On the reconstruction of functional coefficients for a quasi-stable population dynamics model

For a population dynamics model with age structuring in a quasi-stable version, the inverse problem of restoring two coefficients of the model is considered. In the framework of the inverse problem, the intensity of cell mortality that depends only on time and is uniform in terms of cell age, which is included in the transfer equation, and the density of cell reproduction that depends only on their age, located in a boundary condition of the integral form, are determined. To determine the two desired coefficients of the model, an additional information is required in the form of solution of the direct problem for fixed values of one of its arguments. The uniqueness theorems of solutions to inverse problems of determining coefficients in the equation and in the integral form 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 pre-established. The integral formulas obtained during the analysis of the statements of direct and inverse problems allow us to organize various types of iterative algorithms for numerical solutions of the direct problem and the coefficient inverse problems for obtaining approximate solutions of both direct and inverse problems. The possibilities of using such an iterative numerical solution of coefficient inverse problems should be linked to the ill posedness of the inverse tasks.