In the present paper, sufficient conditions for the exponential stability of nonlinear diffusion systems with infinite distributed and discrete time-varying delays are derived. Such systems arise in many applications, e.g. in population dynamics. The existing Lyapunov-based results on the stability of diffusion nonlinear systems treat either systems with infinite delays or the ones with discrete slowly varying delays (with the delay derivatives smaller than 1), where the conditions are delay-independent in the discrete delays. We introduce the Lyapunov-based analysis of systems with fast varying (without any constraints on the delay-derivative) discrete and infinite distributed delays. We derive delay-independent with respect to discrete delays stability criterion via a novel combination of Lyapunov-Krasovskii functionals and of the Halanay inequality in terms of Linear Matrix Inequalities (LMIs). Numerical examples illustrate the efficiency of the method.