
Marotto F.R. (1978). Snap-back repellers imply chaos in R n. J. Math. Anal. Appl. 63(1): 199–223
May R.M. (1975). Biological populations obeying difference equations: stable points, stable cycles and chaos. J. Theor. Biol. 51(2): 511–524
May R.M. (1976). Simple mathematical models with very complicated dynamics. Nature 261(5560): 459–467
May R.M. and Anderson R.M. (1983). Epidemiology and genetics in the coevolution of parasites and hosts. Proc. R. Soc. Lond. B Biol. Sci. 219(1216): 281–313
Maynard Smith J. and Hofbauer J. (1987). The battle of the sexes a genetic model with limit cycle behavior. Theor. Popul. Biol. 32(1): 1–14
Pearce G.P. and Spencer H.G. (1992). Population genetic models of genomic imprinting. Genetics 130(4): 899–907
Penrose L.S. (1949). The meaning of fitness in human populations. Ann. Eugen. 14(4): 301–304
Pollak E. (1978). With selection for fecundity the mean fitness does not necessarily increase. Genetics 90(2): 383–389
Reik W. and Walter J. (2001). Genomic imprinting: parental influence on the genome. Nat. Rev. Genet. 2(1): 21–32
Scheuer P. and Mandel S. (1959). An inequality in population genetics. Heredity 13: 519–524
Selgrade J.F. and Namkoong G. (1984). Dynamical behavior of differential equation models of frequency and density dependent populations. J. Math. Biol. 19(1): 133–146
Selgrade J.F. and Ziehe M. (1987). Convergence to equilibrium in a genetic model with differential viability between the sexes. J. Math. Biol. 25(5): 477–490
Spencer H.G. (2003). Further properties of Gavrilets’ one-locus two-allele model of maternal selection. Genetics 164(4): 1689–1692
Spencer H.G., Feldman M.W. and Clark A.G. (1998). Genetic conflicts, multiple paternity and the evolution of genomic imprinting. Genetics 148(2): 893–904
Spencer H.G., Dorn T. and LoFaro T. (2006). Population models of genomic imprinting. II. Maternal and fertility selection. Genetics 173(4): 2391–2398
Wood A.J. and Oakey R.J. (2006). Genomic imprinting in mammals: emerging themes and established theories. PLoS Genet. 2(11): e147
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Dedicated to the memory of Samuel Karlin.
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Van Cleve, J., Feldman, M.W. Stable long-period cycling and complex dynamics in a single-locus fertility model with genomic imprinting. J. Math. Biol. 57, 243–264 (2008). https://doi.org/10.1007/s00285-008-0156-4
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DOI: https://doi.org/10.1007/s00285-008-0156-4
Keywords
- Population genetics
- Fertility selection
- Generalized Hopf bifurcation
- Gene frequency cycling
- Frequency dependence