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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

References

  1. Abbott MR (1993) Phytoplankton patchiness: ecological implications and observation methods. In: Levin SA, Powell TM, Steele JH (eds) Patch dynamics. Lecture notes in biomathematics, vol 96. Springer, Berlin, pp 37–49

  2. Addy K, Green L (1997) Dissolved oxygen and temperature. Fact Sheet No. 96–3, Natural Resources Facts, University of Rhodes Island

  3. Allegretto W, Mocenni C, Vicino A (2005) Periodic solutions in modelling lagoon ecological interactions. J Math Biol 51:367–388

    Article  MathSciNet  MATH  Google Scholar 

  4. Andersson A, Haecky P, Hagstrom A (1994) Effect of temperature and light on the growth of micro- nano- and pico-plankton: impact on algal succession. Mar Biol 120:511–520

    Article  Google Scholar 

  5. Behrenfeld MJ, Falkowski PG (1997) A consumers guide to phytoplankton primary productivity models. Limnol Oceanogr 42:1479–1491

    Article  Google Scholar 

  6. Berestycki H, Desvillettes L, Diekmann O (2014) Can climate change lead to gap formation? Ecol Complex 20:264–270

    Article  Google Scholar 

  7. Bonnefon O, Coville J, Garnier J, Hamel F, Roques L (2014) The spatio-temporal dynamics of neutral genetic diversity. Ecol Complex 20:282–292

    Article  Google Scholar 

  8. Breitburg DL, Loher T, Pacey CA, Gerstein A (1997) Varying effects of low dissolved oxygen on trophic interactions in an estuarine food web. Ecol Monogr 67:489–507

    Article  Google Scholar 

  9. Chapelle A, Ménesguen A, Deslous-Paoli J-M et al (2000) Modelling nitrogen, primary production and oxygen in a Mediterranean lagoon: impact of oysters farming and inputs from the watershed. Ecol Model 127:161–181

    Article  Google Scholar 

  10. Charlson RJ, Lovelock JE, Andreae MO, Warren SG (1987) Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate. Nature 326:655–661

    Article  Google Scholar 

  11. Childress JJ (1975) The respiratory rates of midwater crustaceans as a function of depth of occurrence and relation to the oxygen minimum layer of Southern California. Compar Biochem Physiol Part A Physiol 50:787–799

    Article  Google Scholar 

  12. Childress JJ (1976) Effects of pressure, temperature and oxygen on the oxygen consumption rate of the midwater copepod Gaussia princeps. Mar Biol 39:19–24

    Article  Google Scholar 

  13. Cosner C (2014) Challenges in modeling biological invasions and population distributions in a changing climate. Ecol Complex 20:258–263

    Article  Google Scholar 

  14. Culos GJ, Tyson RC (2014) Response of poikilotherms to thermal aspects of climate change. Ecol Complex 20:293–306

    Article  Google Scholar 

  15. Cushing DH (1975) Marine ecology and fisheries. Cambridge University Press, Cambridge

    Google Scholar 

  16. Davenport J, Trueman ER (1985) Oxygen uptake and buoyancy in zooplanktonic organisms from the tropical Eastern Atlantic. Compar Biochem Physiol Part A Physiol 81:857–863

    Article  Google Scholar 

  17. Decker MB, Breitburg DL, Purcell JE (2004) Effects of low dissolved oxygen on zooplankton predation by the ctenophore Mnemiopsis leidyi. Mar Ecol Progr Ser 280:163–172

    Article  Google Scholar 

  18. Denman K, Hofmann E, Marchant H (1996) Marine biotic responses and feedbacks to environmental change and feedbacks to climate. In: Houghton JT et al (eds) Climate change 1995. The science of climate change. Cambridge University Press, Cambridge, pp 483–516

    Google Scholar 

  19. Devol AH (1981) Vertical distribution of zooplankton respiration in relation to the intense oxygen minimum zones in two British Columbia fjords. J Plankt Res 3:593–602

    Article  Google Scholar 

  20. Enquist BJ, Economo EP, Huxman TE et al (2003) Scaling metabolism from organisms to ecosystems. Nature 423:639–642

    Article  Google Scholar 

  21. Eppley RW (1972) Temperature and phytoplankton growth in the sea. Fish Bull 70:1063–1085

    Google Scholar 

  22. Fasham M (1978) The statistical and mathematical analysis of plankton patchiness. Oceanogr Mar Biol Ann Rev 16:43–79

    Google Scholar 

  23. Fasham MJR, Ducklow HW, McKelvie SM (1990) A nitrogen-based model of plankton dynamics in the oceanic mixed layer. J Marine Res 48:591–639

    Article  Google Scholar 

  24. Ferrarini A, Rossi G, Mondoni A, Orsenigo S (2014) Prediction of climate warming impacts on plant species could be more complex than expected. Evidence from a case study in the Himalaya. Ecol Complex 20:307–314

    Article  Google Scholar 

  25. Franke U, Hutter K, Johnk K (1999) A physical-biological coupled model for algal dynamics in lakes. Bull Math Biol 61:239–272

    Article  Google Scholar 

  26. Franssen SU, Gu J, Bergmann N et al (2011) Transcriptomic resilience to global warming in the seagrass Zostera marina, a marine foundation species. Proc Natl Acad Sci USA 108:19276–19281

    Article  Google Scholar 

  27. Gliwicz MZ (1986) Predation and the evolution of vertical migration in zooplankton. Nature 320:746–748

    Article  Google Scholar 

  28. Greene CH, Widder EA, Youngbluth MJ, Tamse A, Johnson GE (1992) The migration behavior, fine structure, and bioluminescent activity of krill sound-scattering layers. Limnol Oceanogr 37:650–658

    Article  Google Scholar 

  29. Hamme RC, Keeling RF (2008) Ocean ventilation as a driver of interannual variability in atmospheric potential oxygen. Tellus B 60:706–717

    Article  Google Scholar 

  30. Hancke K, Glud RN (2004) Temperature effects on respiration and photosynthesis in three diatom-dominated benthic communities. Aquat Microb Ecol 37:265–281

    Article  Google Scholar 

  31. Harris GP (1986) Phytoplankton ecology: structure, function and fluctuation. Springer, Berlin

    Google Scholar 

  32. Hastings A (2001) Transient dynamics and persistence of ecological systems. Ecol Lett 4:215–220

    Article  Google Scholar 

  33. Hastings A (2004) Transients: the key to long-term ecological understanding? Trends Ecol Evolut 19:39–45

    Article  Google Scholar 

  34. Hein B, Viergutz C, Wyrwa J, Kirchesch V, Schöl A (2014) Modelling the impact of climate change on phytoplankton dynamics and the oxygen budget of the Elbe river and estuary (Germany). In: Kopmann R (ed) Lehfeldt R. Hamburg, ICHE, pp 1035–1042

  35. Hoppe H-G, Gocke K, Koppe R, Begler C (2002) Bacterial growth and primary production along a North-South transect of the Atlantic Ocean. Nature 416:168–171

    Article  Google Scholar 

  36. Huisman J, Weissing FJ (1995) Competition for nutrients and light in a mixed water column: a theoretical analysis. Am Nat 146:536–564

    Article  Google Scholar 

  37. Huisman J, van Oostveen P, Weissing FJ (1999) Species dynamics in phytoplankton blooms: incomplete mixing and competition for light. Am Nat 154:46–68

    Article  Google Scholar 

  38. Huisman J, Sharples J, Stroom JM, Visser PM, Kardinaal WEA, Verspa JM, Sommeijer B (2004) Changes in turbulent mixing shift competition for light between phytoplankton species. Ecology 85:2960–2970

    Article  Google Scholar 

  39. Hull V, Mocenni C, Falcucci M, Marchettini N (2000) A trophodynamic model for the Lagoon of Fogliano (Italy) with ecological dependent modifying parameters. Ecol Model 134:153–167

    Article  Google Scholar 

  40. Hull V, Parrella L, Falcucci M (2008) Modelling dissolved oxygen dynamics in coastal lagoons. Ecol Model 211:468–480

    Article  Google Scholar 

  41. Intergovernmental Panel on Climate Change (2014) Climate change 2014: synthesis report. In: Team Core Writing, Pachauri RK, Meyer LA (eds) Contribution of working groups I, II and III to the fifth assessment report of the intergovernmental panel on climate change. IPCC, Geneva

  42. Jones RI (1977) The importance of temperature conditioning to the respiration of natural phytoplankton communities. Brit Phycol J 12:277–285

    Article  Google Scholar 

  43. Keeling RF, Kortzinger A, Gruber N (2010) Ocean deoxygenation in a warming world. Mar Sci 2:199–229

    Article  Google Scholar 

  44. Kremer J, Nixon SW (1978) A coastal marine ecosystem: simulation and analysis. Springer, Berlin

    Google Scholar 

  45. Kyriazopoulos P, Nathan J, Meron E (2014) Species coexistence by front pinning. Ecol Complex 20:271–281

    Article  Google Scholar 

  46. Li W, Smith J, Platt T (1984) Temperature response of photosynthetic capacity and carboxylase activity in arctic marine phytoplankton. Mar Ecol Progr Ser 17:237–243

    Article  Google Scholar 

  47. Long A, Tyson RC (2014) Integrating Homo sapiens into ecological models: imperatives of climate change. Ecol Complex 20:325–334

    Article  Google Scholar 

  48. Mackas DL, Boyd CM (1979) Spectral analysis of zooplankton spatial heterogeneity. Science 204:62–64

    Article  Google Scholar 

  49. Malchow H, Petrovskii SV, Hilker FM (2003) Models of spatiotemporal pattern formation in plankton dynamics. Nova Acta Leopoldina NF 88:325–340

    Google Scholar 

  50. Malchow H, Petrovskii SV, Venturino E (2008) Spatiotemporal patterns in ecology and epidemiology: theory, models, and simulation. CRC Press, Boca Raton

    Google Scholar 

  51. Matear R, Hirst A, McNeil B (2000) Changes in dissolved oxygen in the SouthernOcean with climate change. Geochem Geophys Geosyst 1(11):2000GC000086

  52. Martin AP (2003) Phytoplankton patchiness: the role of lateral stirring and mixing. Progr Oceanogr 57:125–174

    Article  Google Scholar 

  53. Medvinsky AB, Petrovskii SV, Tikhonova IA, Malchow H, Li B-L (2002) Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev 44:311–370

    Article  MathSciNet  MATH  Google Scholar 

  54. Misra A (2010) Modeling the depletion of dissolved oxygen in a lake due to submerged macrophytes. Nonlin Anal Model Cont 15:185–198

    MATH  Google Scholar 

  55. Monin AS, Yaglom AM (1971) Statistical fluid mechanics: mechanics of turbulence, vol 1. MIT Press, Cambridge

    Google Scholar 

  56. Moss BR (2009) Ecology of fresh waters: man and medium, past to future. Wiley, London

    Google Scholar 

  57. Najjar RG, Walker HA, Anderson PJ et al (2000) The potential impacts of climate change on the mid-Atlantic coastal region. Clim Res 14:219–233

    Article  Google Scholar 

  58. Najjar RG, Pyke CR, Adams MB et al (2010) Potential climate-change impacts on the Chesapeake Bay. Estuar Coastal Shelf Sci 86:1–20

    Article  Google Scholar 

  59. Nguyen KDT, Morley SA, La CH et al (2011) Upper temperature limits of tropical marine ectotherms: global warming implications. PLoS ONE 6(12):e29340

    Article  Google Scholar 

  60. Okubo A (1980) Diffusion and ecological problems: mathematical models. Springer, Berlin

    Google Scholar 

  61. Petrovskii SV, Malchow H (2001) Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics. Theor Popul Biol 59:157–174

    Article  MATH  Google Scholar 

  62. Petrovskii SV, Malchow H (2004) Mathematical models of marine ecosystems. In: The encyclopedia of life support systems (EOLSS). EOLSS Publishers, Oxford

  63. Petrovskii SV, Li B-L, Malchow H (2004) Transition to spatiotemporal chaos can resolve the paradox of enrichment. Ecol Complex 1:37–47

    Article  Google Scholar 

  64. Petrovskii SV, Morozov AY, Venturino E (2002) Allee effect makes possible patchy invasion in a predator-prey system. Ecol Lett 5:345–352

    Article  Google Scholar 

  65. Petrovskii SV, Kawasaki K, Takasu F, Shigesada N (2001) Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species. Japan J Ind Appl Maths 18:459–481

    Article  MathSciNet  MATH  Google Scholar 

  66. Pinel-Alloul B (1995) Spatial heterogeneity as a multiscale characteristic of zooplankton community. Hydrobiologia 300(301):17–42

    Article  Google Scholar 

  67. Prosser CL (1961) Oxygen: respiration and metabolism. In: Prosser CL, Brown FA (eds) Comparative animal physiology. WB Saunders, Philadelphia, pp 165–211

    Google Scholar 

  68. Raven JA, Geider RJ (1988) Temperature and algal growth. New Phytol 110:441–461

    Article  Google Scholar 

  69. Robinson C (2000) Plankton gross production and respiration in the shallow water hydrothermal systems of Milos, Aegean Sea. J Plankt Res 22:887–906

    Article  Google Scholar 

  70. Shaffer G, Leth O, Ulloa O et al (2000) Warming and circulation change in the Eastern South Pacific Ocean. Geophys Res Lett 27:1247–1250

    Article  Google Scholar 

  71. Sekerci Y, Petrovskii S (2015) Mathematical modelling of spatiotemporal dynamics of oxygen in a plankton system. Math Model Nat Phenom 7:96–114

    Article  MathSciNet  Google Scholar 

  72. Steele JH (1978) Spatial pattern in plankton communities. Plenum, London

    Google Scholar 

  73. Steel JA (1980) Phytoplankton models. In: LeCren ED, Lowe-McConnell RH (eds) Functioning of freshwater ecosystems, vol 2. Cambridge University Press, Cambridge, pp 220–227

    Google Scholar 

  74. Williamson P, Gribbin J (1991) How plankton change the climate. N Sci 129:48–52

Download references

Acknowledgments

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sergei Petrovskii.

Appendix

Appendix

The Jacobian matrix J, i.e., the matrix of the linearized system (1921), 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. (2627).

\(\bullet \) 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. (2830).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sekerci, Y., Petrovskii, S. Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change. Bull Math Biol 77, 2325–2353 (2015). https://doi.org/10.1007/s11538-015-0126-0

Download citation

Keywords

  • Phytoplankton
  • Global warming
  • Oxygen depletion
  • Extinction
  • Pattern formation