Bifurcations in haploid and diploid sequence space models

Journal of Mathematical Biology

Volumn 35, Issue 3, 1997, Pages 321-343

  Baake, Ellen ^a,b   Wiehe, Thomas ^c  

^a MAX PLANCK INSTITUTE FOR DEVELOPMENTAL BIOLOGY   (Germany)

^b UNIVERSITY OF MUNICH   (Germany)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Error threshold; Hopf bifurcations; Mutation; Selection; Sequence space

Indexed keywords

EID: 0040552620     PISSN: 03036812     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002850050054     Document Type: Article

References (36)

1. Akin, E.: The Geometry of Population Genetics (Lect. Notes Biomath. 31). New York: Springer 1979
2. Akin, E.: Cycling in simple genetic systems. J. Math. Biol. 13, 305-324 (1982)
3. Baake, E.: Diploid sequence space, dominance, and evolution. Proc. 4th Int. Conf. on Math. Population Dynamics, in press (1996)
4. Baake, E.: Diploid models on sequence space. J. Biol. Syst. 3, 343-349 (1995)
5. Bürger, R.: On the evolution of dominance modifiers I. A nonlinear analysis. J. Theor. Biol. 101, 585-598 (1983)
6. Crow, J. F., Kimura, M.: An Introduction to Population Genetics Theory. New York: Harper & Row 1970
7. Eigen, M.: Selforganization of matter and the evolution of biological macromolecules. Naturwiss. 58, 465-523 (1971)
8. Eigen, M., McCaskill, J., Schuster, P.: The molecular quasi-species. Adv. Chem. Phys. 75, 149-263 (1989)
9. Ewens, W. J.: Mathematical Population Genetics. Berlin: Springer 1979
10. Fisher, R. A.: On the dominance ratio. Proc. Roy. Soc. Edinb. 42, 321-341 (1922)
11. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer 1983
12. Hadeler, K. P.: On the equilibrium states in certain selection models. J. Math. Biol. 1, 51-56 (1974)
13. Hadeler, K. P.: Stable polymorphisms in a selection model with mutation. SIAM J. Appl. Math. 41, 1-7 (1981)
14. Haldane, J. B. S.: A mathematical theory of natural and artificial selection. Part V: Selection and mutation. Proc. Camb. Phil. Soc. 23, 838-844 (1928)
15. Hastings, A.: Stable cycling in discrete-time genetic models. Proc. Natl. Acad. Sci. USA 78, 7224-7225 (1981)
16. Higgs, P. G.: Error thresholds and stationary mutant distributions in multi-locus diploid genetics models. Genet. Res., Camb. 63, 63-78 (1994)
17. Hofbauer, J.: The selection mutation equation. J. Math. Biol. 23, 41-53 (1985)
18. Hofbauer, J., Sigmund, K.: The Theory of Evolution and Dynamical Systems. Cambridge: CUP 1988
19. Kimura, M.: The number of heterozygous nucleotide sites maintained in a finite population due to steady flux of mutations. Genetics 61, 893-903 (1969)
20. Kimura, M., Crow, J. F.: The number of alleles that can be maintained in a finite population. Genetics 49, 725-738 (1964)
21. Kingman, J. F. C.: A mathematical problem in population genetics. Proc. Camb. Phil. Soc. 57, 574-582 (1961)
22. Lang, S.: Linear Algebra. Reading (Mass.): Addison-Wesley 1970
23. Moran, P. A. P.: Global stability of genetic systems governed by mutation and selection. Math. Proc. Camb. Phil. Soc. 80, 331-336 (1976)
24. Nowak, M., Schuster, P.: Error thresholds of replication in finite populations. Mutation frequencies and the onset of Muller's ratchet. J. Theor. Biol. 137, 375-395 (1989)
25. Reidys, C., Stadler, P., Schuster, P.: Generic properties of combinatory maps: neutral networks of RNA secondary structures. Bull. Math. Biol., in press (1996)
26. Richter, O.: Simulation des Verhaltens ökologischer Systeme: Mathematische Methoden und Modelle. Weinheim: Verlag Chemie 1985
27. Schuster, P., Swetina, J.: Stationary mutant distributions and evolutionary optimization. Bull. Math. Biol. 50, 635-660 (1988)
28. Stadler, P. F., Schuster, R: Mutation in autocatalytic reaction networks. An analysis based on perturbation theory. J. Math. Biol. 30, 597-632 (1992)
29. Stadler, P. F., Schnabl, W., Forst, C. V., Schuster, P.: Dynamics of small autocatalytic reaction networks. II: Replication, mutation and catalysis. Bull. Math. Biol. 57, 21-61 (1995)
30. Szucs, J. M.: Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population. J. Math. Biol. 31, 317-349 (1993)
31. Szucs, J. M.: Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population with random mating II. The case of constant fertilities. J. Math. Biol. 31, 601-609 (1993)
32. Thompson, C. J. and McBride, J. L.: On Eigen's theory of the self-organization of matter and the evolution of biological macromolecules. Math. Biosci. 21, 127-142 (1974)
33. van Lint, J. H.: Introduction to Coding Theory. New York: Springer 1982
34. Wagner, G. P., Krall, P.: What is the difference between models of error thresholds and Muller's ratchet. J. Math. Biol. 32, 33-44 (1993)
35. Wiehe, T., Baake, E., Schuster, P.: Error propagation in reproduction of diploid organisms. A case study on single peaked landscapes. J. Theor. Biol. 177, 1-15 (1995)
36. Yodzis, P.: Introduction to theoretical ecology. New York: Harper & Row 1989