The Reconstruction of Functional Coefficients for a Quasi-Stable Population Dynamics’ Model

摘要

Abstract For a model of population dynamics with age structuring in a quasi-stable version, the inverse problem of restoring two coefficients of the model is considered: the intensity of cell death that depends only on time and is uniform in cell age, which is included in the transfer equation, and the density of cell reproduction, which depends only on their age, located in the nonlocal boundary condition of the integral form. To determine the two required coefficients of the model in terms of the inverse problem statement, an additional specification of the solution of the direct problem with fixed values of one of its arguments is required. Uniqueness theorems for solutions of inverse problems of determining coefficients in an equation and in a boundary condition are formulated and proved. In this case, the properties of the solution of the direct problem and the conditions for its solvability are preliminarily established. The integral formulas obtained in the analysis of direct and inverse problem statements make it possible to organize various iterative algorithms for numerical solutions of the direct problem and inverse problems to obtain approximate solutions to problems. The possibility of using such an iterative numerical solution of coefficient inverse problems should be linked to the incorrect nature of the inverse formulations.