CONTINUOUS-TIME DYNAMICS OF GAMETE FREQUENCY EVOLUTION: A DIFFERENTIAL GEOMETRIC APPROACH TO SELECTION–RECOMBINATION SYSTEMS
Keywords:
Population genetics · Continuous-time dynamics · Linkage disequilibrium · Replicator equations · Selection–recombination balance · Lyapunov stabilityAbstract
We present a rigorous mathematical framework for analyzing the continuous-time evolution of gamete frequencies in diploid populations under the concurrent action of natural selection and genetic recombination. Utilizing dynamical systems theory and differential geometry, we derive the fundamental equations governing four-gamete haplotype dynamics and characterize their asymptotic behavior. We establish conditions for the existence and stability of equilibrium manifolds in the 3-simplex Δ₃, analyses the decay rates of linkage disequilibrium under various selection regimes, and provide explicit closed-form solutions for two limiting cases. The mathematical treatment employs Lyapunov-function analysis to demonstrate global convergence and characterizes the eigen spectrum of the linearized flow near interior equilibria. Numerical integration of the full nonlinear system confirms every theoretical prediction and reveals the multi-timescale geometric structure of solution trajectories.Downloads
References
Bürger R. (2000). The Mathematical Theory of Selection, Recombination, and Mutation. Wiley, Chichester.
Ewens W. J. (2004). Mathematical Population Genetics I: Theoretical Introduction. Springer, New York.
Hofbauer J. & Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press.
Nagylaki T. (1992). Introduction to Theoretical Population Genetics. Springer, Berlin.
Barton N. H. & Turelli, M. (1991). Natural and sexual selection on many loci. Genetics, 127(1), 229–255.
Kirkpatrick M., Johnson, T. & Barton, N. (2002). General models of multilocus evolution. Genetics, 161(4), 1727–1750.
Slatkin M. (2008). Linkage disequilibrium — understanding the evolutionary past and mapping the medical future. Nature Reviews Genetics, 9(6), 477–485.
Diyorov A.M. (2024) Uncountable linear operators of a quadratic stochastic operator. International scientific and practical conference on the topic “The role of digital technologies in the economy and education” (April 26-27, 2024) Samarkand, Uzbekistan
Bürger R. & Lynch, M. (1997). Evolution and extinction in finite populations subject to mutation, natural selection, and genetic drift. Evolution, 51(2), 569–587.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
All content published in the Journal of Applied Science and Social Science (JASSS) is protected by copyright. Authors retain the copyright to their work, and grant JASSS the right to publish the work under a Creative Commons Attribution License (CC BY). This license allows others to distribute, remix, adapt, and build upon the work, even commercially, as long as they credit the author(s) for the original creation.
