Abstract
Processes of replication and mutation pivotal to molecular evolution may be modelled by a set of coupled nonlinear differential equations descriptive of autocatalytic networks. Solutions of the four dimensional system reveal aperiodic behaviours and chaos, punctuated by regions of periodic oscillations of the population variables. This complicated dynamics is encapsulated in terms of polynomial mappings which cast the relevant features of these behaviours in compact form and reproduces many of the fine details of the sequences of bifurcations. The equations descriptive of replication are topologically equivalent to generalized Lotka-Volterra equations, and thus the present map dynamics analysis finds a corresponding broader range of potential future application.
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Arneodo, A., Collet, P. H., Spiegel, E. A., Tressor, C: Asymptotic chaos. Physica14D, 327–347 (1985)
Arneodo, A., Coullet, P., Tressor, C.: Occurrence of strange attractors in three-dimensional Volterra equations. Phys. Lett.A79, 259–263 (1980)
Fraser, S., Kapral, R.: Analysis of flow hysterersis by a one-dimensional map. Phys. Rev.A25, 3223–3233 (1982)
Gaspard, P.: Generation of a countable set of homoclinic flows through bifurcation. Phys. Lett.97A, 1–4 (1983)
Gaspard, P., Kapral, R., Nicolis, G.: Bifurcation phenomena near homoclinic orbits. J. Stat. Phys.35, 697–727 (1984)
Glendinning, P., Sparrow, C.: Local and global behaviour near homoclinic orbits. J. Stat. Phys.35, 645–696 (1984)
Guckenheimer, J., Oster, I., Ipaktchi, A.: The dynamics of density dependent population models. J. Math. Biol.4, 101–147 (1977)
Hastings, A, Powell, T.: Chaos in a three-species food chain. Ecology72, 896–903 (1991)
Hénon, M.: On the numerical computation of Poincare maps. Physica5D, 412–414 (1982)
Hénon, M.: A two dimensional mapping with a strange attractor. Commun. Math. Phys.50, 69–77 (1976)
Hofbauer, J., Sigmund, K.: The Theory of Evolution and Dynamical Systems. New York: Cambridge University Press 1988
Hofbauer, J.: On the occurrence of limit cycles in the Volterra-Lotka Equation. Anal. Theor. Methods Appl.5, 1003–1007 (1981)
Jackson, E. A.: Perspectives in nonlinear dynamics, vol. 1. New York: Cambridge University Press 1990
Kingman, J. F. C.: A simple model for the balance between selection and mutation. J. Appl. Probab.15, 1–12 (1978)
Li, T-Y., Yorke, J. A.: Period three implies chaos. Am. Math. Mon.82, 985–992 (1975)
Lorenz, E. N.: Deterministic nonperiodic flow. J. Atmos. Sci.20, 130–141 (1963)
May, R. M.: Bifurcations and dynamical complexity in ecological systems. Ann. N.Y. Acad. Sci.316, 517–529 (1979)
Metropolis, N., Stein, M. L., Stein, P. R.: On finite limit sets for transformations on the unit interval. J. Comb. Theory15, 25–44 (1973)
Milnor, J.: Remarks on iterated cubic maps. SUNY Stony Brook Institute for Mathematical Sciences (Preprint #1990/6, 1990)
Mira, C: Chaotic Dynamics. New Jersey: World Scientific 1987
Neimark, Ju., Sil'nikov, L.: A case of generation of periodic motions. Sov. Math. Dokl.6, 305–309 (1965)
Petrov, V., Scott, S. K., Showalter, K.: Mixed-mode oscillations in chemical systems. J. Chem. Phys.97, 6191–6198 (1992)
Phillipson, P. E.: Map models for the emergence of ordered states out of chaos. Phys. Lett.A133, 383–390 (1988)
Rössler, O. E.: Continuous chaos-four prototype equations. Ann. N.Y. Acad. Sci.316, 376–392 (1979)
Schnabl W., Stadler, P. F., Forst, C., Schuster, P.: Full characterization of a strange attractor. Physica48D, 65–90 (1991)
Schuster, P., Sigmund, K.: Replicator dynamics. J. Theor. Biol.100, 533–538 (1983)
Sil'nikov, L.; A case of the existence of a countable number of periodic motions. Sov. Math. Dokl6, 163–166 (1965)
Skjolding, B., Branner-Jorgensen, B, Christiansen, P. L., Jensen, H. E.: Bifurcations in discrete dynamical systems with cubic maps. SIAM J. Appl. Math.43, 520–534 (1983)
Sparrow, C.: The Lorenz Equations: bifurcations, chaos and strange attractors. Berlin Heidelberg New York: Springer 1982
Testa, J., Held, G. A.: Study of a one dimensional map with multiple basins. Phys. Rev.A28, 3085–3089 (1983)
Wiggins, S.: Global bifurcations and chaos. Berlin Heidelberg New York: Springer 1988
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Phillipson, P.E., Schuster, P. Map dynamics of autocatalytic networks and the replicator equations. J. Math. Biology 32, 545–562 (1994). https://doi.org/10.1007/BF00573460
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DOI: https://doi.org/10.1007/BF00573460