Abstract: In this work, we study the qualitative properties for complex biological population processes; on its basis, we solve the problem of choosing an initial approximation for the iterative process, which leads to rapid convergence to the solution of the Cauchy problem, depending on the values of the numerical parameters and the initial data. For this purpose, an asymptotic representation of the solution was used as an initial approximation, which made it possible to conduct a numerical experiment and visualize the process described by a system of quasilinear parabolic equations, depending on the values included in the system of numerical parameters. The properties of solutions to the problem in the case of double nonlinearity are studied. The main research method is the self-similar approach.