# Biomathematical Model for Water Quality Assessment: Macroinvertebrate Population Dynamics and Dissolved Oxygen

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Conceptualization

#### Mathematical Formulation

## 3. Qualitative Analysis

#### 3.1. Absence, Presence, and Coexistence of AMS

**Proposition 1.**

- (i)
- AM-free equilibrium: ${E}_{0}=(m,0,0,0)$.
- (ii)
- Presence equilibrium: ${E}_{1}=({o}^{*},{x}_{1}^{*},0,0)$, ${E}_{2}=({o}^{*},0,{x}_{2}^{*},0)$, ${E}_{3}=({o}^{*},0,0,{x}_{3}^{*})$,${E}_{4}=({o}^{*},{x}_{1}^{*},{x}_{2}^{*},0)$, ${E}_{5}=({o}^{*},{x}_{1}^{*},0,{x}_{3}^{*})$ and ${E}_{6}=({o}^{*},0,{x}_{2}^{*},{x}_{3}^{*})$.
- (iii)
- Coexistence equilibrium: ${E}_{7}=({o}^{*},{x}_{1}^{*},{x}_{2}^{*},{x}_{3}^{*})$ with$${x}_{i}^{*}={\displaystyle \frac{{k}_{i}}{m{r}_{i}}}{o}^{*}\left({r}_{i}+{\gamma}_{i}(m-{o}^{*})\right)\phantom{\rule{1.em}{0ex}}for\phantom{\rule{1.em}{0ex}}i=1,2,3.$$

**Proof.**

#### 3.2. Stability Analysis

**Proposition**

**2.**

**Proof.**

## 4. Results

^{−1}, there is no significant difference in the behavior of the population densities (blue band in b–d of Figure 2).

#### Validation

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mathematical Calculation

#### Appendix A.1. Jacobian Matrix

#### Appendix A.2. Coefficients

## Appendix B. Parameters

**Figure A1.**Impacts of hypoxia on reproduction and respiration of aquatic invertebrates. Data digitized at different oxygen concentrations, scaled to control conditions (normoxia) and the Michaelis–Menten model fit (±95% CI). Adapted from Galic et al. [23].

## References

- United Nations. Transforming our world: The 2030 agenda for sustainable development. In General Assembley 70 Session; United Nations: New York, NY, USA, 2015. [Google Scholar]
- Uddin, M.G.; Nash, S.; Olbert, A.I. A review of water quality index models and their use for assessing surface water quality. Ecol. Indic.
**2021**, 122, 3400–3411. [Google Scholar] - Almeida, C.; González, S.O.; Mallea, M.; González, P. A recreational water quality index using chemical, physical and microbiological parameters. Environ. Sci. Pollut. Res.
**2012**, 19, 3400–3411. [Google Scholar] [CrossRef] [PubMed] - Farhat, N.; Kim, L.H.; Vrouwenvelder, J.S. Online characterization of bacterial processes in drinking water systems. Npj Clean Water
**2020**, 3, 1–7. [Google Scholar] [CrossRef] - Jørgensen, S.E.; Xu, F.L.; Salas, F.; Marques, J.C. Handbook of Ecological Indicators for Assessment of Ecosystem Health; CRC Press: Boca Raton, FL, USA, 2016; pp. 9–64. [Google Scholar]
- Hernández-Mira, F.A.; Rosas-Acevedo, J.L.; Reyes-Umaña, M.; Violante-González, J.; Sigarreta-Almira, J.M.; Vakhania, N. Multimetric Index to Evaluate Water Quality in Lagoons: A Biological and Geomorphological Approach. Sustainability
**2021**, 13, 4631. [Google Scholar] [CrossRef] - Bain, M.B.; Stevenson, N.J. (Eds.) Aquatic Habitat Assessment: Commom Methods; Asian Fisheries Society: Bethesda, MD, USA, 1999; p. 186. ISBN 1-888569-18-2. [Google Scholar]
- Ejigu, M.T. Overview of water quality modeling. Cogent Eng.
**2021**, 8, 1891711. [Google Scholar] [CrossRef] - Zhen-Gang, J. Hydrodynamics and Water Quality: Modeling Rivers, Lakes, and Estuaries; John Wiley & Sons: Hoboken, NJ, USA, 2017. [Google Scholar]
- Loucks, D.P.; Van Beek, E. Water Resource Systems Planning and Management: An Introduction to Methods, Models, and Applications; Springer: Cham, Switzerland, 2017. [Google Scholar]
- Gonenc, I.E.; Wolflin, J.P. (Eds.) Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development; CRC Press: Boca Raton, FL, USA, 2004. [Google Scholar]
- Jacoby, J.; Welch, E. Pollutant Effects in Freshwater: Applied Limnology, 3rd ed.; CRC Press: Boca Raton, FL, USA, 2004. [Google Scholar]
- Sánchez, E.; Colmenarejo, M.F.; Vicente, J.; Rubio, A.; García, M.G.; Travieso, L.; Borja, R. Use of the water quality index and dissolved oxygen deficit as simple indicators of watersheds pollution. Ecol. Indic.
**2007**, 7, 315–328. [Google Scholar] [CrossRef] - Kannel, P.R.; Lee, S.; Lee, Y.S.; Kanel, S.R.; Khan, S.P. Application of water quality indices and dissolved oxygen as indicators for river water classification and urban impact assessment. Environ. Monit. Assess.
**2007**, 132, 93–110. [Google Scholar] [CrossRef] - Best, M.A.; Wither, A.W.; Coates, S. Dissolved oxygen as a physico-chemical supporting element in the Water Framework Directive. Mar. Pollut. Bull.
**2007**, 55, 53–64. [Google Scholar] [CrossRef] - Kannel, P.R.; Kanel, S.R.; Lee, S.; Lee, Y.S.; Gan, T.Y. A review of public domain water quality models for simulating dissolved oxygen in rivers and streams. Environ. Model. Assess.
**2011**, 16, 183–204. [Google Scholar] [CrossRef] - Kaller, M.D.; Kelso, W.E. Association of macroinvertebrate assemblages with dissolved oxygen concentration and wood surface area in selected subtropical streams of the southeastern USA. Aquat. Ecol.
**2007**, 41, 95–110. [Google Scholar] [CrossRef] - Wilhm, J.; McClintock, N. Dissolved oxygen concentration and diversity of benthic macroinvertebrates in an artificially destratified lake. Hydrobiologia
**1978**, 57, 163–166. [Google Scholar] [CrossRef] - Wang, J.; Fu, Z.; Qiao, H.; Liu, F. Assessment of eutrophication and water quality in the estuarine area of Lake Wuli, Lake Taihu, China. Sci. Total Environ.
**2019**, 650, 1392–1402. [Google Scholar] [CrossRef] [PubMed] - Meier, H.E.M.; Eilola, K.; Almroth-Rosell, E.; Schimanke, S.; Kniebusch, M.; Höglund, A.; Pemberton, P.; Liu, Y.; Väli, G.; Saraiva, S. Disentangling the impact of nutrient load and climate changes on Baltic Sea hypoxia and eutrophication since 1850. Clim. Dyn.
**2019**, 53, 1145–1166. [Google Scholar] [CrossRef] - Zhang, J.; Wang, C.; Jiang, X.; Song, Z.; Xie, Z. Effects of human-induced eutrophication on macroinvertebrate spatiotemporal dynamics in Lake Dianchi, a large shallow plateau lake in China. Environ. Sci. Pollut. Res.
**2020**, 27, 1–15. [Google Scholar] [CrossRef] - Bazzanti, M.; Mastrantuono, L.; Solimini, A.G. Selecting macroinvertebrate taxa and metrics to assess eutrophication in different depth zones of Mediterranean lakes. Fundam. Appl. Limnol. Hydrobiol.
**2012**, 180, 133–143. [Google Scholar] [CrossRef] - Galic, N.; Hawkins, T.; Forbes, V.E. Adverse impacts of hypoxia on aquatic invertebrates: A meta-analysis. Sci. Total Environ.
**2019**, 652, 736–743. [Google Scholar] [CrossRef] - Etemi, F.Z.; Bytyçi, P.; Ismaili, M.; Fetoshi, O.; Ymeri, P.; Shala–Abazi, A.; Muja-Bajraktari, N.; Czikkely, M. The use of macroinvertebrate based biotic indices and diversity indices to evaluate the water quality of Lepenci river basin in Kosovo. J. Environ. Sci. Heal. Part A
**2020**, 55, 748–758. [Google Scholar] [CrossRef] - Slimani, N.; Sánchez-Fernández, D.; Guilbert, E.; Boumaiza, M.; Guareschi, S.; Thioulouse, J. Assessing potential surrogates of macroinvertebrate diversity in North-African Mediterranean aquatic ecosystems. Ecol. Indic.
**2019**, 101, 324–329. [Google Scholar] [CrossRef] - Silva, D.R.O.; Herlihy, A.T.; Hughes, R.M.; Macedo, D.R.; Callisto, M. Assessing the extent and relative risk of aquatic stressors on stream macroinvertebrate assemblages in the neotropical savanna. Sci. Total Environ.
**2018**, 633, 179–188. [Google Scholar] [CrossRef] - Croijmans, L.; de Jong, J.F.; Prins, H.H. Oxygen is a better predictor of macroinvertebrate richness than temperature—A systematic review. Environ. Res. Lett.
**2020**, 38, 1820–1832. [Google Scholar] [CrossRef] - Su, P.; Wang, X.; Lin, Q.; Peng, J.; Song, J.; Fu, J.; Wang, S.; Cheng, D.; Bai, H.; Li, Q. Variability in macroinvertebrate community structure and its response to ecological factors of the Weihe River Basin, China. Ecol. Eng.
**2019**, 140, 105595. [Google Scholar] [CrossRef] - Mezgebu, A.; Lakew, A.; Lemma, B. Water quality assessment using benthic macroinvertebrates as bioindicators in streams and rivers around Sebeta, Ethiopia. Afr. J. Aquat. Sci.
**2019**, 44, 361–367. [Google Scholar] [CrossRef] - Liu, Z.; Fan, B.; Huang, Y.; Yu, P.; Li, Y.; Chen, M.; Cai, M.; Lv, W.; Jiang, Q.; Zhao, Y. Assessing the ecological health of the Chongming Dongtan Nature Reserve, China, using different benthic biotic indices. Mar. Pollut. Bull.
**2018**, 146, 76–84. [Google Scholar] - Liu, Z.; Chen, M.; Li, Y.; Huang, Y.; Fan, B.; Lv, W.; Yu, P.; Wu, D.; Zhao, Y. Different effects of reclamation methods on macrobenthos community structure in the Yangtze Estuary, China. Mar. Pollut. Bull.
**2018**, 127, 429–436. [Google Scholar] - Pineda-Pineda, J.J.; Rosas-Acevedo, J.L.; Sigarreta, J.M.; Hernández-Gómez, J.C.; Reyes-Umaña, M. Biotic Indices to Evaluate Water Quality: BMWP. Int. J. Environ. Ecol. Fam. Urban Stud. IJEEFUS
**2018**, 8, 23–36. [Google Scholar] - Zhou, X.D.; Xu, M.Z.; Lei, F.K.; Zhang, J.H.; Wang, Z.Y.; Luo, Y.Y. Responses of Macroinvertebrate Assemblages to Flow in the Qinghai-Tibet Plateau: Establishment and Application of a Multi-metric Habitat Suitability Model. Water Resour. Res.
**2022**, 58, e2021WR030909. [Google Scholar] [CrossRef] - Pineda-Pineda, J.J.; Martínez-Martínez, C.T.; Méndez-Bermúdez, J.A.; Muñoz-Rojas, J.; Sigarreta, J.M. Application of Bipartite Networks to the Study of Water Quality. Sustainability
**2020**, 12, 5143. [Google Scholar] [CrossRef] - Schleiter, I.M.; Borchardt, D.; Wagner, R.; Dapper, T.; Schmidt, K.D.; Schmidt, H.H.; Werner, H. Modelling water quality, bioindication and population dynamics in lotic ecosystems using neural networks. Ecol. Model.
**1999**, 120, 271–286. [Google Scholar] [CrossRef] - Maier, H.R.; Dandy, G.C. The use of artificial neural networks for the prediction of water quality parameters. Water Resour. Res.
**1996**, 32, 1013–1022. [Google Scholar] [CrossRef] - Villamarín, C.; Rieradevall, M.; Paul, M.J.; Barbour, M.T.; Prat, N. A tool to assess the ecological condition of tropical high Andean streams in Ecuador and Peru: The IMEERA index. Ecol. Indic.
**2013**, 29, 79–92. [Google Scholar] [CrossRef] - Karaouzas, I.; Smeti, E.; Kalogianni, E.; Skoulikidis, N.T. Ecological status monitoring and assessment in Greek rivers: Do macroinvertebrate and diatom indices indicate same responses to anthropogenic pressures? Ecol. Indic.
**2019**, 101, 126–132. [Google Scholar] [CrossRef] - El Sayed, S.M.; Hegab, M.H.; Mola, H.R.; Ahmed, N.M.; Goher, M.E. An integrated water quality assessment of Damietta and Rosetta branches (Nile River, Egypt) using chemical and biological indices. Environ. Monit. Assess.
**2020**, 192, 1–16. [Google Scholar] [CrossRef] [PubMed] - Streeter, H.W.; Phelps, E.B. A Study of the Pollution and Natural Purification of the Ohio River; Public Health Bulletin No 146; Public Health Service: Washington, DC, USA, 1925.
- Wang, Q.; Li, S.; Jia, P.; Qi, C.; Ding, F. A review of surface water quality models. Sci. World J.
**2013**. [Google Scholar] [CrossRef] - da Silva Burigato Costa, C.M.; Leite, I.R.; Almeida, A.K.; de Almeida, I.K. Choosing an appropriate water quality model—A review. Environ. Monit. Assess.
**2021**, 193, 1–15. [Google Scholar] [CrossRef] - Park, R.A.; Clough, J.S.; Wellman, M.C. AQUATOX: Modeling environmental fate and ecological effects in aquatic ecosystems. Ecol. Model.
**2008**, 213, 1–15. [Google Scholar] [CrossRef] - West Virginia Department of Environmental Protection. Available online: https://dep.wv.gov/WWE/getinvolved/sos/Documents/Benthic/VisualMacroGuide.pdf (accessed on 1 September 2022).
- Researchgate. Available online: https://www.researchgate.net/profile/Pablo-Gutierrez-Fonseca/publication/295854904_Guia_fotografica_de_familias_de_macroinvertebrados_acuaticos_de_Puerto_Rico/links/56ce23e508aeb52500c36b4f/Guia-fotografica-de-familias-de-macroinvertebrados-acuaticos-de-Puerto-Rico.pdf (accessed on 1 September 2022).
- Everaert, G.; De Neve, J.; Boets, P.; Dominguez-Granda, L.; Mereta, S.T.; Ambelu, A.; Hoang, T.H.; Goethals, P.L.M.; Thas, O. Comparison of the abiotic preferences of macroinvertebrates in tropical river basins. PLoS ONE
**2014**, 9, e108898. [Google Scholar] [CrossRef] [PubMed] - Weiwei, L.; Youhui, H.; Zhiquan, L.; Yang, Y.; Bin, F.; Yunlong, Z. Application of macrobenthic diversity to estimate ecological health of artificial oyster reef in Yangtze Estuary, China. Mar. Pollut. Bull.
**2016**, 103, 137–143. [Google Scholar] - Weiwei, L.; Zhiquan, L.; Yang, Y.; Youhui, H.; Bin, F.; Qichen, J.; Yunlong, Z. Loss and self-restoration of macrobenthic diversity in reclamation habitats of estuarine islands in Yangtze Estuary, China. Mar. Pollut. Bull.
**2016**, 103, 128–136. [Google Scholar] - Pearl, R.; Reed, L.J. On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proc. Natl. Acad. Sci. USA
**1920**, 6, 275. [Google Scholar] [CrossRef] - Hutchinson, G.E. An Introduction to Population Ecology; Yale University Press: New Haven, CT, USA, 1978. [Google Scholar]
- Chapman, E.J.; Byron, C.J. The flexible application of carrying capacity in ecology. Glob. Ecol. Conserv.
**2018**, 13, e00365. [Google Scholar] - Gunderson, L.H. Ecological resilience-in theory and application. Annu. Rev. Ecol. Syst.
**2000**, 31, 425–439. [Google Scholar] [CrossRef] - Rosas-Acevedo, J.L.; Ávila-Pérez, H.; Sánchez-Infante, A.; Rosas-Acevedo, Y.; García-Ibañez, S.; Sampedro-Rosas, L.; Granados-Ramírez, J.G.; Juárez-López, A.L. índice BMWP, FBI y EPT para determinar la calidad del agua en la laguna de Coyuca de Benítez, Guerrero, México. Rev. Iberoam. Cienc.
**2014**, 1, 81–88. [Google Scholar] - Rosińska, J.; Kozak, A.; Dondajewska, R.; Kowalczewska-Madura, K.; Gołdyn, R. Water quality response to sustainable restoration measures–Case study of urban Swarzędzkie Lake. Ecol. Indic.
**2018**, 84, 437–449. [Google Scholar] [CrossRef] - Gołdyn, R.; Podsiadłowski, S.; Dondajewska, R.; Kozak, A. The sustainable restoration of lakes—Towards the challenges of the Water Framework Directive. Ecohydrol. Hydrobiol.
**2014**, 14, 68–74. [Google Scholar] [CrossRef] - Kail, J.; Brabec, K.; Poppe, M.; Januschke, K. The effect of river restoration on fish, macroinvertebrates and aquatic macrophytes: A meta-analysis. Ecol. Indic.
**2015**, 58, 311–321. [Google Scholar] [CrossRef]

**Figure 1.**Conceptual diagram of the mathematical model. The state variables o, ${x}_{1}$, ${x}_{2}$, and ${x}_{3}$ represent the DO concentrations in the water and the AM populations—intolerant, regularly tolerant, and tolerant to organic contamination, respectively. The processes are: (a) oxygenation/aeration, (b) DO consumption, (c) AM natality, (d) DO utilization, and (e) interspecific competition.

**Figure 2.**Influence of oxygenation/aeration rate r on AM population densities. (

**a**) Variation of DO through time for different values of r. The population fluctuations of the different classes of AMs are shown in (

**b**), (

**c**), and (

**d**) for ${x}_{1},{x}_{2}$, and ${x}_{3}$, respectively. Note that, in all cases, if r is smaller, then AM reproduction is smaller and slower than when r is larger.

**Figure 3.**Behavior of AM population densities. In all cases; $m=14.5$, ${r}_{1}=0.1$, ${r}_{2}=0.2$, ${r}_{3}=0.3$, ${\beta}_{1}=1*{10}^{-3}$, ${\beta}_{2}=1*{10}^{-4}$, ${\beta}_{3}=1*{10}^{-5}$, ${\gamma}_{1}=1*{10}^{-4}$, ${\gamma}_{2}=1*{10}^{-5}$, ${\gamma}_{3}=1*{10}^{-6}$, and the initial conditions are the same, except (

**a**) ${k}_{1}<{k}_{2}<{k}_{3}$ and $r=0.2$ (poor water quality), (

**b**) ${k}_{1}<{k}_{3}<{k}_{2}$ and $r=2.5$ (regular water quality), and (

**c**) ${k}_{3}<{k}_{2}<{k}_{1}$ and $r=10.5$ (good water quality).

**Figure 4.**Population growth curves of observed vs. simulated data. The dotted lines do not consider the influence of the DO, while the continuous line curves do. Data from [53].

Parameter | Description | Value | Reference |
---|---|---|---|

r | oxygenation/aeration rate | 0–14.5 mgL^{−1} | Galic et al. [23] |

m | oxygen saturation constant ^{a} | 0 < m ≤ 14.5 mgL^{−1} | |

β_{i} | average respiration rate of AMs ^{a} | 0–0.918 | Galic et al. [23] |

r_{i} | average reproduction rate of AMs ^{a} | 0–1.02 | Galic et al. [23] |

k_{i} | average carrying capacity of AMs | varies | |

γ_{i} | average DO utilization rate | varies |

^{a}—Assuming an oxygen saturation response for each of the rates (see Figure A1 in Appendix B).

**Table 2.**Relationships between the cardinalities of different AM classes, meaning, and classification of water quality.

Relation | Meaning | Water Quality |
---|---|---|

x_{3} < x_{2} < x_{1} | Low pollution with a tendency to increase | good |

x_{3} < x_{1} < x_{2} | Regular pollution with a tendency to decrease | moderate |

x_{2} < x_{3} < x_{1} | Low pollution with a tendency to increase | good |

x_{2} < x_{1} < x_{3} | High pollution with a tendency to decrease | poor |

x_{1} < x_{3} < x_{2} | Regular pollution with a tendency to increase | moderate |

x_{1} < x_{2} < x_{3} | High pollution with a tendency to decrease | poor |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pineda-Pineda, J.J.; Muñoz-Rojas, J.; Morales-García, Y.E.; Hernández-Gómez, J.C.; Sigarreta, J.M.
Biomathematical Model for Water Quality Assessment: Macroinvertebrate Population Dynamics and Dissolved Oxygen. *Water* **2022**, *14*, 2902.
https://doi.org/10.3390/w14182902

**AMA Style**

Pineda-Pineda JJ, Muñoz-Rojas J, Morales-García YE, Hernández-Gómez JC, Sigarreta JM.
Biomathematical Model for Water Quality Assessment: Macroinvertebrate Population Dynamics and Dissolved Oxygen. *Water*. 2022; 14(18):2902.
https://doi.org/10.3390/w14182902

**Chicago/Turabian Style**

Pineda-Pineda, Jair J., Jesús Muñoz-Rojas, Y. Elizabeth Morales-García, Juan C. Hernández-Gómez, and José M. Sigarreta.
2022. "Biomathematical Model for Water Quality Assessment: Macroinvertebrate Population Dynamics and Dissolved Oxygen" *Water* 14, no. 18: 2902.
https://doi.org/10.3390/w14182902