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

• We separate the fast-slow dynamics in general first-order initial value problems with slowly varying parameters.
• We give the asymptotic solution, which are explicit, analytical and uniformly valid on O(1/εε) time interval.
• We verify the high accuracy of the asymptotic solutions by comparing them with the numerically integrated ones.
• We give the error estimate between the approximate solutions and the exact solutions.

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/∊)O(1/∊) large time interval with O(∊)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.