# Global Warming Can Lead to Depletion of Oxygen by Disrupting Phytoplankton Photosynthesis: A Mathematical Modelling Approach

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## Abstract

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## 1. Introduction

## 2. Parametrization of the Oxygen-Phytoplankton System

#### 2.1. Model 1

#### 2.2. Model 2

#### 2.3. Model 3

#### 2.4. Model 4

#### 2.5. Model 5

#### 2.6. Model 6

#### 2.7. Model 7

## 3. Oxygen-Phyto-Zooplankton Model

## 4. Effect of Environmental Stochasticity: Simulations

## 5. Discussion and Concluding Remarks

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Flowchart diagram of mass flows between the components of the system (quantified by the growth rates $Af\left(c\right)u$ and $g(c,u)$) and losses of oxygen and phytoplankton due to their flows out of the system (quantified by $M(c,u)$ and $Q(c,u)$, accordingly); see the details in the text.

**Figure 2.**(

**a**) The (null)-isoclines of the oxygen-phytoplankton system for Model 1. Black curves show the oxygen isocline (${u}_{I}$) for $A=3.5,8,9,10$ and 11 (from left to right, respectively), and the red curve shows the phytoplankton isocline (${u}_{II}$). (

**b**) Positive steady states of Equations (5) and (6) shown as functions of A. Other parameters are ${c}_{1}=1$, $B=12.5$, $\gamma =2.5$, $m=1$ and $\sigma =0.1$.

**Figure 3.**(

**a**) The (null-)isoclines of the oxygen-phytoplankton system for Model 2. Black curves show the oxygen isocline (${u}_{I}$) for $A=3.5,8,9,10$ and 11 (from left to right, respectively), and the red curve shows the phytoplankton isocline (${c}_{II}$). (

**b**) Positive steady states of Equations (8) and (9) shown as functions of A. Here, $v=0.3$, and other parameters are the same as in Figure 2.

**Figure 4.**(

**a**) The (null-)isoclines of the oxygen-phytoplankton system for Model 3. Black curves show the oxygen isocline (${u}_{I}$) for $A=3.5,8,9,10$ and 11 (from left to right, respectively), and the red curve shows the phytoplankton isocline (${c}_{II}$). (

**b**) Positive steady states of Equations (11) and (12) shown as functions of A. Here, $v=0.3$ and $h=0.5$, and other parameters are the same as in Figure 2.

**Figure 5.**(

**a**) The (null-)isoclines of the oxygen-phytoplankton system for Model 4. Black curves show the oxygen isocline (${u}_{I}$) for $A=3.5,8,9,10$ and 11 (from left to right, respectively), and the red curve shows the phytoplankton isocline (${c}_{II}$). (

**b**) Positive steady states of Equations (14) and (15) shown as functions of A. Here ${c}_{2}=0.5$, other parameters are the same as in Figure 4.

**Figure 6.**(

**a**) The (null-)isoclines of the oxygen-phytoplankton system for Model 5. Black curves show the oxygen isocline (${u}_{I}$) for $A=3.5,8,9,10$ and 11 (from left to right, respectively), and the red curve shows the phytoplankton isocline (${c}_{II}$). (

**b**) Positive steady states of Equations (17) and (18) shown as functions of A. Here, ${c}_{2}=0.5$, other parameters are the same as in Figure 4.

**Figure 7.**(

**a**) The (null-)isoclines of the oxygen-phytoplankton system for Model 6. Black curves show the oxygen isocline (${u}_{I}$) for $A=3.5,8,9,10$ and 11 (from left to right, respectively), and the red curve shows the phytoplankton isocline (${u}_{II}$). (

**b**) Positive steady states of Equations (20) and (21) shown as functions of A. Parameters are the same as in Figure 6.

**Figure 8.**(

**a**) The (null-)isoclines of the oxygen-phytoplankton system for Model 7. Black curves show the oxygen isocline (${u}_{I}$) for $A=3.5,8,9,10$ and 11 (from left to right, respectively), and the red curve shows the phytoplankton isocline (${c}_{II}$). (

**b**) Positive steady states of Equation (23) and (24) shown as functions of A. Here, $\nu =0.01$ and ${c}_{3}=1$, and other parameters are the same as in Figure 6.

**Figure 10.**Oxygen concentration (blue), phytoplankton density (green) and zooplankton density (black) vs. time obtained for ${A}_{D}={w}_{A}={w}_{C}=0$, ${A}_{Q}=2.02$, ${c}_{1Q}=0.7$ and ${c}_{1D}=0.1$. The presence of noise distorts the otherwise periodical oscillations. Results shown in (

**a**,

**b**) are obtained in different simulation runs performed for the same parameters and initial conditions; hence, the apparent difference between (

**a**,

**b**) is the effect of the stochasticity.

**Figure 11.**Average amplitude of the oscillations of the zooplankton density obtained for different values of ${c}_{1D}$; other parameters are the same as in Figure 10. The black dashed line shows the best-fit of the data by a cubic polynomial, $y=-0.12{x}^{3}+8.3{x}^{2}-0.0087x+0.015$.

**Figure 12.**Phytoplankton density (green) and the concentration of oxygen (blue) vs. time obtained for parameters ${A}_{Q}=0.94$, ${A}_{D}=0.3$, ${w}_{A}=2\xb7{10}^{-5}$, ${c}_{1Q}=0.75$, ${c}_{1D}=0.1$ and ${w}_{C}={10}^{-6}$. (

**a**,

**b**) show two simulation runs obtained for the same parameters and the initial conditions.

**Figure 13.**Average extinction time calculated for different values of ${c}_{1D}$ for the oxygen-phytoplankton model (${v}_{0}=0$, black circles) and the full oxygen-phyto-zooplankton model (${v}_{0}=0.01$, red circles). Other parameters are the same as in Figure 12. The black dashed-and-dotted straight line shows the best-fit of the data by a linear function.

**Figure 14.**Population densities of phytoplankton (green), zooplankton (black) and the oxygen concentration (blue) vs. time obtained for parameters ${A}_{D}=0.3$, ${w}_{A}=0.7\xb7{10}^{-6}$, ${A}_{Q}=2.0965$, ${c}_{1Q}=0.7$, ${w}_{C}=0$ and ${c}_{1D}=0$. Results shown in (

**a**,

**b**) are obtained in different simulation runs performed for the same parameters and initial conditions.

**Figure 15.**Probability of the oxygen depletion and plankton extinction shown as a map in the parameter plane $({w}_{A},{A}_{D})$. Parameters are the same as in Figure 14.

**Table 1.**Summary of the biological factors accounted for and of the types of nonlinear responses used in Models 1–7.

Model | Type of Function | Biological Processes Accounted for | Existence |
---|---|---|---|

Used for M and Q | of ${\mathit{A}}_{\mathbf{cr}}$ | ||

1 | M: bilinear | unbounded oxygen uptake | yes |

Q: linear | natural mortality, viral infection | ||

2 | M: bilinear | unbounded oxygen uptake | yes |

Q: Holling II | predation | ||

3 | M: bilinear | unbounded oxygen uptake | yes |

Q: linear and Holling II | predation, natural mortality, viral infection | ||

4 | M: linear and Monod | saturated oxygen uptake | yes |

Q: Holling II | predation | ||

5 | M: linear and Monod | saturated oxygen uptake | yes |

Q: linear and Holling II | predation, natural mortality, viral infection | ||

6 | M: linear and Monod | saturated oxygen uptake | yes |

Q: linear | natural mortality, viral infection | ||

7 | M: linear and Monod $\times 2$ | saturated oxygen uptake, zooplankton breathing | yes |

Q: linear and Holling II | predation, natural mortality, viral infection |

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**MDPI and ACS Style**

Sekerci, Y.; Petrovskii, S.
Global Warming Can Lead to Depletion of Oxygen by Disrupting Phytoplankton Photosynthesis: A Mathematical Modelling Approach. *Geosciences* **2018**, *8*, 201.
https://doi.org/10.3390/geosciences8060201

**AMA Style**

Sekerci Y, Petrovskii S.
Global Warming Can Lead to Depletion of Oxygen by Disrupting Phytoplankton Photosynthesis: A Mathematical Modelling Approach. *Geosciences*. 2018; 8(6):201.
https://doi.org/10.3390/geosciences8060201

**Chicago/Turabian Style**

Sekerci, Yadigar, and Sergei Petrovskii.
2018. "Global Warming Can Lead to Depletion of Oxygen by Disrupting Phytoplankton Photosynthesis: A Mathematical Modelling Approach" *Geosciences* 8, no. 6: 201.
https://doi.org/10.3390/geosciences8060201