Flow-Based Modelling of Population Dynamics with Consecutive Continuous Mutations

We develop a continuous mathematical model of population dynamics that describes the sequential emergence of new genotypes under limited resources. The framework models genotype density as a nonlinear flow in mutation space, combining transport driven by a time-dependent mutation rate with logistic growth and nonlocal competition. For the advection-reaction regime without reverse mutations, we derive analytical solutions using the method of characteristics and obtain explicit expressions for time-varying carrying capacities and mutation velocities. We analyze how decaying and accelerating mutation rates shape the saturation and propagation of population fronts through level-set geometry. When reverse mutations are included, the system becomes a quasilinear parabolic equation with diffusion in genotype space; numerical experiments show that backward mutation flows stabilize the dynamics and smooth the evolving fronts. The proposed model generalizes classical quasispecies and Crow-Kimura formulations by incorporating logistic regulation, variable mutation rates, and reversible transitions, offering a unified approach to evolutionary processes relevant to virology, bacterial adaptation, and tumor progression.