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

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• 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.

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

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

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 $\theta = 2$

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 $b = 2.3$

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 $b = 2.01$

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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