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Coupled nonlinear system of reaction-diffusion equations describing multi-component (species) interactions with heterogeneous coefficients is considered. Finite volume method based approximation for the space is used to construct semi-discrete form for the computation of numerical solutions. Two techniques for time approximations, namely, a fully implicit (FI) and a semi-implicit (SI) schemes are examined. The fully implicit scheme is constructed using Newton's method and leads to the coupled system of equations on each nonlinear and time iterations which is computationally rather expensive. In order to minimize the latter hurdle, an efficient and fast multiscale solver is proposed for reaction-diffusion systems in heterogeneous media. To construct fast solver, we apply a semi-implicit scheme that leads to an uncoupled system for each individual component. To reduce the size of the discrete system, we present a multiscale model reduction technique. Multiscale solver is based on the uncoupled operator of the problem and constructed by the use of Generalized Multiscale Finite Element Method (GMsFEM). In GMsFEM we use a diffusion part of the operator and construct multiscale basis functions. We collect multiscale basis functions to construct a projection/prolongation matrix and generate reduced order model on the coarse grid for fast solution. Moreover, the prolongation operator is used to reconstruct a fine-scale solution and accurate approximation of the reaction part of the problem which then leads to a very accurate and computationally effective multiscale solver. We provide numerical results for two species competition test problems in two-dimensional domain with heterogeneous inclusions. We investigate the influence of number of the multiscale basis functions to the method accuracy and ability to work with different values of the diffusion coefficients.