In this work we considered a parabolic system of two quasilinear reaction-diffusion equations for a biological population problem of the Kolmogorov-Fisher type describes the process of a biological population in a nonlinear two-component medium. We studied the qualitative properties of the solution to Cauchy problem based on self-similar analysis and its numerical solutions using the methods of modern computer technologies, to study the methods of linearization to the convergence of the iterative process with further visualization.