# Pattern Formation in a Model Oxygen-Plankton System

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

## 3. Simulation Results: Pattern Formation

## 4. Discussion and Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Allegretto, W.; Mocenni, C.; Vicino, A. Periodic solutions in modelling lagoon ecological interactions. J. Math. Biol.
**2005**, 51, 367–388. [Google Scholar] [CrossRef] [PubMed] - Hull, V.; Mocenni, C.; Falcucci, M.; Marchettini, N. A trophodynamic model for the lagoon of Fogliano (Italy) with ecological dependent modifying parameters. Ecol. Model.
**2000**, 134, 153–167. [Google Scholar] [CrossRef] - Hull, V.; Parrella, L.; Falcucci, M. Modelling dissolved oxygen dynamics in coastal lagoons. Ecol. Model.
**2008**, 211, 468–480. [Google Scholar] [CrossRef] - Marchettini, N.; Mocenni, C.; Vicino, A. Integrating slow and fast dynamics in a shallow water coastal lagoon. Ann. Di Chim.
**1999**, 89, 505–514. [Google Scholar] - Misra, A. Modeling the depletion of dissolved oxygen in a lake due to submerged macrophytes. Nonlinear Anal. Model. Control
**2010**, 15, 185–198. [Google Scholar] - Misra, A.K.; Chandra, P.V.; Raghavendra, V. Modeling the depletion of dissolved oxygen in a lake due to algal bloom: Effect of time delay. Adv. Water Resour.
**2011**, 34, 1232–1238. [Google Scholar] [CrossRef] - Caruso, A.; Gargano, M.E.; Valenti, D.; Fiasconaro, A.; Spagnolo, B. Cyclic fluctuations, climatic changes and role of noise in Planktonic Foraminifera in the Mediterranean Sea. Fluct. Noise Lett.
**2005**, 5, L349–L355. [Google Scholar] [CrossRef] - Ito, T.; Minobe, S.; Long, M.C.; Deutsch, C. Upper ocean O
_{2}trends: 1958–2015. Geophys. Res. Lett.**2017**, 44, 4214–4223. [Google Scholar] [CrossRef] - Richardson, K.; Bendtsen, J. Photosynthetic oxygen production in a warmer ocean: The Sargasso Sea as a case study. Phil. Trans. R. Soc. A
**2017**, 375. [Google Scholar] [CrossRef] [PubMed] - Petrovskii, S.; Sekerci, Y.; Venturino, E. Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change. J. Theor. Biol.
**2017**, 424, 91–109. [Google Scholar] [CrossRef] [PubMed] - Sekerci, Y.; Petrovskii, S. Mathematical modelling of spatiotemporal dynamics of oxygen in a plankton system. Math. Model. Nat. Phenom.
**2015**, 10, 96–114. [Google Scholar] [CrossRef] - Sekerci, Y.; Petrovskii, S. Mathematical modelling of plankton–oxygen dynamics under the climate change. Bull. Math. Biol.
**2015**, 77, 2325–2353. [Google Scholar] [CrossRef] [PubMed] - Conte, F.; Cubbage, J. Phytoplankton and recreational ponds. West. Reg. Aquacult. Center
**2001**, 5, 1–6. [Google Scholar] - Franke, U.; Hutter, K.; Jöhnk, K. A physical-biological coupled model for algal dynamics in lakes. Bull. Math. Biol.
**1999**, 61, 239–272. [Google Scholar] [CrossRef] [PubMed] - Okubo, A. Diffusion and ecological problems: Mathematical models. In Biomathematics; Springer: Berlin, Germany, 1980; Volume 10. [Google Scholar]
- Abbott, M.R. Phytoplankton patchiness: Ecological implications and observation methods. Patch Dyn.
**1993**, 96, 37–49. [Google Scholar] - Denman, K.L. Covariability of chlorophyll and temperature in the sea. In Deep Sea Research and Oceanographic Abstracts; Elsevier: Amsterdam, The Netherlands, 1976; Volume 23, pp. 539–550. [Google Scholar]
- Fasham, M. The statistical and mathematical analysis of plankton patchiness. Oceanogr. Mar. Biol. Annu. Rev.
**1978**, 16, 43–79. [Google Scholar] - Platt, T. Local phytoplankton abundance and turbulence. Deep Sea Res. Oceanogr. Abstr.
**1972**, 19, 183–187. [Google Scholar] [CrossRef] - Malchow, H.; Petrovskii, S.V.; Venturino, E. Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation; Chapman & Hall/CRC Press: London, UK, 2008. [Google Scholar]
- Medvinsky, A.B.; Petrovskii, S.V.; Tikhonova, I.A.; Malchow, H.; Li, B.-L. Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev.
**2002**, 44, 311–370. [Google Scholar] [CrossRef] - Battaglia, G.; Joos, F. Hazards of decreasing marine oxygen: The near-term and millennial-scale benefits of meeting the Paris climate targets. Earth Syst. Dyn. Discuss.
**2017**. [Google Scholar] [CrossRef] - Breitburg, D.; Levin, L.A.; Oschlies, A.; Grégoire, M.; Chavez, F.P.; Conley, D.J.; Garçon, V.; Gilbert, D.; Gutiérrez, D.; Isensee, K.; et al. Declining oxygen in the global ocean and coastal waters. Science
**2018**, 359, eaam7240. [Google Scholar] [CrossRef] [PubMed] - Shaffer, G.; Olsen, S.M.; Pedersen, J.O.P. Long-term ocean oxygen depletion in response to carbon dioxide emissions from fossil fuels. Nat. Geosci.
**2009**, 2, 105–109. [Google Scholar] [CrossRef] - Sekerci, Y.; Petrovskii, S.V. Global warming can lead to depletion of oxygen by disrupting phytoplankton photosynthesis: A mathematical modelling approach. Geosciences
**2018**, 8, 201. [Google Scholar] [CrossRef] - Hancke, K.; Glud, R.N. Temperature effects on respiration and photosynthesis in three diatom-dominated benthic communities. Aquat. Microb. Ecol.
**2004**, 37, 265–281. [Google Scholar] [CrossRef] [Green Version] - López-Sandoval, D.C.; Rodriguez-Ramos, T.; Cermeño, P.; Sobrino, C.; Marañón, E. Photosynthesis and respiration in marine phytoplankton: Relationship with cell size, taxonomic affiliation, and growth phase. J. Exp. Mar. Biol. Ecol.
**2014**, 457, 151–159. [Google Scholar] [CrossRef] - McKinnon, A.D.; Duggan, S.; Logan, M.; Lonborg, C. Plankton respiration, production, and trophic state in tropical coastal and shelf waters adjacent to Northern Australia. Front. Mar. Sci.
**2017**, 4, 346. [Google Scholar] [CrossRef] - Robinson, C. Plankton gross production and respiration in the shallow water hydrothermal systems of Milos, Aegean Sea. J. Plankt. Res.
**2000**, 22, 887–906. [Google Scholar] [CrossRef] [Green Version] - Bardey, P.; Garnesson, P.; Moussu, G.; Wald, L. Joint analysis of temperature and ocean colour satellite images for mesoscale activities in the Gulf of Biscay. Int. J. Remote Sens.
**1999**, 20, 1329–1341. [Google Scholar] [CrossRef] - Wyatt, T. The biology of Oikopleura dioica and Fritillaria borealis in the southern bight. Mar. Biol.
**1973**, 22, 137–158. [Google Scholar] [CrossRef] - Neufeld, Z.; Tel, T. Advection in chaotically time-dependent open flows. Phys. Rev. E
**1998**, 57, 2832–2842. [Google Scholar] [CrossRef] [Green Version] - Bracco, A.; Provenzale, A.; Scheuring, I. Mesoscale vortices and the paradox of the plankton. Proc. R. Soc. Ser. B
**2000**, 267, 1795–1800. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Károlyi, G.; Péntek, A.; Scheuring, I.; Tél, T.; Toroczkai, Z. Chaotic flow: The physics of species coexistence. Proc. Natl. Acad. Sci. USA
**2000**, 97, 13661–13665. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**The structure of the conceptual model describing the interactions between oxygen (c), phytoplankton (u) and zooplankton (v). Arrows show the flows of matter through the system, and the parameterizations of the rates are as labelled. Phytoplankton produces oxygen through photosynthesis during the day-time then consumes it during the night [13]. Zooplankton feeds on phytoplankton and consumes oxygen through breathing; see further details in the text.

**Figure 2.**Snapshots of the oxygen spatial distribution obtained for $A=2.09$ and shown for for $t=1$, $t=200$, $t=500$ and $t=1100$ (left to right, top to bottom), other parameters are given in the text. The initial conditions are given by Equations (10)–(12) where ${c}_{0}=0.4638$, ${u}_{0}=0.4179$ and ${v}_{0}=0.1343$.

**Figure 3.**Spatial distribution of the oxygen concentration obtained for $A=2.3$ and shown for $t=1$, $t=200$, $t=800$ and $t=2000$ (left to right, top to bottom). Other parameters are the same as above. The initial conditions are given by Equations (10)–(15) with ${c}_{0}=0.4696$, ${u}_{0}=0.3775$ and ${v}_{0}=0.1501$.

**Figure 4.**Spatial distribution of the oxygen concentration obtained for $A=2.06$ and shown for $t=1$, $t=300$, $t=500$ and $t=900$ (left to right, top to bottom). Other parameters are the same as above. The initial conditions are given by Equations (13)–(15) with ${c}_{0}=0.4630$, ${u}_{0}=0.4245$ and ${v}_{0}=0.1314$.

**Figure 5.**Spatial distribution of the oxygen concentration obtained for $A=2.4$ and shown for $t=1$, $t=120$, $t=200$ and $t=1200$ (left to right, top to bottom). Other parameters are the same as above. The initial conditions are given by Equations (13)–(15) with ${c}_{0}=0.4723$, ${u}_{0}=0.3612$ and ${v}_{0}=0.1554$.

**Figure 6.**Spatial distribution of the oxygen concentration obtained for $A=2.7$ and shown for $t=1$, $t=70$, $t=140$ and $t=500$ (left to right, top to bottom). Other parameters are the same as above. The initial conditions are given by Equations (13)–(15) with ${c}_{0}=0.4802$, ${u}_{0}=0.3206$ and ${v}_{0}=0.1658$.

**Figure 7.**Spatial distribution of the oxygen concentration obtained for $A=2.04$ in a larger domain $900\times 900$ and shown for $t=1$, $t=350$, $t=500$, $t=900$, $t=1400$ and $t=1600$ (left to right, top to bottom). Other parameters are the same as above. The initial conditions are given by Equations (16)–(18) with ${c}_{0}=0.4624$, ${u}_{0}=0.4290$ and ${v}_{0}=0.1293$.

**Figure 8.**Spatial distribution of the oxygen concentration obtained for $A=2.07$ and shown for $t=1$, $t=180$, $t=220$, $t=250$, $t=290$ and $t=500$ (left to right, top to bottom). Other parameters and the initial conditions are the same as in Figure 7.

**Figure 9.**Spatial distribution of the oxygen concentration obtained for $A=2.4$ and shown for $t=1$, $t=120$, $t=200$ and $t=3000$ (left to right, top to bottom). The other parameters are the same as above. The initial conditions are given by Equations (13)–(15) with ${c}_{0}=0.4723$, ${u}_{0}=0.3612$${v}_{0}=0.1554$.

**Figure 10.**Spatial distribution of the oxygen concentration obtained for $A=2.5$ and shown for $t=1$, $t=50$, $t=150$ and $t=600$ (left to right, top to bottom). Other parameters are the same as above. The initial conditions are given by Equations (16)–(18) with ${c}_{0}=0.4750$, ${u}_{0}=0.3464$ and ${v}_{0}=0.1596$.

**Figure 11.**Spatial distribution of the oxygen concentration in $600\times 600$ computational domain obtained for $A=2.4$ and shown for $t=1$, $t=50$, $t=200$ and $t=400$ (left to right, top to bottom). Note that all parameters are the same as in Figure 5, but the initial conditions are slightly different, i.e., $c(x,y,0)=0.4723$, $u(x,y,0)=0.3612-2\times {10}^{-7}(x-180)(x-420)-6\times {10}^{-7}(y-90)(y-210)$ and $v(x,y,0)=0.1554-6\times {10}^{-7}(x-250)-3\times {10}^{-5}(y-250)$. Contrary to the case shown in Figure 5, now the evolution of the initial conditions results in oxygen depletion and species extinction.

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**MDPI and ACS Style**

Sekerci, Y.; Petrovskii, S.
Pattern Formation in a Model Oxygen-Plankton System. *Computation* **2018**, *6*, 59.
https://doi.org/10.3390/computation6040059

**AMA Style**

Sekerci Y, Petrovskii S.
Pattern Formation in a Model Oxygen-Plankton System. *Computation*. 2018; 6(4):59.
https://doi.org/10.3390/computation6040059

**Chicago/Turabian Style**

Sekerci, Yadigar, and Sergei Petrovskii.
2018. "Pattern Formation in a Model Oxygen-Plankton System" *Computation* 6, no. 4: 59.
https://doi.org/10.3390/computation6040059