Fast–slow dynamics in first-order initial value problems with slowly varying parameters and application to a harvested Logistic model

Abstract

In this paper, by using fast–slow decomposition and matching in singular perturbation theory, we separate the fast–slow dynamics in first-order initial value problems with slowly varying parameters and construct the asymptotic approximations to the solutions. Also we prove that the asymptotic solutions are uniformly valid on O(1/∊) large time interval with O(∊) accuracy by using the method of upper and lower solutions. As an application of the general theory, we consider a Logistic model with slowly varying parameters and linear density dependent harvest, in which, we illustrate the theoretical results through several numerical examples.