Balakirieva O. Stabilization of the population age structure within nonhomogeneous Leslie model

The thesis is devoted to mathematical modeling and numerical analysis of discrete matrix models of the population dynamics. The object of the research are discrete processes of the population dynamics. The target of the work is the construction and research of the nonhomogeneous Leslie matrix model and the study of the stabilizing effect of the population age structure within the framework of this model. Results, obtained in this work, are theoretical and methodical basis of forecasting of population development in course of time within the framework of the nonhomogeneous Leslie model. The use of Leslie model modifications suggested in this work is more adapted to the description of dynamics of actual populations, allows to consider the oscillation of population number and also enables to carry out the management of population number to stabilize the age structure of the population. The nonhomogeneous Leslie model also makes it possible to carry out both the animal and human population forecasting. The work covers the modification of the homogeneous Leslie model, connected with the introduction of nonhomogeneity. The approbations of the suggested nonhomogeneous and homogeneous Leslie models by means of forecasting of various species population changes have been conducted. Within the framework of the tasks solution of the thesis research the following basic new scientific results have been received: - for the first time the homogeneous Leslie model operator description has been offered in case of a positive operator in infinitely dimensional space of an equivalent infinitely dimensional matrix model, that allows to study qualitative properties of this model from the general positions of the functional analysis; - homogeneous Leslie model has got a subsequent development, and it was widespread to a nonhomogeneous case (from the formal point of view a product of positive matrices is examined instead a degree of the positive Leslie matrix ) which enables to take into account the population biological parameters in the course of time. - for the first time the theorems on the weak ergodicity property for the product of positive matrices have been formulated and proven, Leslie nonhomogenous model has been shown to have a weak ergodicity property which leads to the distribution stabilization of the part of every age class individuals in relation to the general numberof population; - for the first time the equivalency of Leslie discrete nonhomogeneous model has been established as well as nonhomogenous Markov chain for which the conditions have been obtained. According to these conditions the limit distribution of state probabilities is stabilized within some time in the neighborhood of values given before.