Mathematical Analysis of the Maltus Population Model in a Variable Environment

Abstract

The construction and application of mathematical models is the main means of knowledge of each science. Currently mathematical models along with physical and chemical are a powerful tool in the study of biological problems. One of the main elements in the construction of mathematical models describing the dynamics of the isolated population is to take info account the population under the influence of factors of different nature. Balance considerations underlie the construction of both discrete and continuous mathematical models of population dynamics. If the development of the population is carried out synchronously then it is convenient to use discrete models to describe its dynamics. At the same time, due to the fact that the death and birth of individuals is continuous, the use of discrete models in practice is limited. Therefore the most widely used in practice and deeply studied is a class of models based on ordinary differential equations. Therefore, the most widely used in practice and sufficiently deeply studied analytically is a class of models built on the basis of ordinary differential equations. The proposed work is devoted to the study of a constant population model of Malthus in a nonstationary environment for different values of the growth coefficient. Exact analytical solutions of the Cauchy problem are found. The schemes of identification of growth parameters on the basis of the interval between two consecutive divisions are constructed. Test calculations on the computer are carried out. Graphs of population size change are constructed. The obtained results allow us to conclude that the Malthus model describing the exponential phase of population development can describe in particular the logistic process of continuous growth.