In this paper we consider a parametrized system of weakly nonlinear equations which corresponds to a nonlinear elliptic boundary-value problem with zero source, homogeneous boundary conditions and a positive parameter in the linear term. Positive solutions of this system are of interest to us. A characterization of this positive solution is given. Such a solution is determined by the Modified Newton-Arithmetic Mean method. This method is well suited for implementation on parallel computers. A theorem about the monotone convergence of the method is proved. An application of the method for solving a real practical problem related to the study of interacting populations is described.