On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density

• We discuss the logistic equation subject to uncertainties.
• We provide the first probability density function of the time-population density.
• We present the probability density function of the inflection point.
• We model uncertainties based on the Maximum Entropy Principle.
• We determine closed-form results in the uniformly distributed case.

It is recognized that handling uncertainty is essential to obtain more reliable results in modeling and computer simulation. This paper aims to discuss the logistic equation subject to uncertainties in two parameters: the environmental carrying capacity, K, and the initial population density, N0. We first provide the closed-form results for the first probability density function of time-population density, N(t), and its inflection point, t*. We then use the Maximum Entropy Principle to determine both K and N0 density functions, treating such parameters as independent random variables and considering fluctuations of their values for a situation that commonly occurs in practice. Finally, closed-form results for the density functions and statistical moments of N(t), for a fixed t > 0, and of t* are provided, considering the uniform distribution case. We carried out numerical experiments to validate the theoretical results and compared them against that obtained using Monte Carlo simulation.