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The paper proposes a new general method allowing us to study the problem on constructing hyperbolicity and stability regions for nonlinear dynamical systems. The method is based on a modification of the method by M. Rozo for studying the stability of linear systems with periodic coefficients depending on a small parameter and on the asymptotic formulae in the perturbation theory of linear operators. We obtain approximate formulae describing the boundary of hyperbolicity and stability regions. As an example, we provide the scheme for constructing the stability regions for Mathieu equation.

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The usage of a non-autonomous discrete model (Ricker model) for describing the dynamics of a biological population is considered. It is shown that in case of periodic changes in parameters, the model can be reduced into equivalent autonomous system. The problems of determining the model parameters in a situation where these parameters depend on time are discussed. As an application, the problem of mathematical modeling of the dynamics of the number of families of the natural population of the Burzyan wild-hive honeybee living on the territory of the Republic of Bashkortostan is considered. The results convincingly demonstrate the fact that the dynamics of the Burzyan Wild-Hive Honeybee is significantly influenced by a combination of natural factors. For example the sum of the precipitation in February is particularly significant here (in particular, the increase in precipitation affects the number of bees negatively) and the temperature values in March, April and June.

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