# Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Maximum Reduction of Predatory Performance: $\alpha =0$ and ${\alpha}^{\prime}=0$

^{3}) versus time (months)—where we observe two patterns: 1—the growth of the prey population at a rate ${\beta}_{1}\left(t\right)$ accompanied by the extinction of predators and 2—the extinction of both species.

#### 3.2. Reduction of Predatory Performance with No Prey Eaten by Predator: α = 0

^{3}) versus time (months)—where we observe the same pattern, characterized by the growth of the prey population at a rate ${\beta}_{1}\left(t\right)$ accompanied by the predator “population explosion”.

#### 3.3. Reduction of Predatory Performance with No Increase in the Number of Predators Due to the Feeding of Prey: α′ = 0

^{3}) versus time (months)—where we observe the same two patterns of the Section 3.1, where $\alpha ={\alpha}^{\prime}=0$.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MP | Microplastic |

LVM | Lotka–Volterra model |

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**Figure 1.**Short-term population dynamics of the predator–prey for $\alpha ={\alpha}^{\prime}=0$. Predator and prey have the same response strength to MP particles, i.e., $\Delta ={r}_{11}/{r}_{21}=1.0$. Abscissa measures time in months, and ordinate measures the number of organisms (No./m

^{3}). The populations of prey (${x}_{1}$) and predator (${x}_{2}$) are represented by the blue dotted line and the orange continuous one, respectively.

**Figure 2.**Short-term population dynamics of the predator–prey for $\alpha =0$. Predator and prey have the same response strength to MP particles, ${r}_{11}={r}_{21}=0.1$, i.e., $\Delta ={r}_{11}/{r}_{21}=1.0$. Abscissa measures time in months and ordinate measures the number of organisms (No./m

^{3}). The populations of prey (${x}_{1}$) and predator (${x}_{2}$) are represented by the blue dotted line and the orange continuous one, respectively.

**Figure 3.**Short-term population dynamics of the predator–prey for ${\alpha}^{\prime}=0$. Predator and prey have the same response strength to MP particles, i.e., $\Delta ={r}_{11}/{r}_{21}=1.0$. Abscissa measures time in months and ordinate measures the number of organisms (No./m

^{3}). The populations of prey (${x}_{1}$) and predator (${x}_{2}$) are represented by the blue dotted line and the orange continuous one, respectively.

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**MDPI and ACS Style**

Dos Santos, L.S.; Alcarás, J.R.; Da Costa, L.M.; Simões, M.M.R.; Martinez, A.S.
Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients. *Math. Comput. Appl.* **2022**, *27*, 66.
https://doi.org/10.3390/mca27040066

**AMA Style**

Dos Santos LS, Alcarás JR, Da Costa LM, Simões MMR, Martinez AS.
Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients. *Mathematical and Computational Applications*. 2022; 27(4):66.
https://doi.org/10.3390/mca27040066

**Chicago/Turabian Style**

Dos Santos, Lindomar Soares, José Renato Alcarás, Lucas Murilo Da Costa, Mateus Mendonça Ramos Simões, and Alexandre Souto Martinez.
2022. "Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients" *Mathematical and Computational Applications* 27, no. 4: 66.
https://doi.org/10.3390/mca27040066