Hopf and Bautin bifurcations in a 3D model for pest leafhopper with stage structure and generalist predatory mite

TL;DR

This paper revisits a complex 3D predator-prey model with stage structure, correcting previous errors and demonstrating the occurrence of Hopf and Bautin bifurcations through analytical and numerical methods.

Contribution

It corrects earlier inaccuracies in the bifurcation analysis of a predator-prey model and establishes the presence of Hopf and Bautin bifurcations with supporting numerical examples.

Findings

Model exhibits Hopf bifurcation.

Model exhibits Bautin bifurcation.

Numerical examples support bifurcation analysis.

Abstract

In a recent paper of Yuan and Zhu (J. Differential Equations 321(2022) 99-129), the nature of dynamics of a generalist predator and prey with stage structure is modeled as a three-dimensional coupled nonlinear differential system. A detailed analysis of the bifurcation shows that the model exhibits a high complexity in its dynamics, which arises from the use of predatory mites as agent for controlling stage structures of tea green leafhopper pest. Unfortunately, there is a mistake in Proposition 2.2, item (3), also the hypothesis in Proposition 3.3 is incorrect. In this paper, we revisit the model and straighten those mentioned errors as in item (3), changed the existing hypothesis by a suitable one and give a corrected proof of Proposition 3.3 in its new form. Also, we show that the model does undergo both Hopf bifurcation and Bautin bifurcation, and numerical examples are given to support the analytic results.