Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy

Zhipeng Qiu; Huaiping  Zhu

doi:10.3934/dcdsb.2016069

Article Contents

2016, Volume 21, Issue 8: 2703-2728. Doi: 10.3934/dcdsb.2016069

This issue Previous Article Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays Next Article The Euler scheme for state constrained ordinary differential inclusions

Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy

Zhipeng Qiu^1, and
Huaiping Zhu^2,

1.
Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094
2.
Laboratory of Mathematical Parallel Systems (Lamps), Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3

Received: August 31, 2015

Revised: January 31, 2016

Published: September 2016

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Abstract

Understanding the plankton dynamics can help us take effective measures to settle the critical issue on how to keep plankton ecosystem balance. In this paper, a nutrient-phytoplankton-zooplankton (NPZ) model is formulated to understand the mechanism of plankton dynamics. To account for the harmful effect of the phytoplankton allelopathy, a prototype for a non-monotone response function is used to model zooplankton grazing, and nonlinear phytoplankton mortality is also included in the NPZ model. Using the model, we will focus on understanding how the phytoplankton allelopathy and nonlinear phytoplankton mortality affect the plankton population dynamics. We first examine the existence of multiple equilibria and provide a detailed classification for the equilibria, then stability and local bifurcation analysis are also studied. Sufficient conditions for Hopf bifurcation and zero-Hopf bifurcation are given respectively. Numerical simulations are finally conducted to confirm and extend the analytic results. Both theoretical and numerical findings imply that the phytoplankton allelopathy and nonlinear phytoplankton mortality may lead to a rich variety of complex dynamics of the nutrient-plankton system. The results of this study suggest that the effects of the phytoplankton allelopathy and nonlinear phytoplankton mortality should receive more attention to understand the plankton dynamics.

Keywords:

Nutrient-phytoplankton-zooplankton model,
allelopathy,
nonlinear phytoplankton mortality,
bifurcation,
complex dynamics.

Mathematics Subject Classification: Primary: 92D25, 92D40; Secondary: 34A47.

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