This work addresses the modeling of algal growth under intermittent light conditions by extending the deterministic Photosynthetic Factory (PSF) model into a probabilistic framework. Motivated by experimental evidence that the frequency of light/dark cycles is not constant due to the inherent randomness of hydrodynamic mixing and other physical processes, the PSF model, representing one of the widely used photosynthetic-unit-based models, is first reduced to a single ordinary differential equation and then reformulated as a random differential equation, where the cycle period is treated as a continuous random variable. A computationally relevant contribution of this study is the partition of the time domain for a piece-wise transformation method that enables the analytical computation of probability density functions for central quantities such as the proportion of active photosynthetic units and the average growth rate. A key contribution of this study is the development of a recursive transformation method that enables the analytical computation of probability density functions for central quantities such as the proportion of active photosynthetic units and the average growth rate. This method leverages the structure of the reduced PSF model and the piecewise nature of the input signal to propagate uncertainty through the system. Beyond computational considerations, a key advantage of the approach is that it provides direct access to the probability density functions and their analytical properties, which are generally not available from Monte Carlo simulations alone. This method leverages the structure of the reduced PSF model and the piecewise nature of the input signal to efficiently propagate uncertainty through the system, avoiding the computational burden of Monte Carlo simulations. Numerical experiments illustrate how uncertainty in cycle frequency can influence both transient and long-term system behavior, and how the proposed approach allows for accurate quantification of these effects. The resulting framework offers a tractable and interpretable way to incorporate randomness into biological models of light-driven growth, with potential applications in the design and analysis of photobioreactors operating under variable environmental conditions.
light/dark cycles, uncertainty, random process, probability density function, random differential equations, photosynthetic factory model, algal growth
93C10, 37N25, 34F05