Global asymptotic stability of a unique positive periodic solution for a discrete hematopoiesis model with unimodal production functions

Abstract

The present paper is concerned with the global dynamics of a discrete hematopoiesis model. This model has several unimodal production functions, and periodic coefficients and periodic time delays whose periods are the same $\omega \in {\mathbb {N}}$. Since these production functions are unimodal, they have no monotonicity. The obtained result guarantees that this model has only one positive $\omega$-periodic solution and its periodic solution is globally asymptotically stable. To prove that the unique positive $\omega$-periodic solution is globally asymptotically stable, the difference between that the periodic solution and an arbitrary other is analyzed in detail. It is necessary to evaluate the upper and lower limit values of all the positive solutions to carry out the detailed analysis. Because the production functions have no monotonicity, it is not easy to evaluate the upper and lower limit values of the solutions. Two suitable examples are included to illustrate the main result. Numerical simulation is presented for one of them. The other example is based on the upper and lower limit values of red blood cells for healthy humans known from clinical laboratory tests, and clinical data obtained from clinical studies.