# A Locust-Inspired Model of Collective Marching on Rings

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## Abstract

**:**

## 1. Introduction

#### Contribution

## 2. Related Work

## 3. Model and Definitions

**conflict**between ${A}_{i}$ and ${A}_{j}$. A conflict is “won” by the locust that successfully converts the other locust to their heading.

- At the beginning of time t, A and ${A}^{\to}$ are not adjacent to each other and $b\left(A\right)\ne b\left({A}^{\to}\right)$.
- Once A moves to E, the updated ${A}^{\leftarrow}$ and ${A}^{\to}$ in the new track will have heading $b\left(A\right)$.
- No locust will attempt to move horizontally to E at time $t+1$.

## 4. Stabilization Analysis

#### 4.1. Locusts on Narrow Ringlike Arenas ($k=1$)

**Theorem**

**1.**

**Definition**

**1.**

**segment**of the locusts ${B}_{0},\dots {B}_{q-1}$ at time t. The locust ${B}_{q-1}$ is called the

**segment head**, and A is called the

**segment tail**of this segment.

**Definition**

**2.**

**winning segment**at time t and is denoted $SW\left(t\right)$. The head of $SW\left(t\right)$ is labelled ${H}_{W}\left(t\right)$. For convenience, if at time ${t}_{0}$ the swarm is stable (i.e., ${t}_{0}\ge {T}_{stable}$), then we define $SW\left({t}_{0}\right)$ as the set that contains all m locusts.

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**blocked**. A location ${E}_{i}$ becomes

**unblocked**at time $t+1$ if all empty locations ${E}_{j}$ such that $j<i$ are unblocked at time t, and a locust from $SW\left(t\right)$ swapped locations with ${E}_{i}$ at time t. Once a location becomes

**unblocked**, it remains that way forever.

**Lemma**

**3.**

- 1.
- Every blocked empty location E is outside $SW\left({t}^{*}\right)$ (if any exist)
- 2.
- At least ${t}^{*}$ empty locations are unblocked.

**Proof.**

**Lemma**

**4.**

**Proof.**

**Proof.**

**Proof.**

#### 4.2. Locusts on Wide Ringlike Arenas ($k>1$)

**Theorem**

**2.**

**Definition**

**5.**

**Definition**

**6.**

**compact**if ${X}_{i+1}={X}_{i}^{\to}$ and either:

- 1.
- every locust in X has a clockwise heading and for every $i<\left|X\right|$, $dis{t}^{c}({X}_{i},{X}_{i+1})\le 2$, or
- 2.
- every locust in X has a counterclockwise heading and for every $i<\left|X\right|$, $dis{t}^{cc}({X}_{i},{X}_{i+1})\le 2$.

**Definition**

**7.**

**in deadlock**if $dis{t}^{c}({X}_{j},{Y}_{k})=1$. (See Figure 4).

**Definition**

**8.**

**Observation**

**1.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Lemma**

**8.**

**all**k walks end is $\mathcal{O}\left({n}^{2}\mathrm{log}\left(k\right)\right)$.

**Proof.**

**Theorem**

**4.**

**Proof.**

#### Erratic Track Switching and Global Consensus

- With probability r, a locust might behave erratically in the horizontal phase, staying in place instead of attempting to move according to its heading.
- With probability p, a locust may behave erratically in the vertical phase, meaning that even if the vertical movement conditions (1)–(3) of the model (see Section 3) are not fulfilled, the locust attempts to move vertically to an adjacent empty space on the track above or below them (if such empty space exists).

**Theorem**

**5.**

**Proof.**

## 5. Simulation and Empirical Evaluation

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Image from locust experiments, courtesy of Amir Ayali. The collective clockwise marching of locusts in a ring arena is shown. Locusts were initiated at random positions and orientations in the arena but converged to clockwise marching over time.

**Figure 2.**One step of the locust model with $k=3$, $n=8$ split into horizontal and vertical movements: (

**a**) shows the initial configuration at the beginning of the current time step t; (

**b**) illustrates changes to the configuration after conflicts and horizontal movements; and (

**c**) is the configuration at the beginning of time $t+1$ (or equivalently the end of time t) after vertical movements. The front and back of the blue locust are the red and green locusts, respectively. The purple locusts conflict with each other. Since conditions (1)–(3) are fulfilled, the blue locust may switch tracks, and it does so in the illustration.

**Figure 3.**A locust configuration with $n=8,k=3$. Locusts are colored based based on the segment they belong to (Definition 1). There are 8 segments in total.

**Figure 5.**A partition into maximal compact subsets as in our construction. In this configuration, ${L}_{1}\left(t\right)=3$, ${L}_{2}\left(t\right)=3$, ${L}_{3}\left(t\right)=1$, and $L\left(t\right)=7$. Note that although ${\mathcal{C}}_{1},{\mathcal{C}}_{2}$ are compact, $P\left(t\right)={\mathcal{C}}_{1}\cup {\mathcal{C}}_{2}$ is not compact, and similarly $Q\left(t\right)$ is not compact; thus $P\left(t\right)$ and $Q\left(t\right)$ are not in deadlock, and $L\left(t\right)\ne 1$.

**Figure 6.**Simulations of the locust model. The y axis is ${T}_{stable}$. Column (

**a**) measures ${T}_{stable}$ for $k=1\dots 30$, with n fixed at 30. Column (

**b**) measures ${T}_{stable}$ for $n=1\dots 60$, with k fixed at 5. Column (

**c**) measures ${T}_{stable}$ with $n=30,k=5$, and p (the probability of erratic behavior) going from 0 to 1. The top row measures ${T}_{stable}$ in sparse locust configurations ($m\approx 0.1n$), while the bottom row does so for dense configurations ($m\approx 0.5n$). The dashed red line estimates ${T}_{stable}$ when locusts never switch tracks (except while behaving erratically in column (

**c**)); the blue line estimates ${T}_{stable}$ when locusts switch tracks as often as the model rules allow. Error bars show the standard deviations.

**Figure 7.**Simulations of the locust model fixing $k=1$ and letting n run from 1 to 100. The y axis is ${T}_{stable}$. The orange line denotes dense locust configurations ($m\approx 0.5n$), and the green line denotes sparse configurations $(m\approx 0.1n)$. Error bars show the standard deviations.

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Amir, M.; Agmon, N.; Bruckstein, A.M. A Locust-Inspired Model of Collective Marching on Rings. *Entropy* **2022**, *24*, 918.
https://doi.org/10.3390/e24070918

**AMA Style**

Amir M, Agmon N, Bruckstein AM. A Locust-Inspired Model of Collective Marching on Rings. *Entropy*. 2022; 24(7):918.
https://doi.org/10.3390/e24070918

**Chicago/Turabian Style**

Amir, Michael, Noa Agmon, and Alfred M. Bruckstein. 2022. "A Locust-Inspired Model of Collective Marching on Rings" *Entropy* 24, no. 7: 918.
https://doi.org/10.3390/e24070918