# Autonomous Population Regulation Using a Multi-Agent System in a Prey–Predator Model That Integrates Cellular Automata and the African Buffalo Optimization Metaheuristic

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Prey–Predator

#### 2.2. Cellular Automata

#### Cellular Automata with Prey–Predator

#### 2.3. Agent System

#### Agent System Applied to Prey–Predator Model

#### 2.4. Purpose of the Investigation

- Use a multi-agent system for the development of the dynamic prey–predator spatial model, to maintain a balance of prey and predators in its environment.
- For the dynamic prey–predator spatial model, perform five experiments for each type of approach, with and without multi-agents.

## 3. Cellular Automata

- Cells live on a grid of a finite number of dimensions n. Generally, the grid is called a lattice.
- Each cell in the lattice has a state. The number of state possibilities is typically finite. For example, a cell can have two or three options: 1 and 0; a, b, or c; or prey or predator.
- Each cell in the lattice has a neighborhood. The neighborhood can be defined in different forms, but it is typically a list of adjacent cells.

#### Formal Definition

- $d\in {\mathbb{Z}}^{+}$, is the d-dimension of the euclidean lattice L$\subseteq {\mathbb{Z}}^{d}$.
- $\mathbb{Q}$ is a finite set of states.
- N is a d-dimensional neighborhood vector $N=({n}_{1},{n}_{2},\dots ,{n}_{m})$, where each ${n}_{i}\in {\mathbb{Z}}^{d}$ and ${n}_{i}\ne {n}_{j}$.
- $f:\mathbb{Q}\to \mathbb{Q}$ is a local transition function that specifies the new state of a cell, taking into account the states of its neighbors.

## 4. African Buffalo Optimization

- The warning sound “waaa” indicates the presence of hazards or lack of good grazing fields. It also enables buffaloes to be able to explore other places that may be beneficial to the herd.
- The “maaa” warning sound is used to indicate that a grazing area has a benefit for the herd. It is also an indication that animals continue to take advantage of available resources.

#### African Buffalo Optimization Algorithm

Algorithm 1: African buffalo optimization algorithm. |

- A number of N buffaloes, in which each buffalo m will represent a solution.
- The Index k for the buffaloes, with k in $\{1,\dots ,N\}$.
- The lambda $\lambda $ value with a domain in $[-1,1]$, excluding the zero.
- The learning factors $lp1$ and $lp2$ with a real number domain in $[0,1]$.

## 5. Dynamic Prey–Predator Spatial Model via African Buffalo Optimization

#### 5.1. Lattice Definition

#### 5.2. Cellular Automata Definition

#### 5.3. Season Definition

#### 5.3.1. Intraspecific Competition

Algorithm 2: Intraspecific competition. |

Algorithm 3: Calculate $bgmax$. |

#### 5.3.2. Migration

Algorithm 4: Migration. |

#### 5.4. Reproduction of Predators

Algorithm 5: Reproduction of predators. |

#### 5.4.1. Death of Predators

Algorithm 6: Death of predators. |

#### 5.4.2. Predation

Algorithm 7: Predation. |

#### 5.4.3. Reproduction of Preys

Algorithm 8: Reproduction of prey. |

#### 5.5. Multi-Agent System

#### 5.5.1. Supervising Agent

#### 5.5.2. Regulating Agent

#### 5.5.3. Prey–Predator Module

## 6. Experiments and Discussion

#### Results

- For the parameters $lp1$ and $lp2$ equal to 1, the number of final prey was 80 and the number of predators was 1332 (see Table 4).
- For the parameters $lp1$ and $lp2$ equal to 2, the number of final prey was 93 and the number of predators was 1322 (see Table 5).
- For the parameters $lp1$ and $lp2$ equal to 3, the number of final prey was 21 and the number of predators was 1364 (see Table 6).
- For the parameters $lp1$ and $lp2$ equal to 4, the number of final prey was 115 and the number of predators was 1315 (see Table 7).
- For the parameters $lp1$ and $lp2$ equal to 5, the number of final prey was 505 and the number of predators was 1343 (see Table 8).

- For the parameters $lp1$ and $lp2$ equal to 1, the number of final prey was 1492 and the number of predators was 871 (see Table 9).
- For the parameters $lp1$ and $lp2$ equal to 2, the number of final prey was 1489 and the number of predators was 878 (see Table 10).
- For the parameters $lp1$ and $lp2$ equal to 3, the number of final prey was 1491 and the number of predators was 876 (see Table 11).
- For the parameters $lp1$ and $lp2$ equal to 4, the number of final prey was 1494 and the number of predators was 876 (see Table 12).
- For the parameters $lp1$ and $lp2$ equal to 5, the number of final prey was 1494 and the number of predators was 875 (see Table 13).

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ABO | African Buffalo Optimization |

MAS | Multi-Agent System |

GIS | Geographic Information System |

CA | Cellular Automaton |

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Parameter | Abbreviation | Range | |
---|---|---|---|

Prey | Initial density | Y | ${\mathbb{N}}^{+}$ |

(glass) | Reproductive capacity | ${y}_{b}$ | ${\mathbb{N}}^{+}$ |

Radius reproduction neighborhood | ${r}_{y}$ | ${\mathbb{N}}^{+}$ | |

Intraspecific competition coefficient | ${p}_{\alpha}$ | $[0,1]$ | |

Radius competition neighborhood | ${r}_{c}$ | ${\mathbb{N}}^{+}$ | |

Number of prey in the neighborhood | ${y}_{c}$ | ${\mathbb{N}}^{+}$ | |

Predator | ABO—Initial density | N | ${\mathbb{N}}^{+}$ |

(buffaloes) | Reproductive capacity | ${z}_{b}$ | ${\mathbb{N}}^{+}$ |

Radius reproduction neighborhood | ${r}_{z}$ | ${\mathbb{N}}^{+}$ | |

Radius update $bgmax$ | ${r}_{bgmax}$ | ${\mathbb{N}}^{+}$ | |

ABO—Learning factor $lp1$ | $lp1$ | ${\mathbb{Z}}^{+}$ | |

ABO—Learning factor $lp2$ | $lp2$ | ${\mathbb{Z}}^{+}$ | |

ABO—Lambda | $\lambda $ | $[-1,0[$ and $]0,1]$ |

Prey | Value | Predator | Value |
---|---|---|---|

Initial density | 1000 | Initial density | 5 |

Reproductive capacity | 1 | Reproductive capacity | 3 |

Radius reproduction neighborhood | 3 | Radius reproduction neighborhood | 1 |

Intraspecific competition coefficient | 0.05 | Radius update $bgmax$ | 1 |

Radius competition neighborhood | 3 |

Parameter | Case A | Case B | Case C | Case D | Case E |
---|---|---|---|---|---|

$lp1$ | 1 | 2 | 3 | 4 | 5 |

$lp2$ | 1 | 2 | 3 | 4 | 5 |

$lambda$ | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 674 | 5 |

2 | 1 | 2 | Migration | 1 | 674 | 5 |

3 | 1 | 3 | Migration | 2 | 674 | 5 |

4 | 1 | 4 | Migration | 3 | 674 | 5 |

5 | 1 | 5 | Migration | 4 | 674 | 5 |

6 | 1 | 6 | Migration | 5 | 674 | 5 |

7 | 1 | 7 | Reproduction of Predators | - | 674 | 986 |

8 | 1 | 8 | Death of Predators | - | 674 | 569 |

9 | 1 | 9 | Predation | - | 412 | 569 |

10 | 1 | 10 | Reproduction of Preys | - | 873 | 569 |

11 | 50 | 1 | Intraspecific Competition | - | 72 | 1327 |

12 | 50 | 2 | Migration | 1 | 72 | 1327 |

13 | 50 | 3 | Migration | 2 | 72 | 1327 |

14 | 50 | 4 | Migration | 3 | 72 | 1327 |

15 | 50 | 5 | Migration | 4 | 72 | 1327 |

16 | 50 | 6 | Migration | 5 | 72 | 1327 |

17 | 50 | 7 | Reproduction of Predators | - | 72 | 2043 |

18 | 50 | 8 | Death of Predators | - | 72 | 1332 |

19 | 50 | 9 | Predation | - | 72 | 1332 |

20 | 50 | 10 | Reproduction of Preys | - | 80 | 1332 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 94 | 1339 |

2 | 1 | 2 | Migration | 1 | 94 | 1339 |

3 | 1 | 3 | Migration | 2 | 94 | 1339 |

4 | 1 | 4 | Migration | 3 | 94 | 2088 |

5 | 1 | 5 | Migration | 4 | 94 | 1343 |

6 | 1 | 6 | Migration | 5 | 87 | 1343 |

7 | 1 | 7 | Reproduction of Predators | - | 105 | 1343 |

8 | 1 | 8 | Death of Predators | - | 87 | 1343 |

9 | 1 | 9 | Predation | - | 87 | 1343 |

10 | 1 | 10 | Reproduction of Preys | - | 87 | 1343 |

11 | 50 | 1 | Intraspecific Competition | - | 84 | 1323 |

12 | 50 | 2 | Migration | 1 | 84 | 1323 |

13 | 50 | 3 | Migration | 2 | 84 | 1323 |

14 | 50 | 4 | Migration | 3 | 84 | 1323 |

15 | 50 | 5 | Migration | 4 | 84 | 1323 |

16 | 50 | 6 | Migration | 5 | 84 | 1323 |

17 | 50 | 7 | Reproduction of Predators | - | 84 | 2031 |

18 | 50 | 8 | Death of Predators | - | 84 | 1322 |

19 | 50 | 9 | Predation | - | 83 | 1322 |

20 | 50 | 10 | Reproduction of Preys | - | 93 | 1322 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 671 | 5 |

2 | 1 | 2 | Migration | 1 | 671 | 5 |

3 | 1 | 3 | Migration | 2 | 671 | 5 |

4 | 1 | 4 | Migration | 3 | 671 | 5 |

5 | 1 | 5 | Migration | 4 | 671 | 5 |

6 | 1 | 6 | Migration | 5 | 671 | 5 |

7 | 1 | 7 | Reproduction of Predators | - | 671 | 1153 |

8 | 1 | 8 | Death of Predators | - | 671 | 666 |

9 | 1 | 9 | Predation | - | 365 | 666 |

10 | 1 | 10 | Reproduction of Preys | - | 766 | 666 |

11 | 50 | 1 | Intraspecific Competition | - | 18 | 1367 |

12 | 50 | 2 | Migration | 1 | 18 | 1367 |

13 | 50 | 3 | Migration | 2 | 18 | 1367 |

14 | 50 | 4 | Migration | 3 | 18 | 1367 |

15 | 50 | 5 | Migration | 4 | 18 | 1367 |

16 | 50 | 6 | Migration | 5 | 18 | 1367 |

17 | 50 | 7 | Reproduction of Predators | - | 18 | 2140 |

18 | 50 | 8 | Death of Predators | - | 18 | 1364 |

19 | 50 | 9 | Predation | - | 18 | 1364 |

20 | 50 | 10 | Reproduction of Preys | - | 21 | 1364 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 673 | 5 |

2 | 1 | 2 | Migration | 1 | 673 | 5 |

3 | 1 | 3 | Migration | 2 | 673 | 5 |

4 | 1 | 4 | Migration | 3 | 673 | 5 |

5 | 1 | 5 | Migration | 4 | 673 | 5 |

6 | 1 | 6 | Migration | 5 | 673 | 5 |

7 | 1 | 7 | Reproduction of Predators | - | 673 | 931 |

8 | 1 | 8 | Death of Predators | - | 673 | 536 |

9 | 1 | 9 | Predation | - | 424 | 536 |

10 | 1 | 10 | Reproduction of Preys | - | 903 | 536 |

11 | 50 | 1 | Intraspecific Competition | - | 102 | 1312 |

12 | 50 | 2 | Migration | 1 | 102 | 1312 |

13 | 50 | 3 | Migration | 2 | 102 | 1312 |

14 | 50 | 4 | Migration | 3 | 102 | 1312 |

15 | 50 | 5 | Migration | 4 | 102 | 1312 |

16 | 50 | 6 | Migration | 5 | 102 | 1312 |

17 | 50 | 7 | Reproduction of Predators | - | 102 | 1962 |

18 | 50 | 8 | Death of Predators | - | 102 | 1315 |

19 | 50 | 9 | Predation | - | 102 | 1315 |

20 | 50 | 10 | Reproduction of Preys | - | 115 | 1315 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 673 | 5 |

2 | 1 | 2 | Migration | 1 | 673 | 5 |

3 | 1 | 3 | Migration | 2 | 673 | 5 |

4 | 1 | 4 | Migration | 3 | 673 | 5 |

5 | 1 | 5 | Migration | 4 | 673 | 5 |

6 | 1 | 6 | Migration | 5 | 673 | 5 |

7 | 1 | 7 | Reproduction of Predators | - | 673 | 1109 |

8 | 1 | 8 | Death of Predators | - | 673 | 638 |

9 | 1 | 9 | Predation | - | 379 | 638 |

10 | 1 | 10 | Reproduction of Preys | - | 794 | 638 |

11 | 50 | 1 | Intraspecific Competition | - | 457 | 1243 |

12 | 50 | 2 | Migration | 1 | 457 | 1243 |

13 | 50 | 3 | Migration | 2 | 457 | 1243 |

14 | 50 | 4 | Migration | 3 | 457 | 1243 |

15 | 50 | 5 | Migration | 4 | 457 | 1243 |

16 | 50 | 6 | Migration | 5 | 457 | 1243 |

17 | 50 | 7 | Reproduction of Predators | - | 457 | 1714 |

18 | 50 | 8 | Death of Predators | - | 457 | 1243 |

19 | 50 | 9 | Predation | - | 455 | 1243 |

20 | 50 | 10 | Reproduction of Preys | - | 505 | 1243 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 672 | 5 |

2 | 1 | 2 | Migration | 1 | 672 | 5 |

3 | 1 | 3 | Migration | 2 | 672 | 5 |

4 | 1 | 4 | Migration | 3 | 672 | 5 |

5 | 1 | 5 | Migration | 4 | 672 | 5 |

6 | 1 | 6 | Migration | 5 | 672 | 5 |

7 | 1 | 7 | Reproduction of Predators | - | 672 | 13 |

8 | 1 | 8 | Death of Predators | - | 672 | 8 |

9 | 1 | 9 | Predation | - | 668 | 8 |

10 | 1 | 10 | Reproduction of Preys | - | 1483 | 8 |

11 | 50 | 1 | Intraspecific Competition | - | 1291 | 874 |

12 | 50 | 2 | Migration | 1 | 1291 | 874 |

13 | 50 | 3 | Migration | 2 | 1291 | 874 |

14 | 50 | 4 | Migration | 3 | 1291 | 874 |

15 | 50 | 5 | Migration | 4 | 1291 | 874 |

16 | 50 | 6 | Migration | 5 | 1291 | 874 |

17 | 50 | 7 | Reproduction of Predators | - | 1291 | 1402 |

18 | 50 | 8 | Death of Predators | - | 1291 | 871 |

19 | 50 | 9 | Predation | - | 875 | 871 |

20 | 50 | 10 | Reproduction of Preys | - | 1492 | 871 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 669 | 5 |

2 | 1 | 2 | Migration | 1 | 669 | 5 |

3 | 1 | 3 | Migration | 2 | 669 | 5 |

4 | 1 | 4 | Migration | 3 | 669 | 5 |

5 | 1 | 5 | Migration | 4 | 669 | 5 |

6 | 1 | 6 | Migration | 5 | 669 | 5 |

7 | 1 | 7 | Reproduction of Predators | - | 669 | 15 |

8 | 1 | 8 | Death of Predators | - | 669 | 8 |

9 | 1 | 9 | Predation | - | 666 | 8 |

10 | 1 | 10 | Reproduction of Preys | - | 1483 | 8 |

11 | 50 | 1 | Intraspecific Competition | - | 1294 | 874 |

12 | 50 | 2 | Migration | 1 | 1294 | 874 |

13 | 50 | 3 | Migration | 2 | 1294 | 874 |

14 | 50 | 4 | Migration | 3 | 1294 | 874 |

15 | 50 | 5 | Migration | 4 | 1294 | 874 |

16 | 50 | 6 | Migration | 5 | 1294 | 874 |

17 | 50 | 7 | Reproduction of Predators | - | 1294 | 1409 |

18 | 50 | 8 | Death of Predators | - | 1294 | 878 |

19 | 50 | 9 | Predation | - | 869 | 878 |

20 | 50 | 10 | Reproduction of Preys | - | 1489 | 878 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 674 | 5 |

2 | 1 | 2 | Migration | 1 | 674 | 5 |

3 | 1 | 3 | Migration | 2 | 674 | 5 |

4 | 1 | 4 | Migration | 3 | 674 | 5 |

5 | 1 | 5 | Migration | 4 | 674 | 5 |

6 | 1 | 6 | Migration | 5 | 674 | 5 |

7 | 1 | 7 | Reproduction of Predators | - | 674 | 13 |

8 | 1 | 8 | Death of Predators | - | 674 | 8 |

9 | 1 | 9 | Predation | - | 670 | 8 |

10 | 1 | 10 | Reproduction of Preys | - | 1476 | 8 |

11 | 50 | 1 | Intraspecific Competition | - | 1296 | 870 |

12 | 50 | 2 | Migration | 1 | 1296 | 870 |

13 | 50 | 3 | Migration | 2 | 1296 | 870 |

14 | 50 | 4 | Migration | 3 | 1296 | 870 |

15 | 50 | 5 | Migration | 4 | 1296 | 870 |

16 | 50 | 6 | Migration | 5 | 1296 | 870 |

17 | 50 | 7 | Reproduction of Predators | - | 1296 | 1404 |

18 | 50 | 8 | Death of Predators | - | 1296 | 876 |

19 | 50 | 9 | Predation | - | 872 | 876 |

20 | 50 | 10 | Reproduction of Preys | - | 1491 | 876 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 673 | 5 |

2 | 1 | 2 | Migration | 1 | 673 | 5 |

3 | 1 | 3 | Migration | 2 | 673 | 5 |

4 | 1 | 4 | Migration | 3 | 673 | 5 |

5 | 1 | 5 | Migration | 4 | 673 | 5 |

6 | 1 | 6 | Migration | 5 | 673 | 5 |

7 | 1 | 7 | Reproduction of Predators | - | 673 | 13 |

8 | 1 | 8 | Death of Predators | - | 673 | 8 |

9 | 1 | 9 | Predation | - | 670 | 8 |

10 | 1 | 10 | Reproduction of Preys | - | 1481 | 8 |

11 | 50 | 1 | Intraspecific Competition | - | 1299 | 874 |

12 | 50 | 2 | Migration | 1 | 1299 | 874 |

13 | 50 | 3 | Migration | 2 | 1299 | 874 |

14 | 50 | 4 | Migration | 3 | 1299 | 874 |

15 | 50 | 5 | Migration | 4 | 1299 | 874 |

16 | 50 | 6 | Migration | 5 | 1299 | 874 |

17 | 50 | 7 | Reproduction of Predators | - | 1299 | 1405 |

18 | 50 | 8 | Death of Predators | - | 1299 | 876 |

19 | 50 | 9 | Predation | - | 875 | 876 |

20 | 50 | 10 | Reproduction of Preys | - | 1494 | 876 |

ID | ID-S | ID-SS | Name-SS | ID-M | N-Prey | N-Predator |
---|---|---|---|---|---|---|

1 | 1 | 1 | Intraspecific Competition | - | 664 | 5 |

2 | 1 | 2 | Migration | 1 | 664 | 5 |

3 | 1 | 3 | Migration | 2 | 664 | 5 |

4 | 1 | 4 | Migration | 3 | 664 | 5 |

5 | 1 | 5 | Migration | 4 | 664 | 5 |

6 | 1 | 6 | Migration | 5 | 664 | 5 |

7 | 1 | 7 | Reproduction of Predators | - | 664 | 15 |

8 | 1 | 8 | Death of Predators | - | 664 | 8 |

9 | 1 | 9 | Predation | - | 661 | 8 |

10 | 1 | 10 | Reproduction of Preys | - | 1483 | 8 |

11 | 50 | 1 | Intraspecific Competition | - | 1307 | 868 |

12 | 50 | 2 | Migration | 1 | 1307 | 868 |

13 | 50 | 3 | Migration | 2 | 1307 | 868 |

14 | 50 | 4 | Migration | 3 | 1307 | 868 |

15 | 50 | 5 | Migration | 4 | 1307 | 868 |

16 | 50 | 6 | Migration | 5 | 1307 | 868 |

17 | 50 | 7 | Reproduction of Predators | - | 1307 | 1400 |

18 | 50 | 8 | Death of Predators | - | 1307 | 875 |

19 | 50 | 9 | Predation | - | 880 | 875 |

20 | 50 | 10 | Reproduction of Preys | - | 1494 | 875 |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Almonacid, B.; Aspée, F.; Yimes, F.
Autonomous Population Regulation Using a Multi-Agent System in a Prey–Predator Model That Integrates Cellular Automata and the African Buffalo Optimization Metaheuristic. *Algorithms* **2019**, *12*, 59.
https://doi.org/10.3390/a12030059

**AMA Style**

Almonacid B, Aspée F, Yimes F.
Autonomous Population Regulation Using a Multi-Agent System in a Prey–Predator Model That Integrates Cellular Automata and the African Buffalo Optimization Metaheuristic. *Algorithms*. 2019; 12(3):59.
https://doi.org/10.3390/a12030059

**Chicago/Turabian Style**

Almonacid, Boris, Fabián Aspée, and Francisco Yimes.
2019. "Autonomous Population Regulation Using a Multi-Agent System in a Prey–Predator Model That Integrates Cellular Automata and the African Buffalo Optimization Metaheuristic" *Algorithms* 12, no. 3: 59.
https://doi.org/10.3390/a12030059