Flow-Based Modelling of Population Dynamics with Consecutive Continuous Mutations
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arXiv
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| Format: | Preprint |
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2025
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| author | Bratus, Alexander Yakushkina, Tatiana Posvyanski, Vladimir |
| author_facet | Bratus, Alexander Yakushkina, Tatiana Posvyanski, Vladimir |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Flow-Based Modelling of Population Dynamics with Consecutive Continuous Mutations Bratus, Alexander Yakushkina, Tatiana Posvyanski, Vladimir Populations and Evolution Dynamical Systems 92D15, 35Q92, 92D25 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. |
| title | Flow-Based Modelling of Population Dynamics with Consecutive Continuous Mutations |
| topic | Populations and Evolution Dynamical Systems 92D15, 35Q92, 92D25 |
| url | https://arxiv.org/abs/2512.08660 |