Global Hopf bifurcation for two zooplankton-phytoplankton model with two delays

Renxiang Shi,Wenguo Yang

doi:10.1186/s13662-020-03060-7

Renxiang Shi, Wenguo Yang

Open Access

https://doi.org/10.1186/s13662-020-03060-7

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Abstract

In this paper, we study the global existence of a bifurcating periodic solution for a two zooplankton-phytoplankton model with two delays. First, we demonstrate that the bifurcating periodic solution exists when one delay increases and the other delay remains unchanged. Second, we give simulation to describe the bifurcating periodic solution when one delay increases. Our work answers the question in Sect. 5 (Shi and Yu in Chaos Solitons Fractals 100:62–73, 2017).

Highlights

The dynamics of a plankton model is important for an aquatic system: phytoplankton produce oxygen by photosynthesis and absorb nearly half carbon dioxide, so phytoplankton exert great influence on our ecosystem
The delayed phytoplankton-zooplankton model has attracted much interest [1,2,3,4,5,6,7,8,9,10]; the model contains two delays, that is, gestation delay for zooplankton and the maturity delay for toxic phytoplankton, which have been studied by many researchers [1, 4, 6, 8,9,10] in recent years
In [4, 6, 8], the author took the delay required for the maturation of toxic phytoplankton as a parameter, and the dynamics of system was studied

Summary

Introduction

The dynamics of a plankton model is important for an aquatic system: phytoplankton produce oxygen by photosynthesis and absorb nearly half carbon dioxide, so phytoplankton exert great influence on our ecosystem. There have been many works about the dynamics of a phytoplankton-zooplankton model in recent years. In [4, 6, 8], the author took the delay required for the maturation of toxic phytoplankton as a parameter, and the dynamics of system was studied. The author [6] discussed Hopf bifurcation of the following phytoplankton-zooplankton system with two delays:. In paper [11], the global existence of a bifurcating periodic solution of a neural network model with two delays was studied by limiting τ1 ∈ (0, τ10) and taking τ2 as a bifurcation parameter. Based on the above works, we shall study the global Hopf bifurcation of system (1.2) with delays, which is the second question given in Sect.

The mathematical model

Conclusion