On the generalized logistic random differential equation: Theoretical analysis and numerical simulations with real-world data

• Published in: Communications in nonlinear science & numerical simulation Vol. 116; p. 106832
• Main Authors: Bevia, V., Calatayud, J., Cortés, J.-C., Jornet, M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2023
• Subjects: Convergence; Generalized logistic differential equation; Probability density function; Random differential equation; Real-world application; Sample-path and mean-square solution; Probability density function; Real-world application; Generalized logistic differential equation; Random differential equation; Sample-path and mean-square solution; Convergence
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2022.106832

Based on the previous literature about the random logistic and Gompertz models, the aim of this paper is to extend the investigations to the generalized logistic differential equation in the random setting. First, this is done by rigorously constructing its solution in two different ways, namely, the sample-path approach and the mean-square calculus. Secondly, the probability density function at each time instant is derived in two ways: by applying the random variable transformation technique and by solving the associated Liouville’s partial differential equation. It is also proved that both the stochastic solution and its density function converge, under specific conditions, to the corresponding solution and density function of the logistic and Gompertz models, respectively. The investigation finishes showing some examples, where a number of computational techniques are combined to construct reliable approximations of the probability density of the stochastic solution. In particular, we show, step-by-step, how our findings can be applied to a real-world problem. •The generalized logistic model with randomness is fully studied.•Uncertainty considered in all input parameters with arbitrary distributions.•RVT and Liouville PDE are applied to determine the 1-PDF under general assumptions.•Theoretical results are applied to a biological real-world problem.•PSO and Maximum Entropy used to fit the model to real data.•Wavelet-based AMR and lagrangian PDE methods combined to compute the 1-PDF.