Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects

Juping Ji; Lin Wang

doi:10.3934/dcdss.2020135

Article Contents

2020, Volume 13, Issue 11: 3073-3081. Doi: 10.3934/dcdss.2020135

This issue Previous Article Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes Next Article Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity

Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects

Juping Ji and
Lin Wang^,

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B5A3, Canada

^* Corresponding author
^* Corresponding author

Received: December 31, 2018

Revised: January 31, 2019

Early access: November 2019

Published: July 2020

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Abstract

In this paper, we analyze a nutrient-phytoplankton model with toxic effects governed by a Holling-type Ⅲ functional. We show the model can undergo two saddle-node bifurcations and a Hopf bifurcation. This results in very interesting dynamics: the model can have at most three positive equilibria and can exhibit relaxation oscillations. Our results provide some insights on understanding the occurrence and control of phytoplankton blooms.

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Mathematics Subject Classification: 37C10, 37G10, 92B05.

Citation:

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Figure 1. The saddle-node bifurcation curves in ($ \theta,b) $ space

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Figure 2. Bifurcation diagram of Model (1). Parameter values used here are: $ a = 0.5 $, $ \varepsilon = 1 $, $ \delta = 0.05 $, $ \mu = 0.08 $, $ \xi = 0.18 $ and $ \theta = 2 $

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Figure 3. A phase portrait of Model (1). Parameter values are: $ a = 0.5 $, $ \varepsilon = 1 $, $ \delta = 0.05 $, $ \mu = 0.08 $, $ \theta = 2 $, $ \xi = 0.18 $ and $ b = 2.3 $

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Figure 4. A numerical solution of Model (1): relaxation oscillations are observed. Parameter values are: $ a = 0.5 $, $ \varepsilon = 1 $, $ \delta = 0.05 $, $ \mu = 0.08 $, $ \xi = 0.18 $, $ \theta = 2 $.and $ b = 2.01 $

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Figure 5. Bifurcation diagram of Model (1) using $ \theta $ as the bifurcation parameter. Other parameter values are: $ a = 0.5 $, $ \varepsilon = 1 $, $ \delta = 0.05 $, $ \mu = 0.08 $, $ \xi = 0.18 $ and $ b = 2.5 $

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