Analysis of random non-autonomous logistic-type differential equations via the Karhunen-Loeve expansion and the Random Variable Transformation technique

Abstract

[EN] This paper deals with the study, from a probabilistic point of view, of logistic-type differential equations with uncertainties. We assume that the initial condition is a random variable and the diffusion coefficient is a stochastic process. The main objective is to obtain the first probability density function, f_1(p,t), of the solution stochastic process, P(t,¿). To achieve this goal, first the diffusion coefficient is represented via a truncation of order N of the Karhunen¿Loève expansion, and second, the Random Variable Transformation technique is applied. In this manner, approximations, say f^N_1(p,t), of f_1(p,t) are constructed. Afterwards, we rigorously prove that f^N_1(p,t) ¿¿ f_1(p,t) as N ¿ ¿ under mild conditions assumed on input data (initial condition and diffusion coefficient). Finally, three illustrative examples are shown.