Stability analysis of a mathematical model in a microcosm with piecewise constant arguments

Abstract

In this paper, we have modeled a population density of a bacteria species in a microcosm by using a differential equation, [Formula in text] where t ≥ 0, the parameters r, α, β(0) and β(1) denote positive numbers ann [t] denotes the integer part of [Formula in text]. First, to obtain the local and global behaviors, the boundedness character and the periodic nature of the population density for bacteria, discrete solutions of differential Eq. (A) is investigated. Examinations of the stability characterization of (A) show that increasing of the population growth rate decreases the local stability of the positive equilibrium point. Due to this result we need to consider a second approximation to obtain stability of population density. This can be performed at low density by incorporating an Allee function to (A) at time t. For the theoretical results obtained here we give an example by taking some parameter values from experimental data of bacteria populations [8] and show that the experimental and theoretical results for both models with and without Allee effect are in good agreement.