Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool

TL;DR

This paper provides a comprehensive guide for using DDE-Biftool to analyze a predator-prey model with time delay, illustrating how to perform bifurcation analysis and compare numerical results with theoretical predictions.

Contribution

It introduces the latest version of DDE-Biftool for beginners and demonstrates its application on a delayed predator-prey model, highlighting key features and potential pitfalls.

Findings

Successful bifurcation analysis of the predator-prey model

Comparison between numerical and theoretical results

Guidance for new users of DDE-Biftool

Abstract

Time delay has been incorporated in models to reflect certain physical or biological meaning. The theory of delay differential equations (DDEs), which has seen extensive growth in the last seventy years or so, can be used to examine the effects of time delay in the dynamical behavior of systems being considered. Numerical tools to study DDEs have played a significant role not only in illustrating theoretical results but also in discovering interesting dynamics of the model. DDE-Biftool, which is a Matlab package for numerical continuation and numerical bifurcation analysis of DDEs, is one of the most utilized and popular numerical tools for DDEs. In this paper, we present a guide to using the latest version of DDE-Biftool targeted to researchers who are new to the study of time delay systems. A short discussion of an example application, which is a harvested predator-prey model with a single discrete time delay, will be presented first. We then implement this example model in DDE-Biftool, pointing out features where beginners need to be cautious. We end with a comparison of our theoretical and numerical results.