In recent years, the development and analysis of reaction-diffusion models have gained increasing attention due to their applicability in various sustainability-related contexts. Notable examples include the modeling of ecological dynamics, such as vegetation patterns in arid and semi-arid ecosystems. These models are typically described by systems of parabolic partial differential equations, where reaction-diffusion processes are used to describe spatially distributed growth, competition, and resource transport. Solving such problems typically requires the repeated long-time integration of large highly stiff systems of ordinary differential equations (e.g. for parameters calibration). Therefore fast and reliable adapted time integration methods are required. In this talk, we propose two classes of linearly implicit time integration schemes based on generalizations and modifications of the so-called Time-Accurate and highly-Stable-Explicit (TASE) Runge-Kutta (RK) methods. We also illustrate suitable splitting techniques for extension to the efficient solution of multidimensional problems. Numerical experiments on reaction–diffusion models related to battery degradation and vegetation dynamics show that the proposed approaches significantly reduce computational effort compared to the standard TASE-RK schemes, while preserving accuracy and stability. This research activity falls within the activities of PRIN-MUR 2022 project 20229P2HEA Stochastic numerical modelling for sustainable innovation, CUP E53D23017940001, granted by the Italian Ministry of University and Research within the scrolling of the final rankings of the PRIN 2022 call.