The purpose of this paper is to construct a model for the process of generation and degeneration of red blood cells, taking into account biological-medical experimental data. The paper contains four sections. The first section presents basic biological facts enabling the reader who is not a specialist to understand this process. Section 2 is devoted to constructing an analytic model which uses a first order linear partial differential equation and a nonlinear integral equation. Section 3 lists some of the simplest mathematical properties of this model and the biological consequences arising from it. Section 4 gives a simplified model which can be described in terms of a nonlinear ordinary differential equation with a delayed parameter. The properties of this simple equation are of interest from both a mathematical and a biological point of view. In particular, the proof of the existence of periodic equations requires the application of a nontrivial version of a theorem on fixed points. The problem of the stability of this periodic solution is open. The problem is important in that the existence of a stable periodic solution gives a theoretical explanation of certain types of blood diseases.
We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.