Stability and Bifurcation Analysis of a Phytoplankton-Zooplankton Model with Linear Functional Responses

TL;DR

This paper analyzes the stability and bifurcations of a phytoplankton-zooplankton model with linear responses, demonstrating stability results and identifying a Neimark-Sacker bifurcation with an attracting invariant curve.

Contribution

It provides a comprehensive stability analysis for both continuous and discrete models and establishes the occurrence of a Neimark-Sacker bifurcation in the system.

Findings

Global asymptotic stability of fixed points proved for continuous model

Local and global dynamics characterized for discrete model

Neimark-Sacker bifurcation with attracting invariant curve identified

Abstract

In this paper, the dynamics of a phytoplankton-zooplankton system with linear functional responses are examined. For the continuous-time model, the global asymptotic stability of the fixed points is demonstrated by constructing Lyapunov functions. For the discrete version of the model, both local and global dynamics are investigated using LaSalle's Invariance Principle. Furthermore, the occurrence of a Neimark-Sacker bifurcation at the positive fixed point is established, and it is proved that the resulting invariant closed curve is attracting.