This presentation focuses on the study of the long-term behavior of nonlinear reaction-diffusion PDEs with uncertain parameters. Key outcomes of interest include equilibria and their stability, possible destabilization mechanisms and corresponding bifurcation points, and the existence of solutions with specific spatio-temporal structures. As a demonstrative example, we examine the Klausmeier model, a system of PDEs describing vegetation-water interactions in semi-arid environments. It exhibits a variety of nonlinear phenomena, including the formation of vegetation patterns, making it a suitable framework for illustrating how uncertainty in PDE parameters propagates to system features. To develop a comprehensive framework from both theoretical and numerical perspectives, we also investigate variants of the Allen--Cahn equation, a prototypical PDE for bistability, accounting for random, spatially-heterogeneous effects.