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A Markov chain model of evolution in asexually reproducing populations: insight and analytical tractability in the evolutionary process

The evolutionary process has been modelled in many ways using both stochastic and deterministic models. We develop an algebraic model of evolution in a population of asexually reproducing organisms in which we represent a stochastic walk in phenotype space, constrained to the edges of an underlying graph representing the genotype, with a time-homogeneous Markov Chain. We show its equivalence to a more standard, explicit stochastic model and show the algebraic model's superiority in computational efficiency. Because of this increase in efficiency, we offer the ability to simulate the evolution of much larger populations in more realistic genotype spaces. Further, we show how the algebraic properties of the Markov Chain model can give insight into the evolutionary process and allow for analysis using familiar linear algebraic methods.