# 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$${x}_{i}^{*}={\displaystyle \frac{{k}_{i}}{m{r}_{i}}}{o}^{*}\left(\right)open="("\; close=")">{r}_{i}+{\gamma}_{i}(m-{o}^{*})$$

**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].

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**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 |

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**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