# American Institute of Mathematical Sciences

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

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

* Corresponding author

Received  January 2019 Revised  February 2019 Published  November 2019

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.

Citation: Juping Ji, Lin Wang. Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020135
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The saddle-node bifurcation curves in ($\theta,b)$ space
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 $\theta = 2$
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 $b = 2.3$
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 $b = 2.01$
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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