# Numerical Analysis of the Transfer Dynamics of Heavy Metals from Soil to Plant and Application to Contamination of Honey

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Development

_{n}+ nH

^{+}⟷ Me

^{n+}+ nH

_{2}O,

^{2}),

^{n+}in plant (mg/kg),

^{n+}in the soil (mg/L),

^{2}·year)). The parameter $p\left(t\right)$ signifies the amount of water accessible to plant roots, measured in millimeters (mm). The coefficients $\alpha $, $\beta $, and $\phi $ correspond to the rates of absorption (in liters per kilogram per year, L/(kg·year)), leakage (per year, 1/year), and reaction (per year, 1/year), respectively.

^{n+}within the plants.

_{3}+ 3H

^{+}⟷ Al

^{3+}+ 3H

_{2}O.

^{2}/kg) are fixed constants. Moreover, a bimodal growth function is proposed in [6] as

## 3. Mathematical Analysis

**Theorem**

**1.**

**Proof.**

- ${\mu}^{\prime}\left(S\right)=0$ when $d=f$, where $d:=\frac{c}{e}$,
- ${\mu}^{\prime}\left(S\right)<0$ when $d<f$,
- ${\mu}^{\prime}\left(S\right)>0$ when $d>f$.

## 4. Parameter Identification

#### 4.1. Formulation of the Inverse Problem

#### 4.2. Numerical Solution to the Inverse Problem

## 5. Computational Simulations

^{2}and an initial concentration of aluminium ions in the plant ${S}^{0}=0$ mg/kg. The concentration of Al${}^{3+}$ ions in the soil is considered to be ${A}^{0}=615$ mg/L, while the concentration of H${}^{+}$ ions in the soil solution is set at ${H}^{0}=0.001$ mg/L.

^{2}·year). The available water for roots is defined by a periodic function that reflects seasonal variation, expressed as $p\left(t\right)=7cos\left(3t\pi \right)+10$ mm.

^{2}/kg for the biomass coefficient, $c=25$ year${}^{-1}$ for the metabolic rate, $e=500$ mg/kg as the critical concentration threshold, and $f={10}^{-12}$ year${}^{-1}$ for the inefficiency factor. These parameters are crucial in modeling the dynamics of plant growth and their interaction with the heavy metal contaminants in the soil, providing a detailed simulation of the environmental processes at play.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Reconstructed parameters $W\left(t\right)$ and $p\left(t\right)$ with perturbed measurements.

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**MDPI and ACS Style**

Atanasov, A.; Georgiev, S.; Vulkov, L.
Numerical Analysis of the Transfer Dynamics of Heavy Metals from Soil to Plant and Application to Contamination of Honey. *Symmetry* **2024**, *16*, 110.
https://doi.org/10.3390/sym16010110

**AMA Style**

Atanasov A, Georgiev S, Vulkov L.
Numerical Analysis of the Transfer Dynamics of Heavy Metals from Soil to Plant and Application to Contamination of Honey. *Symmetry*. 2024; 16(1):110.
https://doi.org/10.3390/sym16010110

**Chicago/Turabian Style**

Atanasov, Atanas, Slavi Georgiev, and Lubin Vulkov.
2024. "Numerical Analysis of the Transfer Dynamics of Heavy Metals from Soil to Plant and Application to Contamination of Honey" *Symmetry* 16, no. 1: 110.
https://doi.org/10.3390/sym16010110