CONTINUOUS-TIME DYNAMICS OF GAMETE FREQUENCY EVOLUTION: A DIFFERENTIAL GEOMETRIC APPROACH TO SELECTION–RECOMBINATION SYSTEMS

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.