Existence of multiple positive periodic solutions for discrete hematopoiesis models with a unimodal production function

Abstract

This paper is devoted to elucidating a sufficient condition under which Mackey-Glass type discrete hematopoiesis models have at least two positive periodic solutions. This model has periodic coefficients and time delays, and includes several production function terms that act as feedback. Our result is obtained by applying the Krasnosel’skii fixed point theorem and is represented by a relationship between period, coefficients and the production function. Example and its simulations are attached to show how to apply our result. In this example, there are exactly two positive 3-periodic solutions in the hematopoiesis model. Simulation shows that one periodic solution is stable and the other is unstable. It also shows that our result can be improved by weakening assumptions about the production function.