The Effect of Toxin and Human Impact on Marine Ecosystem

Abstract

We formed a plankton-nutrient interaction model which consists of phytoplankton, herbivorous zooplankton, dissolved limiting nutrient with general nutrient uptake functions, instantaneous nutrient recycling, and harvesting on plankton population. Afterward, we modified and expanded the primary model by considering the effect of sunlight, additional nutrients, harmful chemicals, and carbon dioxide into account. Phytoplankton obtain carbohydrate supply from the carbon dioxide in the air, and the overall nutrient uptake rate increases in the presence of sunlight. So, the effect of sunlight and additional food is discussed. Hypothetically, carbon dioxide accelerates the growth of phytoplankton but we considered some limiting factors which abate the process. Some assumptions were made to construct the system equations. We assumed that phytoplankton releases toxic substances to defend themselves from the predation of zooplankton. The entire system was studied analytically, and the threshold values for the existence and stability of various steady states were discussed. Further, we discussed whether pollution emissions can variate the dynamics of the primary system, bring in recurrence bloom therein toxic phytoplankton can be applied to a great extent to sustain system stability.

Appendix

1.1 Stability Analysis of the System

We construct the 3 × 3 Jacobian matrix:

$$\displaystyle \begin{aligned} \left( \begin{array}{ccc} -\alpha - \frac{m_{1}a_{1}p}{(a_{1}+s)^{2}} & \gamma_{1} - \frac{m_{1}s}{a_{1}+s} & \gamma_{2} \\ \frac{m_{1}a_{1}p}{(a_{1}+s)^{2}} & \frac{m_{1}s}{a_{1}+s} - \frac{m_{2}a_{2}z}{(a_{2}+p)^{2}} - \beta_{1}- h_{1}c_{1} & -\frac{m_{2}p}{a_{2}+p} \\ 0 & \frac{m_{2}a_{2}z}{(a_{2}+p)^{2}}-\frac{\mu a_{3}z}{(a_{3}+p)^{2}} & \frac{m_{2}p}{a_{2}+p} - \beta_{2} - h_{2}c_{2}- \frac{\mu p}{a_{3}+p} \end{array} \right)\end{aligned}$$

We check the eigen values at every equilibrium point E ₀, E ₁, E ^∗.

The Jacobian at the equilibrium point $E_{0}= (\frac {M}{\alpha }, 0, 0)$ has the eigen values − α, $\frac {m_{1}s}{a_{1}+s}-\beta _{1}-c_{1}h_{1}$, and − (β ₂ + c ₂ h ₂). So, we can say that E ₀ is locally asymptotically stable if $\frac {m_{1}s}{a_{1}+s}-\beta _{1}-c_{1}h_{1}< 0$.

The variational matrix of the system around the positive equilibrium E ^∗ = (s ^∗, p ^∗, z ^∗) is

$$\displaystyle \begin{aligned} \left( \begin{array}{ccc} n_{11} & n_{12} & n_{13} \\ n_{21} & n_{22} & n_{23} \\ 0 & n_{32} & 0 \end{array} \right)\end{aligned}$$

where $n_{11}= -\alpha - \frac {m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}}< 0$, $n_{12} = \gamma _{1} - \frac {m_{1}s^{*}}{a_{1}+s^{*}}< 0$, n ₁₃ = γ ₂ > 0, $n_{21}= \frac {m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}}> 0$, $n_{22}= \frac {m_{1}s^{*}}{a_{1}+s^{*}} - \frac {m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}} - \beta _{1}- h_{1}c_{1}= \frac {m_{2}z}{a_{2}+p}- \frac {m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}}> 0$, $n_{23}= -\frac {m_{2}p^{*}}{a_{2}+p^{*}}< 0$, and $n_{32}= \frac {m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}}-\frac {\mu a_{3}z^{*}}{(a_{3}+p^{*})^{2}}> 0$

The characteristic equation is of the form λ ³ + A ₁ λ ² + A ₂ λ + A ₃,

where A ₁ = −(n ₁₁ + n ₂₂), A ₂ = n ₁₁ n ₂₂ − n ₁₂ n ₂₁ − n ₂₃ n ₃₂, and A ₃ = n ₁₁ n ₂₃ n ₃₂ − n ₁₃ n ₂₁ n ₃₂.

By the Routh-Hurwitz criteria, all roots of above equation have negative real parts if and only if A ₁ > 0, A ₃ > 0, and A ₁ A ₂ − A ₃ > 0.

$A_{1}=-(n_{11}+n_{22})= -(-\alpha - \frac {m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}}+\frac {m_{2}z^*}{a_{2}+p^*} - \frac {m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}}) >0$

if $\alpha + \frac {m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}}+\frac {m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}}> \frac {m_{2}z^*}{a_{2}+p^*}.$

A ₂ = n ₁₁ n ₂₂ − n ₁₂ n ₂₁ − n ₂₃ n ₃₂ > 0

if n ₁₁ n ₂₂ > n ₁₂ n ₂₁ + n ₂₃ n ₃₂

$$\displaystyle \begin{aligned} & A_{3}= n_{11}n_{23}n_{32}- n_{13}n_{21}n_{32}\\ =&(-\alpha - \frac{m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}})(-\frac{m_{2}p^{*}}{a_{2}+p^{*}})(\frac{m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}} -\frac{\mu a_{3}z^{*}}{(a_{3}+p^{*})^{2}})\\ &\quad - \gamma_{2} \frac{m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}} (\frac{m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}}-\frac{\mu a_{3}z^{*}}{(a_{3}+p^{*})^{2}})\\ =&[(\alpha + \frac{m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}})(\frac{m_{2}p^{*}}{a_{2}+p^{*}})-\gamma_{2} \frac{m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}}] (\frac{m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}}-\frac{\mu a_{3}z^{*}}{(a_{3}+p^{*})^{2}})\\ =&[\alpha (\frac{m_{2}p^{*}}{a_{2}+p^{*}})+ (\frac{m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}})(\frac{m_{2}p^{*}}{a_{2}+p^{*}})- \gamma_{2} \frac{m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}}] (\frac{m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}}\\ &\quad -\frac{\mu a_{3}z^{*}}{(a_{3}+p^{*})^{2}}) =[\alpha (\frac{m_{2}p^{*}}{a_{2}+p^{*}})+ (\frac{m_{1}a_{1}p^{*}}{(a_{1}+s^{*})^{2}})\\ &\qquad (\frac{m_{2}p^{*}}{a_{2}+p^{*}}- \gamma_{2})] (\frac{m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}}-\frac{\mu a_{3}z^{*}}{(a_{3}+p^{*})^{2}}) > 0 \end{aligned} $$

Since, $\frac {m_{2}p^{*}}{a_{2}+p^{*}}>\gamma _{2}$ and $\frac {m_{2}a_{2}z^{*}}{(a_{2}+p^{*})^{2}}> \frac {\mu a_{3}z^{*}}{(a_{3}+p^{*})^{2}}$

Finally, A ₁ A ₂ −A ₃ = −(n ₁₁ +n ₂₂)(n ₁₁ n ₂₂ −n ₁₂ n ₂₁ −n ₂₃ n ₃₂)−(n ₁₁ n ₂₂ −n ₁₂ n ₂₁ −n ₂₃ n ₃₂) > 0 since, − (n ₁₁ + n ₂₂)(n ₁₁ n ₂₂ − n ₁₂ n ₂₁ − n ₂₃ n ₃₂) > n ₁₁ n ₂₂ − n ₁₂ n ₂₁ − n ₂₃ n ₃₂.