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Peer-reviewedCC BYMathematical and Computational Applications

Stochastic Mutation Semigroups: From Deterministic Collapse to Probabilistic Evolutionary Dynamics

  • Marshal I. Sampson1
  • Christiana F. Igiri2
  • Reny George3
  • Julie S. George4
4 affiliations
  1. 1Department of Mathematics, Faculty of Physical Sciences, Akwa Ibom State University, Ikot Akpaden 534111, Nigeria
  2. 2Department of Mathematics, University of Calabar, Calabar 540281, Nigeria
  3. 3Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia
  4. 4Department of Health and Rehabilitation Sciences, College of Applied Medical Sciences, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia
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Abstract

The deterministic framework of mutation semigroups provides algebraic conditions for evolutionary collapse, but real mutation processes are inherently stochastic. This paper develops a comprehensive theory of stochastic mutation semigroups, where elementary mutations occur with empirically measured probabilities. A stochastic mutation semigroup is introduced as a Markov chain on the transformation semigroup generated by elementary mutation operators. We prove a stochastic transitivity threshold theorem under a positivity condition on the probability of rank reduction, correcting a logical gap in previous formulations, and we characterize collapse via the spectral properties of the associated Markov chain. Using HIV-1 sequence data from public databases and empirical mutation rates reported in the literature, the framework is illustrated through conceptual examples. Complete pseudocode is provided for the probabilistic pair-graph algorithm, along with convergence criteria and numerical examples demonstrating performance. The framework is further extended to infinite state spaces via topological semigroup theory and to time-varying mutation rates through dynamic L∗-classes. These results bridge the gap between abstract semigroup theory and evolutionary biology, offering a theoretical foundation for understanding stochastic mutation dynamics.

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Cite

BibTeXRIS
BibTeX
@article{sampson2026stochastic,
  title = {{Stochastic Mutation Semigroups: From Deterministic Collapse to Probabilistic Evolutionary Dynamics}},
  author = {Marshal I. Sampson and Christiana F. Igiri and Reny George and Julie S. George},
  journal = {Mathematical and Computational Applications},
  year = {2026},
  month = sep,
  doi = {10.3390/mca31050195},
  url = {https://doi.org/10.3390/mca31050195},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Stochastic Mutation Semigroups: From Deterministic Collapse to Probabilistic Evolutionary Dynamics
AU  - Marshal I. Sampson
AU  - Christiana F. Igiri
AU  - Reny George
AU  - Julie S. George
PY  - 2026
DA  - 2026/09/01
JO  - Mathematical and Computational Applications
DO  - 10.3390/mca31050195
UR  - https://doi.org/10.3390/mca31050195
AB  - The deterministic framework of mutation semigroups provides algebraic conditions for evolutionary collapse, but real mutation processes are inherently stochastic. This paper develops a comprehensive theory of stochastic mutation semigroups, where elementary mutations occur with empirically measured probabilities. A stochastic mutation semigroup is introduced as a Markov chain on the transformation semigroup generated by elementary mutation operators. We prove a stochastic transitivity threshold theorem under a positivity condition on the probability of rank reduction, correcting a logical gap in previous formulations, and we characterize collapse via the spectral properties of the associated Markov chain. Using HIV-1 sequence data from public databases and empirical mutation rates reported in the literature, the framework is illustrated through conceptual examples. Complete pseudocode is provided for the probabilistic pair-graph algorithm, along with convergence criteria and numerical examples demonstrating performance. The framework is further extended to infinite state spaces via topological semigroup theory and to time-varying mutation rates through dynamic L∗-classes. These results bridge the gap between abstract semigroup theory and evolutionary biology, offering a theoretical foundation for understanding stochastic mutation dynamics.
SN  - 2297-8747
SN  - 1300-686X
ER  -