Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change

Abstract

Ocean dynamics is known to have a strong effect on the global climate change and on the composition of the atmosphere. In particular, it is estimated that about 70 % of the atmospheric oxygen is produced in the oceans due to the photosynthetic activity of phytoplankton. However, the rate of oxygen production depends on water temperature and hence can be affected by the global warming. In this paper, we address this issue theoretically by considering a model of a coupled plankton–oxygen dynamics where the rate of oxygen production slowly changes with time to account for the ocean warming. We show that a sustainable oxygen production is only possible in an intermediate range of the production rate. If, in the course of time, the oxygen production rate becomes too low or too high, the system’s dynamics changes abruptly, resulting in the oxygen depletion and plankton extinction. Our results indicate that the depletion of atmospheric oxygen on global scale (which, if happens, obviously can kill most of life on Earth) is another possible catastrophic consequence of the global warming, a global ecological disaster that has been overlooked.

Appendix

The Jacobian matrix J, i.e., the matrix of the linearized system (19–21), is as follows:

$$\begin{aligned} J = \begin{pmatrix} -\frac{Au}{(1+c)^{2}}-1-\frac{u c_2}{(c+c_2)^2}-\frac{\nu v c_3}{(c+c_3)^2} &{} \frac{A}{1+c}-\frac{c}{c+c_2} &{} -\frac{\nu c}{c+c_3}\\ \frac{Bc_1 u}{(c+c_1)^{2}} &{} \frac{Bc}{c+c_1}-2u-\frac{vh}{(u+h)^{2}}-\sigma &{} -\frac{u}{u+h} \\ \frac{uv}{u+h}\frac{2c {c_4}^2}{(c^2 +{c_4}^2)^2}&{} \frac{vh}{(u+h)^{2}}\frac{c^2}{(c^2+{c_4}^2)} &{} \frac{uc^2}{(u+h)(c^2+{c_4}^2)}-\mu \\ \end{pmatrix}\nonumber \\ \end{aligned}$$

(39)

Its specific form for each of the steady states is given below along with the corresponding characteristic equation.

$\bullet$ Extinction state $E_1=(0,0,0)$

The Jacobian matrix (39) takes the following form:

$$\begin{aligned} J_{(0,0,0)} = \begin{pmatrix} -1 &{}A &{}0\\ 0 &{}-\sigma &{}0\\ 0 &{}0 &{}-\mu \end{pmatrix}, \end{aligned}$$

(40)

and the characteristic equation is

$$\begin{aligned} (1+\lambda )(\sigma +\lambda )(\mu +\lambda )=0. \end{aligned}$$

(41)

$\bullet$ Zooplankton-free states $E_2^{(1)}$ and $E_2^{(1)}$

The Jacobian matrix is:

$$\begin{aligned} J_{(c,u,0)} = \begin{pmatrix} -\frac{Au}{(1+c)^{2}}-1-\frac{uc_2}{(c+c_2)^2}-\lambda &{} \frac{A}{c+1} -\frac{c}{c+c_2} &{} -\frac{\nu c}{c+c_3} \\ \frac{Bc_1 u}{(c+c_1)^{2}} &{} \frac{Bc}{c+c_1}-2u-\sigma -\lambda &{} -\frac{u}{u+h} \\ 0 &{} 0 &{} \frac{u}{u+h}\frac{c^2}{c^2+{c_4}^2}-\mu -\lambda \end{pmatrix}, \nonumber \\ \end{aligned}$$

(42)

and the characteristic equation is:

$$\begin{aligned}&\Bigg [\Bigg (-\frac{Au}{(1+c)^{2}}-1-\frac{u c_2}{(c +c_2)^2}-\lambda \Bigg )\Bigg (\frac{Bc}{c+c_1} - 2u-\sigma -\lambda \Bigg ) \nonumber \\&\quad -\Bigg (\frac{A}{1+c}-\frac{c}{c+c_2} \Bigg )\Bigg (\frac{Bc_1 u}{(c+c_1)^{2}}\Bigg )\Bigg ] \cdot \Bigg (\frac{u}{u+h}\left( \frac{c^2}{c^2 + {c_4}^2}\right) -\mu -\lambda \Bigg )=0,\qquad \end{aligned}$$

(43)

where c and u are defined by Eqs. (26–27).

$\bullet$ Oxygen–phytoplankton–zooplankton coexistence state $E_3$

$$\begin{aligned}&J_{(c,u,v)} \nonumber \\&\quad = \begin{vmatrix} -\frac{Au}{(1+c)^{2}}-1-\frac{uc_2}{(c+c_2)^2}-\frac{\nu vc_3}{(c+c_3)^2}-\lambda&\frac{A}{1+c}-\frac{c}{c+c_2}&-\frac{\nu c}{c+c_3}\\ \frac{Bc_1 u}{(c+c_1)^{2}}&\frac{Bc}{c+c_1}-2u-\frac{vh}{(u+h)^{2}}-\sigma -\lambda&-\frac{u}{u+h} \\ \frac{uv}{u+h}\frac{2c{c_4}^2}{(c^2+{c_4}^2)^2}&\frac{vh}{(u+h)^{2}}\frac{c^2}{(c^2+{c_4}^2)}&\frac{uc^2}{(u+h)(c^2+{c_4}^2)}-\mu -\lambda \end{vmatrix}\nonumber \\ \end{aligned}$$

(44)

where c, u, and v are defined by Eqs. (28–30).