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Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change

Bulletin of Mathematical Biology

December 2015, Volume 77, Issue 12, pp 2325–2353| Cite as

  • Yadigar Sekerci
  • Sergei Petrovskii
Original Article
First Online: 25 November 201

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.

Keywords

Phytoplankton Global warming Oxygen depletion Extinction Pattern formation 
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Notes

Acknowledgments

The authors are thankful to Nikolay Brilliantov (Leicester) for stimulating discussions at the early stage of this study.

Appendix

The Jacobian matrix J, i.e., the matrix of the linearized system (19–21), is as follows:
𝐽=⎛⎝⎜⎜⎜⎜−𝐴𝑢(1+𝑐)2−1−𝑢𝑐2(𝑐+𝑐2)2−𝜈𝑣𝑐3(𝑐+𝑐3)2𝐵𝑐1𝑢(𝑐+𝑐1)2𝑢𝑣𝑢+ℎ2𝑐𝑐42(𝑐2+𝑐42)2𝐴1+𝑐−𝑐𝑐+𝑐2𝐵𝑐𝑐+𝑐1−2𝑢−𝑣ℎ(𝑢+ℎ)2−𝜎𝑣ℎ(𝑢+ℎ)2𝑐2(𝑐2+𝑐42)−𝜈𝑐𝑐+𝑐3−𝑢𝑢+ℎ𝑢𝑐2(𝑢+ℎ)(𝑐2+𝑐42)−𝜇⎞⎠⎟⎟⎟⎟J=(−Au(1+c)2−1−uc2(c+c2)2−νvc3(c+c3)2A1+c−cc+c2−νcc+c3Bc1u(c+c1)2Bcc+c1−2u−vh(u+h)2−σ−uu+huvu+h2cc42(c2+c42)2vh(u+h)2c2(c2+c42)uc2(u+h)(c2+c42)−μ)
(39)
Its specific form for each of the steady states is given below along with the corresponding characteristic equation.

∙∙ Extinction state 𝐸1=(0,0,0)E1=(0,0,0)

The Jacobian matrix (39) takes the following form:
𝐽(0,0,0)=⎛⎝⎜⎜−100𝐴−𝜎000−𝜇⎞⎠⎟⎟,J(0,0,0)=(−1A00−σ000−μ),
(40)
and the characteristic equation is
(1+𝜆)(𝜎+𝜆)(𝜇+𝜆)=0.(1+λ)(σ+λ)(μ+λ)=0.
(41)
∙∙ Zooplankton-free states 𝐸(1)2E2(1) and 𝐸(1)2E2(1)
The Jacobian matrix is:
𝐽(𝑐,𝑢,0)=⎛⎝⎜⎜⎜⎜−𝐴𝑢(1+𝑐)2−1−𝑢𝑐2(𝑐+𝑐2)2−𝜆𝐵𝑐1𝑢(𝑐+𝑐1)20𝐴𝑐+1−𝑐𝑐+𝑐2𝐵𝑐𝑐+𝑐1−2𝑢−𝜎−𝜆0−𝜈𝑐𝑐+𝑐3−𝑢𝑢+ℎ𝑢𝑢+ℎ𝑐2𝑐2+𝑐42−𝜇−𝜆⎞⎠⎟⎟⎟⎟,J(c,u,0)=(−Au(1+c)2−1−uc2(c+c2)2−λAc+1−cc+c2−νcc+c3Bc1u(c+c1)2Bcc+c1−2u−σ−λ−uu+h00uu+hc2c2+c42−μ−λ),
(42)
and the characteristic equation is:
[(−𝐴𝑢(1+𝑐)2−1−𝑢𝑐2(𝑐+𝑐2)2−𝜆)(𝐵𝑐𝑐+𝑐1−2𝑢−𝜎−𝜆)−(𝐴1+𝑐−𝑐𝑐+𝑐2)(𝐵𝑐1𝑢(𝑐+𝑐1)2)]⋅(𝑢𝑢+ℎ(𝑐2𝑐2+𝑐42)−𝜇−𝜆)=0,[(−Au(1+c)2−1−uc2(c+c2)2−λ)(Bcc+c1−2u−σ−λ)−(A1+c−cc+c2)(Bc1u(c+c1)2)]⋅(uu+h(c2c2+c42)−μ−λ)=0,
(43)
where c and u are defined by Eqs. (26–27).
∙∙ Oxygen–phytoplankton–zooplankton coexistence state 𝐸3E3
𝐽(𝑐,𝑢,𝑣)=∣∣∣∣∣∣∣−𝐴𝑢(1+𝑐)2−1−𝑢𝑐2(𝑐+𝑐2)2−𝜈𝑣𝑐3(𝑐+𝑐3)2−𝜆𝐵𝑐1𝑢(𝑐+𝑐1)2𝑢𝑣𝑢+ℎ2𝑐𝑐42(𝑐2+𝑐42)2𝐴1+𝑐−𝑐𝑐+𝑐2𝐵𝑐𝑐+𝑐1−2𝑢−𝑣ℎ(𝑢+ℎ)2−𝜎−𝜆𝑣ℎ(𝑢+ℎ)2𝑐2(𝑐2+𝑐42)−𝜈𝑐𝑐+𝑐3−𝑢𝑢+ℎ𝑢𝑐2(𝑢+ℎ)(𝑐2+𝑐42)−𝜇−𝜆∣∣∣∣∣∣∣J(c,u,v)=|−Au(1+c)2−1−uc2(c+c2)2−νvc3(c+c3)2−λA1+c−cc+c2−νcc+c3Bc1u(c+c1)2Bcc+c1−2u−vh(u+h)2−σ−λ−uu+huvu+h2cc42(c2+c42)2vh(u+h)2c2(c2+c42)uc2(u+h)(c2+c42)−μ−λ|
(44)
where c, u, and v are defined by Eqs. (28–30).

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