This article outlines an attempt to lay the groundwork for understanding stochastic dynamical descriptions of biological processes in terms of a discrete-state space, discrete-time random dynamical system (RDS), or random transformation approach. Such mathematics is not new for continuous systems, but the discrete state space formulation significantly reduces the technical requirements for its introduction to a much broader audiences. In particular, we establish some elementary contradistinctions between Markov chain (MC) and RDS descriptions of a stochastic dynamics. It is shown that a given MC is compatible with many possible RDS, and we study in particular the corresponding RDS with maximum metric entropy. Specifically, we show an emergent behavior of an MC with a unique absorbing and aperiodic communicating class, after all the trajectories of the RDS synchronizes. In biological modeling, it is now widely acknowledged that stochastic dynamics is a more complete description of biological reality than deterministic equations; here we further suggest that the RDS description could be a more refined description of stochastic dynamics than a Markov process. Possible applications of discretestate RDS are systems with fluctuating law of motion, or environment, rather than inherent stochastic movements of individuals.

We study finite state random dynamical systems (RDS) and their induced Markov chains (MC) as stochastic models for complex dynamics. The linear representation of deterministic maps in RDS is a matrix-valued random variable whose expectation corresponds to the transition matrix of the MC. The instantaneous Gibbs entropy, Shannon-Khinchin entropy of a step, and the entropy production rate of the MC are discussed. These three concepts, as key anchoring points in applications of stochastic dynamics, characterize respectively the uncertainties of a system at instant time t, the randomness generated in a step in the dynamics, and the dynamical asymmetry with respect to time reversal. The stationary entropy production rate, expressed in terms of the cycle distributions, has found an expression in terms of the probability of the deterministic maps with single attractor in the maximum entropy RDS. For finite RDS with invertible transformations, the non-negative entropy production rate of its MC is bounded above by the Kullback-Leibler divergence of the probability of the deterministic maps with respect to its time-reversal dual probability.