  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.
