# A reaction telegraph model reveals synergy between motility strategies in Myxococcus xanthus predation

Maxime Estavoyer

<figure><img src="image_1.png" /></figure>

## To cite this version:

Maxime Estavoyer. A reaction telegraph model reveals synergy between motility strategies in Myxococcus xanthus predation. 2025. hal-04699047v3

HAL Id: hal-04699047

https://inria.hal.science/hal-04699047v3

Preprint submitted on 31 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# A reaction telegraph model reveals synergy between motility strategies in *Myxococcus xanthus* predation

Maxime Estavoyer¹

¹ Aix Marseille Univ, Université de Toulon, CNRS, CPT (UMR 7332),

## Abstract

The predatory bacterium *Myxococcus xanthus* can invade prey bacteria using two distinct motility apparatuses. It is commonly acknowledged that adventurous motility is used for isolated bacteria, while social motility corresponds to bacterial clusters. Inspired by recent biological findings, we propose a simple model of predatory invasion focusing on the co-occurrence of these two mechanisms and their possible synergistic effects. At microscopic scale, cell motion is persistent; therefore, we opt for a transport-reaction model, extending previous reaction-diffusion models. Another specificity is the structuration of the bacterial population into clusters with varying speeds and persistence times. In the linear regime, we find a transition from normal speed to anomalous speed, consistent with reaction-diffusion theory but with specificities due to the hyperbolic nature of the model. For the nonlinear regime, we numerically observe and study the existence of transitions between pulled and pushed fronts. Finally, we reproduced biological experiments with mutants lacking each of the motility apparatuses based on relevant modifications of the model. Moreover, we propose a rational basis for the reported synergistic effects. Our work paves the way for a better understanding of the complex waves of bacterial population advance, which are precursors to biofilm formation.
# Contents

<table><tr><td>1 Introduction</td><td>2</td></tr><tr><td>2 Modeling</td><td>4</td></tr><tr><td>3 Results for the simplified case n = 2</td><td>9</td></tr><tr><td>3.1 Linear speeds</td><td>9</td></tr><tr><td>3.2 Nonlinear speed - Pulled and Pushed regime</td><td>10</td></tr><tr><td>3.3 Optimal invasion strategy</td><td>12</td></tr><tr><td>4 Results for the full model n > 2</td><td>15</td></tr><tr><td>4.1 Linear speed</td><td>15</td></tr><tr><td>4.2 Nonlinear speed - Pulled and Pushed regime</td><td>15</td></tr><tr><td>4.3 Biological modeling</td><td>16</td></tr><tr><td>4.3.1 Propagation example</td><td>16</td></tr><tr><td>4.3.2 Study of strain invasion speed</td><td>17</td></tr><tr><td>4.3.3 Optimal motility strategy for invasion</td><td>18</td></tr><tr><td>5 Conclusion</td><td>20</td></tr><tr><td>A Derivation of the linear propagation speed for n = 2</td><td>24</td></tr><tr><td>A.1 Derivation of normal speeds</td><td>24</td></tr><tr><td>A.2 Derivation of anomalous speeds</td><td>26</td></tr><tr><td>A.3 Numerical simulations</td><td>28</td></tr><tr><td>B Derivation of the linear propagation speed for n > 2</td><td>29</td></tr></table>

## 1 Introduction

**Biological background** *Myxococcus xanthus* is a social bacterium that moves and feeds co-operatively in predatory groups. It uses two motility modes: social motility (S) and adventurous motility (A). Social motility depends on type IV pili, filamentous appendages acting as grappling hooks at the bacterial pole, enabling coordinated movement of bacterial clusters [1]. In contrast, adventurous motility relies on gliding, allowing individual bacteria, mainly at the colony edge, to move independently [2]. These motility systems are regulated by distinct gene sets [3] and exhibit fundamentally different characteristics, detailed in this article.

Beyond motility, *M. xanthus* is a predatory bacterium that attacks various prey species [4, 5]. Prey biomass, rich in amino acids and lipids, provides its primary carbon and energy sources. The bacterium employs two predation strategies depending on prey type [6]. The first, a frontal attack, involves bacterial penetration and prey lysis [7, 8]. The second, the wolf-pack strategy, consists of *M. xanthus* cells surrounding and killing the prey colony [8, 9]. This study focuses on modeling the frontal attack.
Prey destruction occurs primarily through contact-dependent killing. Studies show that isolated *M. xanthus* cells approach prey using adventurous motility, halt upon contact, and induce cell death [10, 11, 12]. Cooperation enhances vegetative growth and predation, facilitating hydrolytic enzyme secretion and improving prey biomass degradation within bacterial clusters, making group predation more efficient than solitary attacks [13]. Moreover, isolated *M. xanthus* cells often leave killed *Escherichia coli* prey without degrading it, likely due to insufficient enzyme production [9, 11].

These rod-shaped bacteria move along their main axis using two motility systems. Occasionally, they reverse direction by switching polarity [14]. Studies suggest that *M. xanthus* is not chemotactic towards its prey [15], and its overall chemotactic behavior remains controversial [16, 17]. Additionally, clustered bacteria tend to move faster than isolated ones [18]. We aim to determine how the coexistence of social and adventurous motility influences the predation speed of *M. xanthus*. This question is important because it may explain why *M. xanthus* has evolved to maintain two distinct motility systems, and under which ecological or dynamical conditions their coexistence provides a selective advantage.

**Mathematical model** To analyze the impact of A and S motility coexistence on predation efficiency, we introduce a mathematical model in Section 2. It describes the spatiotemporal dynamics of bacterial clusters, from isolated bacteria to clusters of size $n$, with $n \in N^*$. The model assumes that isolated bacteria lyse prey but do not consume the resulting biomass, which is later assimilated by clusters, consistent with experimental observations [9, 11]. Polarity switching is modeled using a telegraph operator, which corresponds to one-dimensional run-and-tumble process. Unlike diffusion, this approach captures differences in instantaneous speed and movement persistence between the two motility systems. The reaction terms model coagulation, fragmentation, and proliferation, capturing the biological interactions that govern cluster dynamics.

**Mathematical literature** The model introduced in this article extends previous work [19], which proposed a simplified reaction-diffusion model considering only isolated bacteria and two-bacteria clusters to study the impact of the speed advantage of bacterial clusters. Numerical simulations revealed the presence of pulled and pushed fronts. Further analysis provided a rigorous proof that, under the assumption of fast coagulation-fragmentation, a unique transition occurs between pulled and pushed fronts [20].

Previous models describe the spread of *M. xanthus* [21, 22], but they do not account for its predatory behavior or the coexistence of both motility systems. Here, we analyze *M. xanthus* invasion dynamics through traveling wave solutions. Traveling waves in reaction-telegraph equations have been widely studied (see, for example, [23, 24, 25, 26, 27, 28, 29, 30]) as well as in the broader context of hyperbolic equations [31, 32].

**Propagation speed** To investigate whether the coexistence of both motility systems enhances the predation speed of *M. xanthus*, we begin by characterizing the propagation speed predicted by our model. We consider both the linear speed, which corresponds to the asymptotic spreading velocity predicted by a linear stability analysis around the unstable equilibrium, and the nonlinear speed, which represents the actual invasion speed of the front, influenced by nonlinear effects.
Based on the marginal stability conjecture [33, 34] and under certain assumptions, we derive a formula for the linear speed. Two scenarios arise: either the linear speed equals the linear speed obtained from the subsystem modeling only isolated bacteria, when decoupled from the influence of bacterial clusters; or it exceeds this value, in which case the propagation is termed anomalous [35, 36]. A front is classified as pulled when both speeds are equal; otherwise, it is pushed [37]. We also conduct numerical simulations to determine the nonlinear speed. Our results reveal the existence of both pulled and pushed front regimes.

**Comparison with mutant strain experiments** To evaluate the reliability of our model, we compare its predictions to experimental observations on mutant strains of *M. xanthus*. The bacterium has two mutant strains: $A+S^-$ and $A-S^+$, corresponding to populations composed solely of adventurous or social bacteria, respectively. The *Wildtype* strain (or $A+S^+$) possesses both motility systems and represents the bacterium's natural state. Biological experiments show that the *Wildtype* and $A+S^-$ strains invade *E. coli* prey significantly faster than the $A-S^+$ strain [12]. To account for these different strains, we adapt our initial model to define two submodels, each corresponding to one of the mutant strains. Numerical simulations confirm the experimental results, showing that the $A-S^+$ strain exhibits a significantly reduced invasion capacity compared to the other strains.

**Result on motility coexistence** We then investigate how the coexistence of motility types influences the invasion dynamics. Our results show that in a pulled front regime, the coexistence of adventurous and social motility does not affect invasion speed. However, in a pushed front regime, it significantly accelerates invasion, increasing predation speed. This prediction may therefore provide an explanation for the existence of two distinct motility systems rather than a single one.

**Outline** The paper is organized as follows. Section 2 introduces the system modeling *M. xanthus* invasion dynamics. Section 3 analyzes the simplest case, where a bacterium is either isolated or part of a cluster of two bacteria ($n = 2$), serving as a preliminary step to illustrate the methods used. We first derive the linear propagation speed, then numerically explore the nonlinear speed, concluding with a discussion of the biological implications. Section 4 extends this analysis to the general case $n > 2$, applying similar techniques to study the full model dynamics.

## 2 Modeling

**Diffusion versus telegraph dispersal** Reaction-diffusion equations have long been used to model biological dispersion [38, 39]. However, diffusion has a major limitation in this context: it does not account for motion persistence [40]. In one dimension, and only in one dimension, bacterial movement follows a *Run and Tumble* process, also known as *telegraph dispersal*, based on a velocity jump process [41, 42]. This approach is increasingly employed to model biological systems (see, e.g., [43, 44, 45]), particularly for bacterial motility [46, 47, 48, 49].
In one dimension, during the *Run* phase, a particle moves in a straight line to the right or left at speed *v*. After a randomly distributed exponential time, the particle reverses its direction. This motion is modeled by the following kinetic equations,

$$ egin{cases} \partial_t f^+ + \frac{v}{\epsilon} \partial_x f^+ = \frac{1}{2\tau\epsilon^2} (f^- - f^+), \\ \partial_t f^- - \frac{v}{\epsilon} \partial_x f^- = \frac{1}{2\tau\epsilon^2} (f^+ - f^-), \end{cases} $$

where $f^+$ (resp. $f^-$) represents the density of bacteria moving to the right (resp. left). The parameter $\epsilon$ is a scaling parameter that represents the ratio between the spatial scale and the mean free path of bacteria. The transport term models the *Run* phase, while the RHS corresponds to the *Tumble* phase/Reversal phase.

Making the change of variable $f = f^+ + f^-$ and applying the Kac trick [50], system (2) simplifies into a single equation given by

$$ \tau \epsilon^2 \partial_{tt} f + \partial_t f = \tau v^2 \partial_{xx} f. \qquad (1) $$

For $\epsilon = 0$, equation (1) reduces to the classical heat equation, $\partial_t f = D \partial_{xx} f$, where

$$ D := \tau v^2. \qquad (2) $$

This relation, which links the diffusion coefficient to persistence and instantaneous speed, allows us to compare the impact of differences in persistence and speed between isolated bacteria and bacterial clusters. By assuming the same diffusion coefficient for both, we can isolate how these variations specifically influence the predation speed of *M. xanthus*.

**Introduction of the models** The *M. xanthus* population consists of isolated bacteria and clusters of varying sizes. In our model, we distinguish between two contexts of motility: isolated bacteria are assumed to rely predominantly on adventurous motility, while bacteria in clusters are assumed to rely predominantly on social motility. The cluster size, defined as the total number of bacteria in a cluster, is denoted by $i$. We assume that clusters cannot exceed a maximum size $n \in \mathbb{N}$. The density of clusters of size $i$ moving to the right is denoted by $p_i^+$, while $p_i^-$ represents those moving to the left. The total density of clusters of size $i$ is given by $p_i := p_i^+ + p_i^-$.

In this article, we introduce and study two distinct models. The first model is a general model that accounts for the dynamics of isolated bacteria and clusters of varying sizes ($n > 2$). It serves as the main reference for analyzing propagation dynamics. To illustrate the mathematical tools used in this analysis, we introduce a simpler model by restricting cluster size to $n = 2$, where a bacterium is either isolated or part of a two-bacterium cluster. Since the general model is not directly applicable in this case, we propose an alternative variant that preserves its fundamental principles.
## Formulation of the general model $n > 2$

The densities $p_1^+$ and $p_1^-$ are given by the following system of equations,

$$ \left\{ \begin{aligned} \partial_t p_1^+ + \frac{1}{\epsilon} \partial_x p_1^+ &= \frac{1}{2\epsilon^2} (p_1^- - p_1^+) + \mathcal{F}_1^+ [p_j^+] + \mathcal{C}_1^+ [p_j^+] + \frac{\alpha_1 p_1 (1-p)_+}{2}, \\ \partial_t p_1^- - \frac{1}{\epsilon} \partial_x p_1^- &= \frac{1}{2\epsilon^2} (p_1^+ - p_1^-) + \mathcal{F}_1^- [p_j^+] + \mathcal{C}_1^- [p_j^+] + \frac{\alpha_1 p_1 (1-p)_+}{2}, \end{aligned} \right. \quad (3) $$

where $p$ represents the total number of bacteria, given by the following formula, $p := p_1 + 2p_2 + \dots + np_n$ and $(1-p)_+$ corresponds to the positive part of $1-p$, $(1-p)_+ := \max\{1-p, 0\}$.

To account for the cooperative behavior between isolated bacteria and clusters, we modify the structure of the term modeling bacterial proliferation for clusters of size $i \in \{2, 3, \dots, n\}$. In this case, the equations for the pair $(p_i^+, p_i^-)$ are given by

$$ \partial_t p_i^{\pm} \pm \frac{v(i)}{\epsilon} \partial_x p_i^{\pm} = \pm \frac{1}{2\tau(i)\epsilon^2} (p_i^{-} - p_i^{+}) + \mathcal{F}_i^{\pm} [p_j^{\pm}] + \mathcal{C}_i^{\pm} [p_j^{\pm}] \\ + N [d(i-1)p_{i-1}^{\pm} - d(i)p_i^{\pm}] (1-p)_+, \quad (4) $$

where $N$ is a spatiotemporal variable corresponding to the nutrient density.

**Dispersal term** The function $v: i \in \{2, \dots, n\} \mapsto v(i)$ represents the instantaneous speed of bacterial clusters of size $i$ relatively to the isolated bacteria. Similarly, the function $\tau: i \in \{2, \dots, n\} \mapsto \tau(i)$ represents the average time between a change of direction relatively to the isolated bacteria. Through a time and space rescaling, which we do not detail here, the instantaneous speed and persistence of isolated bacteria are normalized to 1, i.e., $v(1) = 1$ and $\tau(1) = 1$. We assume that isolated bacteria and clusters share the same scaling parameter, denoted by $\epsilon$.

**Fragmentation term** The operator $\mathcal{F}_i^{\pm}[p_j^{\pm}]$ represents the fragmentation events of bacterial clusters of size $i$ moving to the right (+) or to the left (-) between times $t$ and $t+dt$. This operator is given by

$$ \mathcal{F}_i^{\pm}[p_j^{\pm}] = \sum_{j=i+1}^{n} \beta(j)\eta(j, i)p_j - \beta(i)p_i^{\pm}, \quad i \in \{1, \dots, n\}, $$

where $\beta(i)$ represents the fragmentation rate of clusters of size $i$. The first sum represents larger clusters that have fragmented into two distinct parts, with one resulting cluster having size $i$. We assume that when a cluster of bacteria fragments, the two daughter clusters randomly choose their direction, either to the right or to the left. The function $\eta$ corresponds to the fragmentation kernel; in other words, $\eta(j, i)$ represents the probability that a cluster of size $j$ fragments into a cluster of size $i$ and a cluster of size $j-i$. The second term corresponds to clusters of size $i$ that have fragmented during the time interval $dt$. Since no experimental data allow us to determine the biologically most relevant form of the fragmentation operator, we
consider a natural case in which we assume that the larger a cluster is in size, the more likely it is to undergo fragmentation. Mathematically, we set the fragmentation rate as $\beta(j) := \beta \times j$ for all $j \in \{1, \dots, n\}$. Regarding possible fragmentation events, we assume them to be uniformly probable, meaning that $\eta(j, i) := 1/(j-1)$, for all $j \in \{1, \dots, n\}$ and $i \in \{1, \dots, j-1\}$.

**Coagulation term** The operator $C_i^{\pm}[p_j^{\pm}]$ corresponds to the coagulation events during the time interval $dt$, and it is given by the following formula

$$C_i^{\pm}[p_j^{\pm}] = \frac{1}{2} \sum_{j=1}^{i-1} \gamma(i-j, j) \left[ p_j^{\pm} p_{i-j}^{\pm} + \frac{1}{2} (p_j^{+} p_{i-j}^{-} + p_j^{-} p_{i-j}^{+}) \right] - p_i^{\pm} \sum_{j=1}^{n-i} \gamma(j, i) p_j,$$

for all $i \in \{1, \dots, n\}$.

The first term corresponds to the coagulation of two smaller clusters forming a cluster of size $i$. We assume that if two clusters are moving in the same direction, the resulting cluster from their coagulation will also move in the same direction. Otherwise, the newly formed cluster has an equal probability of moving left or right. The function $\gamma(i-j, j)$ represents the coagulation rate between a cluster of size $j$ and a cluster of size $i-j$. Since no experimental data provide an exact functional form for $\gamma$, we assume that the rate is proportional to the product of the cluster sizes, leading to the following expression

$$\gamma(j, j') = \gamma \times j \times j', \quad \text{for all } j, j' \in \{1, \dots, n\} \text{ such that } j + j' \le n.$$

We assume that two clusters cannot coagulate if the resulting cluster exceeds size $n$. Mathematically, this constraint is expressed as $\gamma(j, j') = 0$ for all $j$ and $j'$ such that $j + j' > n$. Alternative coagulation and fragmentation assumptions exist; see, for example, [51, 52]. The second term of the coagulation operator represents all clusters of size $i$ that have coagulated with another cluster to form a larger-sized cluster. The fragmentation and coagulation operators preserve the mass, $p$.

**Proliferation term** To model cooperation within *M. xanthus* colonies, we define two distinct proliferation terms that differ significantly between isolated bacteria and those grouped in clusters. The role of isolated bacteria is to kill prey, thereby releasing nutrients into the environment. These nutrients are subsequently consumed by bacterial clusters, enhancing their proliferation. In contrast, isolated bacteria are assumed to divide independently of these nutrients.

More precisely, we assume that when an isolated bacterium divides, both daughter cells remain isolated. For isolated bacteria, we use a logistic proliferation term given by $\alpha_1 p_1(1-p)_+/2$. The parameter $\alpha_1$ represents the proliferation rate. This logistic form ensures that bacterial division slows down as the total bacterial mass $p$ increases, preventing excessive growth. Once the mass exceeds a certain threshold, division events are assumed to no longer occur.

For bacteria within clusters, the division term is given by:

$$\text{Division term: } N \times [d(i-1)p_{i-1}^{\pm} - d(i)p_{i}^{\pm}] \times (1-p)_{+}.$$

Unlike isolated bacteria, clustered bacteria rely on the availability of nutrients for their proliferation. These nutrients are released into the environment through prey lysis driven by isolated
bacteria. To model this, we assume that $N$ evolves according to the following equation:

$$ \partial_t N = \delta p_1 (1 - N/N_M), \qquad (5) $$

where $\delta$ represents the rate of nutrient production, and $N_M$ is the carrying capacity. We assume that the diffusion of these nutrients is negligible. Since we focus solely on the invasion dynamics, we do not model the degradation of these nutrients.

Regarding the division term, we assume that when a bacterium within a cluster divides, both daughter cells remain part of the same cluster. As a result, the evolution of clusters of size $i$ due to division between times $t$ and $t+dt$ depends on clusters of size $i-1$, from which a bacterium has divided, and clusters of size $i$, where a division has occurred. We assume that the division rate, $d$, is given by

$$ d(i) = i \times \alpha_2, \quad \text{for all } i \in \{2, \dots, n-1\}, \quad \text{and} \quad d(1) = d(n) = 0. $$

The linear growth of $d$ reflects the assumption that all bacteria, whether isolated or within a cluster, has the same probability of dividing. Consequently, larger clusters are more likely to undergo division. To ensure that bacterial clusters do not exceed a maximum size of $n$, we assume that division is no longer possible for clusters of size $n$. As before, we assume that the division rate decreases with the total bacterial mass $p$ and that division can only occur if the total mass remains below the critical threshold of 1.

## Formulation of the simplified model $n = 2$

To introduce the mathematical concepts used to study the propagation speed of traveling wave solutions for the model (3)-(4)-(5), we first consider a simpler model where bacteria are either isolated or form clusters of two bacteria. However, in the model (3)-(4)-(5), the choice of the proliferation term for clustered bacteria is only relevant when the maximum cluster size is sufficiently large. For $n = 2$, this term becomes irrelevant, as two-bacteria clusters gradually disappear and nutrients do not influence *M. xanthus* predation behavior. To address this issue, we introduce a nonlinear, nutrient-dependent logistic proliferation term into the equations governing two-bacteria clusters. Under the assumption $n = 2$, we consider the following model:

$$ \left\{ \begin{aligned} \partial_t p_1^+ + \frac{1}{\epsilon} \partial_x p_1^+ &= \frac{1}{2\epsilon^2} (p_1^- - p_1^+) - k(p_1^+)^2 - k p_1^+ p_1^- + k p_2 + \frac{\alpha_1}{2} p_1 (1-p)_+, \\ \partial_t p_1^- + \frac{1}{\epsilon} \partial_x p_1^- &= \frac{1}{2\epsilon^2} (p_1^+ - p_1^-) - k(p_1^-)^2 - k p_1^- p_1^+ + k p_2 + \frac{\alpha_1}{2} p_1 (1-p)_+, \\ \partial_t p_2^+ + \frac{v}{\epsilon} \partial_x p_2^+ &= \frac{1}{2\tau\epsilon^2} (p_2^- - p_2^+) - k p_2^+ + \frac{k}{2} (p_1^+)^2 + \frac{k}{2} p_1^+ p_1^- + \frac{\alpha_2}{2} N p_2 (1-p)_+, \\ \partial_t p_2^- + \frac{v}{\epsilon} \partial_x p_2^- &= \frac{1}{2\tau\epsilon^2} (p_2^+ - p_2^-) - k p_2^- + \frac{k}{2} (p_1^-)^2 + \frac{k}{2} p_1^- p_1^+ + \frac{\alpha_2}{2} N p_2 (1-p)_+, \\ \partial_t N &= \delta p_1 (1 - N/N_M). \end{aligned} \right. \qquad (6) $$
According to the experimental biological data from the article [53], there is a symmetry between the probability of fragmentation and coagulation. To simplify the upcoming calculations, we assume that the fragmentation rate equals the coagulation rate, and denote it by $k$. This choice of proliferation term would be irrelevant for a larger $n$. As in the general model, the parameter $v$ (resp. $\tau$) represents the advantage or disadvantage in speed (resp. average time before a change of direction) of clusters of two bacteria.

# 3 Results for the simplified case $n = 2$

## 3.1 Linear speeds

In this section, we study the critical linear propagation speed of the model (6). Linearizing this model around the zero equilibrium point gives us the following linear system:

$$ \partial_t p_1^+ + \frac{1}{\epsilon} \partial_x p_1^+ = \frac{1}{2\epsilon^2} (p_1^- - p_1^+) + k p_2 + \frac{\alpha_1}{2} p_1, \quad (7a) $$

$$ \partial_t p_1^- - \frac{1}{\epsilon} \partial_x p_1^- = \frac{1}{2\epsilon^2} (p_1^+ - p_1^-) + k p_2 + \frac{\alpha_1}{2} p_1, \quad (7b) $$

$$ \partial_t p_2^+ + \frac{v}{\epsilon} \partial_x p_2^+ = \frac{1}{2\tau\epsilon^2} (p_2^- - p_2^+) - k p_2^+, \quad (7c) $$

$$ \partial_t p_2^- - \frac{v}{\epsilon} \partial_x p_2^- = \frac{1}{2\tau\epsilon^2} (p_2^+ - p_2^-) - k p_2^-. \quad (7d) $$

Note that equations (7c)-(7d) are independent of the variables $p_1^+$ and $p_1^-$. However, through the coupling term, $p_2^+$ influences the behavior of $p_1^+$. Nevertheless, since $p_2^+$ decays exponentially to zero for all $x$ as $t \to +\infty$, one might expect the coupling term in (7a)-(7b) to quickly become negligible. However, we will see that it can permanently increase the wave speed. This phenomenon, known as anomalous propagation, was first introduced in the article [35] and has since been extensively studied [34, 54, 36, 55].

Thus, the linear propagation speed of (7a)-(7d) can follow two distinct scenarios. In the first case, the coupling term has no effect, and the critical asymptotic speed is determined solely by the decoupled system of equations (7a)-(7b), i.e., with $k = 0$. We refer to this as the normal speed. Conversely, if the pairs of bacteria, through the coupling term, influence the propagation speed of the isolated bacteria, we will refer to it as anomalous speed.

We define the linear propagation speed as follows:

**Definition 1** (Linear propagation speed). The linear propagation speed of a linear system (S) is given by

$$ s_{\text{lin}}^* = \sup\{s : D_s(\nu, \lambda) \text{ has a pinched double root with } \text{Re } \lambda \ge 0\}, $$

where $D_s(\nu, \lambda)$ is the dispersion relation associated with the problem (S), and the notion of a pinched double root is introduced in the A.

This definition is based on the distinction between convective instability and absolute instability [34]. In certain specific cases, this definition may prove to be imperfect. It then becomes
necessary to consider the singularities of the pointwise Green's function associated with the problem. The connection between pinched double roots and these singularities has been studied theoretically for a broad class of parabolic systems, by Holzer and Scheel in the article [34]. However, to simplify the calculations in A, we will nonetheless use this definition.

In A, we show that the linear propagation speed, $s_{\text{lin}}^{*}$, can take one of four possible values:

* The two normal speeds: parabolic and hyperbolic

$$s_{\text{para}} := \frac{2\sqrt{\alpha_1}}{1 + \alpha_1 \epsilon^2}, \qquad s_{\text{hyp}} := \frac{1}{\epsilon}.$$

* The two positive anomalous speeds given by:

$$(s_{\text{anom}}^{\pm})^2 = s^{\pm}(0) = \frac{-a_2(0) \pm \sqrt{a_2^2(0) - 4a_0(0)a_4}}{2a_4},$$

provided these two speeds are well-defined. The functions $a_0$, $a_2$, and the coefficient $a_4$ are defined by equations (16), (17), and (18), respectively, in A.

According to Definition 1, the selected linear speed is the largest speed among those associated with pinched double roots. Based on this postulate, one can explicitly derive certain critical transition values, which will not be detailed in this article due to their significant complexity. We illustrate transitions between different possible linear speeds in Figure 9.

When isolated bacteria are faster than clusters, we obtain the following result:

* If $\epsilon < 1/\sqrt{\alpha_1}$, then the linear propagation speed can be the parabolic speed or one of the anomalous speeds: $s_{\text{para}}$, $s_{\text{anom}}^{+}$, and $s_{\text{anom}}^{-}$.

* If $\epsilon > 1/\sqrt{\alpha_1}$, then the linear propagation speed corresponds to the hyperbolic speed.

In A, numerical simulations confirm that this linear propagation speed is indeed selected in the nonlinear model within a pulled front regime, when the initial conditions are given by Heaviside functions.

## 3.2 Nonlinear speed – Pulled and Pushed regime

The propagation speed of the nonlinear model (6) does not always match the linear speed. In this section, we use numerical simulations to compare both speeds and determine whether the front is pulled or pushed [37].

For numerical simulations, we employ the numerical scheme presented in the article [29], based on the nonlinear flux-limiter schemes developed in the articles [56, 57]. To approximate the wave speed, we use an algorithm based on the LeVeque-Yee formula [58, 59]. In this article, we consider only steep initial data. More specifically, for the model (6), we use initial conditions of the form $p_i^{\pm}(t=0, x) = (p_i^{*}/2)H(x-x_0)$, for $i=1, 2$, where $H$ denotes the Heaviside function and $x_0$ is a positive constant.

In Figure 1, we plot with black dots the evolution of the critical speed for the nonlinear model, normalized by the speed obtained for $v = 1$, for different values of $v$. When $v = 1$, the selected
<figure><img src="image_3.png" /><figcaption>Figure 1: Comparison between the numerical critical speed of the model (6) and the theoretical linear speed for different values of parameter $v$. The black points correspond to the numerical critical speed of the model (6) normalized by the speed for $v = 1$. The red curve represents the parabolic speed, and the cyan curve represents the anomalous minus speed. For $v$ small enough, the critical speed corresponds to the parabolic speed, indicating a pulled front regime. For $v > v_c \approx 1.1$, the critical speed differs from the possible linear speeds, indicating a pushed front regime. The other parameter values are given by $k = 1$, $\alpha_1 = 1$, $\alpha_2 = 4$, $\tau = 1$, $N_M = 6$, $\epsilon = 0.9$, and $\delta = 2$.</figcaption></figure>

numerical speed closely matches the parabolic speed. However, a slight numerical error on the order of $10^{-2}$ remains due to the inaccuracies of the numerical simulations (mesh, convergence time, etc.). Since this error persists for other values of $v$, we normalize the propagation speed by the value calculated at $v = 1$ in order to enhance the readability of the figure and facilitate comparison with theoretical speeds. The colored curves correspond to the linear propagation speed normalized by the parabolic speed independent of $v$. We observe two distinct regimes separated by a critical threshold $v_c$. When $v < v_c$, the front is pulled, as the selected numerical speed matches the parabolic linear speed. Conversely, when $v > v_c$, the speed exceeds both the parabolic linear speed and all possible linear speeds, indicating a pushed front. The hyperbolic speed and the anomalous plus speed are not shown in this figure as they are not admissible for the chosen parameter values.

We performed several additional numerical simulations, not presented in this article, for different parameter values. These simulations suggest that the nonlinear speed can match the parabolic or hyperbolic linear speeds but never the anomalous linear speeds. Similar results were reported for the reaction-diffusion model in [19, 20]. Furthermore, when the model is
in the pushed regime, the propagation speed always appears to exceed the linear speed. For single reaction-diffusion equations, the nonlinear speed is always greater than the linear speed. However, in the article [60], the authors present a counterexample demonstrating that this is not always the case for reaction-diffusion systems.

### 3.3 Optimal invasion strategy

The difference in motility between isolated bacteria and bacterial clusters result in distinct persistence and instantaneous speed characteristics. In the article [18], it was shown that bacterial clusters move faster than isolated bacteria. Moreover, based on the contrasting mechanisms of adventurous and social twitching motility [61, 62], we explore the case where A motility is more persistent than the other. Consequently, we assume that the biologically relevant parameter values satisfy $v > 1$ and $\tau < 1$.

In Figure 2, we use numerical simulations to compute the critical speed of model (6) as a function of the motility parameters $(v, \tau)$. As before, we normalize the numerical critical speed by the speed obtained when $v = 1$ and $\tau = 1$. In this case, the front is pulled, and the selected linear speed is the parabolic speed. The intensity of the red color represents the renormalized numerical critical speed. We observe that as either $v$ or $\tau$ increases, this speed also increases. Since the instantaneous speed advantage of clusters is counterbalanced by a persistence disadvantage, it is also relevant to analyze how the critical speed evolves when one parameter increases while the other decreases.

As mentioned in the introduction, the diffusion coefficient is a combination of persistence and instantaneous speed (2). We define $D$ as the ratio between the diffusion coefficient of bacterial clusters and that of isolated bacteria. To focus exclusively on the impact of differences in instantaneous speed and persistence time between clusters and isolated bacteria, we assume that this ratio remains equal to 1, i.e., $D := \tau v^2 = 1$. If this ratio differs from 1, for example $D > 1$, this difference could alone influence the propagation speed of the bacteria [19, 20]. We thus define $s$ as the nonlinear critical speed of the model (6), under the constraint $1 = \tau v^2$,

$$s(\tilde{v}) = s^*(\tau = 1/\tilde{v}^2, v = \tilde{v}),$$

where $s^*$ denotes the critical nonlinear speed of the traveling wave solution of (6).

Figure 3 shows the variations of the speed $s$ for two values of the proliferation rate of isolated bacteria, $\alpha_1$. First, we consider the extreme case where isolated bacteria do not divide, i.e., $\alpha_1 = 0$. In this case, we observe that the speed $s$ is strictly increasing as $v$ increases (and as $\tau$ decreases) under the constraint $1 = \tau v^2$. Under the assumption $\alpha_1 = 0$, we lose the linear proliferation term of isolated bacteria, so propagation can only occur through the nonlinear terms.

In the case $\alpha_1 = 1$, we observe two possible scenarios. When $v$ is small enough, the front is pulled, and the critical nonlinear speed always coincides with the normal critical nonlinear speed for $v = 1$ and $\tau = 1$. Our numerical simulations indicate that anomalous speeds are never selected. When $v$ becomes large enough, the front becomes pushed, and the increase in $v$ under the constraint $1 = \tau v^2$ leads to an increase in invasion speed. In this figure, the critical transition separating pulled and pushed fronts occurs around $v_c \approx 1.1 (\tau_c = 1/v_c^2)$. This critical threshold strongly depends on the parameter values.
<figure><img src="image_4.png" /><figcaption>Figure 2: Evolution of the numerical critical speed of the traveling wave as a function of the parameters $\tau$ and $v$ for the model (6). The colorimetry represents the intensity of the numerical critical speed obtained from our numerical simulations, divided by the speed at $\tau = 1$ and $v = 1$ (parabolic speed). The circles correspond to the constraint $D := \tau v^2 = 1$. We note that under the constraint $\tau v^2 = 1$, the increase of $v$ and the decrease of $\tau$ leads to an increase in the speed ratio. The parameter values are: $\alpha_1 = 0.5$, $\alpha_2 = 4$, $k = 1$, $\epsilon = 0.85$, $\delta = 3$, $N_M = 6$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3: **Evolution of the numerical critical speed of the model (6) under the constraint 1 = τv² for: Top. α₁ = 0, Bottom. α₁ = 1.** The black curves represent the numerical nonlinear speed of the model (6). For the bottom figure, we normalize this numerical speed by the speed calculated at τ = 1 and v = 1 (parabolic speed). We calculate this speed for different values of v under the constraint 1 = τv², so as v increases, the value of the parameter τ, not shown in the figure, will decrease. The rectangles above the two figures illustrate the pushed or pulled regime of the traveling wave. We observe that in both cases, the increase in v coupled with the decrease in τ leads to an increase in the critical speed. The values of the other parameters are given by α₂ = 4, k = 1, ε = 0.85, N<sub>M</sub> = 3 and δ = 2.</figcaption></figure>
More generally, our numerical simulations suggest that for any $\alpha_1 > 0$, a critical threshold $v_c$ exists such that:

* If the advantage of instantaneous speed, coupled with the corresponding disadvantage of persistence, remains below $v_c$, the invasion speed is not affected by these motility differences. In this case, the coexistence of both motility systems does not impact bacterial predation.

* However, if the instantaneous speed $v$ exceeds $v_c$ (resulting in $\tau < \tau_c$), the trade-off leads to an increase in invasion speed. Thus, the coexistence of adventurous and social motility provides a significant advantage for predation.

# 4 Results for the full model $n > 2$

In this extended model, the spatially homogeneous null state remains an unstable equilibrium point. The stable steady state is more complex to characterize explicitly, and we use numerical simulations to determine it.

## 4.1 Linear speed

Following a reasoning similar to the case $n = 2$, and using Definition 1, we show in B that the linear propagation speed, $s_{\text{lin}}^*$, can take the following possible values:

* The two normal speeds: parabolic and hyperbolic

$$s_{\text{para}} := \frac{2\sqrt{\alpha_1}}{1 + \alpha_1 \epsilon^2}, \qquad s_{\text{hyp}} := \frac{1}{\epsilon}.$$

* The different anomalous speeds: $s_{\text{anom},i}^{\pm}$ for all $i \in \{2, \dots, n\}$, where $s_{\text{anom},i}^{\pm}$ are defined in equation (24).

## 4.2 Nonlinear speed – Pulled and Pushed regime

In Figure 4, using numerical simulations, we plot, with black dots, the evolution of the critical speed for the nonlinear model (3)-(4)-(5) for different functions $v$. More precisely, we define $v$ as a threshold function of the following form $v(i) = \mathbf{1}_{i<\lfloor n/2 \rfloor}(i) + \bar{v}\mathbf{1}_{i\ge\lfloor n/2 \rfloor}(i)$, for all $i \in \{2, \dots, n\}$, where $\bar{v}$ is a positive real parameter that we vary in Figure 4. As before, we normalize the numerical speed by the speed obtained at $\bar{v} = 1$. To compare with theoretical linear speeds, we also illustrate the parabolic speed (red curve) and the largest of the admissible anomalous speeds (cyan curve), both normalized by the parabolic speed. When $\bar{v}$ is small enough, the propagation speed of system (3)-(4)-(5) matches the linear prediction. However, when $\bar{v}$ becomes sufficiently large, the speed exceeds the linear speeds, and the front becomes pushed.
<figure><img src="image_6.png" /><figcaption>Figure 4: Comparison between the numerical critical speed of the nonlinear model (3)-(4)-(5) and the theoretical linear speed for different values of the parameter $\bar{v}$. The black points correspond to the numerical critical speed of the nonlinear model (3)-(4)-(5) normalized by the speed for $\bar{v} = 1$. The red horizontal line represents the parabolic linear speed, and the cyan curve represents the largest admissible anomalous speeds, both normalized by the parabolic speed. For $\bar{v}$ small enough, the critical speed of the nonlinear model is equal to the linear speed. Then, when $\bar{v}$ becomes large enough, the nonlinear speed becomes strictly greater than the linear speed. The values of the other parameters are given by $\beta = \gamma = \alpha_1 = \alpha_2 = N_M = 1$, $n = 4$, $\epsilon = 0.95$, $\delta = 1.5$, and the functions are given by $\tau(i) = 1$ for all $i \in \{1, 2, 3, 4\}$, $v(2) = 1$.</figcaption></figure>

## 4.3 Biological modeling

### 4.3.1 Propagation example

In Figure 5, we illustrate the traveling wave solution of the model with multiple cluster sizes (3)-(4)-(5). The dotted lines represent the density of clusters of different sizes, each multiplied by its respective size at a given time $T$. The propagation speeds associated with the densities $p_i$, with $i \in \{1, \dots, n\}$, remain similar across all cluster sizes. The grey area corresponds to the nutrient density at the same time.

Upstream of the front, we only observe the presence of isolated bacteria represented by the black line. These isolated bacteria generate nutrients that subsequently allow the invasion of bacteria clusters. The different cluster sizes are indicated by the following colorimetry: from blue ($i = 2$) to red ($i = n$). We note that the invasion of clusters of increasing sizes occurs in a sequential and ordered manner. After isolated bacteria, we observe the emergence of clusters of two bacteria, followed by clusters of three bacteria, and so on. This heterogeneity in cluster
<figure><img src="image_7.png" /><figcaption>Figure 5: **Example of traveling wave solutions of the model (3)-(4)-(5) at time $t = T$.** The nutrient density, $N$, is represented in grey. The dashed lines correspond to the densities of different sizes of bacterial clusters multiplied by their size, $p_i(x) \times i$ at time $t = T$. The black line represents isolated bacteria, while the lines from blue to red correspond to bacterial clusters and more precisely, from the cluster of two bacteria in blue to the cluster of $n$ bacteria in red. We observe an invasion of isolated bacteria, leading to the creation of nutrients. In the region where nutrients are present, we see the successive invasion of clusters of increasingly larger sizes. This spatial heterogeneity in the distribution of different cluster sizes has been recently observed biologically in the article [53]. The parameter values are given by $\beta = \gamma = \epsilon = \delta = \alpha_1 = \alpha_2 = N_M = 1$, $n = 10$, and the functions are defined as follows: $v(i) = \tau(i) = 1$ for all $i \in \{2, \dots, n\}$.</figcaption></figure>

propagation is also observed in biological predation experiments with *M. xanthus* [53]. Indeed, at the leading edge of the predation front, there are mainly isolated bacteria, called *scouts*. Behind them, small-sized clusters, known as *loners*, begin to appear. Finally, in the wake of these clusters, large clusters, called *swarms*, progressively invade the environment.

### 4.3.2 Study of strain invasion speed

In the article [12], the authors investigated the invasion speed of three strains: *Wildtype* (*A+S+*), *A+S*−, and *A−S+*. The *Wildtype* strain corresponds to the natural state of *M. xanthus*, possessing both motility systems, *A* and *S*. The other two strains are mutants that have lost one of these systems: the *A+S*− strain lacks the *S* motility apparatus, while the *A−S+* strain lacks the *A* motility apparatus.

To study their invasion properties, Seef et al. conducted three independent predation experiments, one for each strain, stopping them at the same time to analyze the progress of bacterial invasion. Their results indicate that the *A−S+* strain exhibits significant difficulty in invading the prey compared to the *Wildtype* and *A+S*− strains.

To adapt our model to these specific strains, we modify the transport terms accordingly. For the *A+S*− strain, we remove the transport term for bacterial clusters, setting $v(i) = 0$ for
all $i \in \{2, \dots, n\}$. Similarly, for the $A-S+$ strain, we remove the transport term for isolated bacteria.

In Figure 6, we present numerical simulations that reproduce the biological experiments described in [12]. Specifically, we perform three independent numerical simulations, one for each strain, and stop them at the same time to compare their respective invasion dynamics. The black curve corresponds to the *Wildtype* strain, the red curve to the $A+S-$ strain, and the green curve to the $A-S+$ strain. Our numerical results align well with the biological findings: the $A-S+$ strain exhibits a significantly slower invasion speed compared to the other two strains.

For these parameter values, the invasion speed of the *Wildtype* strain is similar to the invasion speed of the $A+S-$ strain. However, as explained earlier, for other parameter values, the front can be pushed, and in that case, the speed of the *Wildtype* strain would be greater than the speed of the $A+S-$ strain. In the biological experiments of [12], the $A+S-$ strain exhibited slightly faster predation than the *Wildtype* strain. However, given the high variability of these results, additional experiments are needed to determine whether this difference is statistically significant. Note that, in the absence of prey, the *Wildtype* strain propagates significantly faster than both the $A+S-$ and $A-S+$ strains, demonstrating that $A$ and $S$ motilities are used synergistically [63]

### 4.3.3 Optimal motility strategy for invasion

The role of the coexistence of the two motility systems is a key question in this field. As explained earlier, these two distinct motility systems lead to significant differences in bacterial movement characteristics. We assume that, for the reasons specified before, the diffusion coefficient remains unchanged across different cluster sizes, and only differences in instantaneous speed and persistence need to be considered. The $A$ motility of isolated bacteria induces a relatively low speed but, in return, a very high persistence in the same direction. Conversely, the social motility of clusters generates a higher speed along one direction but a much lower persistence. To study the impact of these motility differences on predation speed, we examine the critical speed of the model (3)-(4)-(5) for different couples of functions ($v_j, \tau_j$) under the constraint $v_j^2 \tau_j = 1$, ensuring that the diffusion coefficient $D$ remains equal to 1 for all cluster sizes (see (2)).

Various function families can be considered for instantaneous speed and persistence. In particular, we analyze two cases: linear functions and Hill functions. First, we assume that the functions $v_j$ are linear. More precisely, we define the family of functions ($v_j$)$_j$ as the set of lines connecting the point (1, 1) to $(n, \bar{v}_{n,j})$, where $\bar{v}_{n,j} \in \mathbb{R}_+^*$ represents the speed of clusters of size $n$. When $\bar{v}_{n,j} < 1$, we have $v'_j < 0$, meaning that clusters move more slowly than isolated bacteria but compensate with higher persistence ($\tau_j = 1/v_j^2$). We have the opposite result for $\bar{v}_j > 1$. Moreover, as $\bar{v}_{n,j}$ deviates further from 1, the difference in motility between isolated bacteria and clusters becomes more pronounced.

In the top of Figure 7, we illustrate the family of functions ($v_j$)$_j$ (on the left) for different values of ($\bar{v}_j$)$_j$, along with their corresponding persistence functions ($\tau_j$)$_j$ (on the right). In the bottom of Figure 7, we also plot the critical speed of the model (3)-(4)-(5), for the different couples of functions ($v_j, \tau_j$) as a function of the coefficients $\bar{v}_j$ associated with these couples. The colors of the squares correspond to the color-coded curves representing the associated functions $v_j$ and $\tau_j$ in the top row. In Figure 8, we follow the same reasoning, but this time taking functions
<figure><img src="image_8.png" /><figcaption>Figure 6: Comparison of the invasion of the three different strains: *Wildtype*, *A+S*−, *A+S*+ at time *t* = *T*. The black curve is a numerical solution of the model (3)-(4)-(5) associated with the *Wildtype* strain. The red (respectively green) curve is a numerical solution of the same model but without transport terms for clusters of bacteria (respectively for isolated bacteria) and corresponds to the *A+S*− strain (respectively *A+S*+ strain). For the *Wildtype* strain and the *A+S*− strain, the functions *v* and *τ* are defined as *v*(*i*) = 1.25 and *τ*(i) = 1/(1.25²) for all *i* ∈ {2, ..., *n*}. For the *A+S*+ strain, the functions *v* and *τ* are defined as *v*(i) = 0 and *τ*(i) = 1/(1.25²) for all *i* ∈ {2, ..., *n*}. The parameter values are: *n* = 10, *β* = *γ* = *δ* = *α*₁ = *α*₂ = *N*<sub>M</sub> = 1, *ε* = 0.5.</figcaption></figure>
$v_j$ in the form of Hill functions.

For both Figures 7 and 8, the results on the critical speed are qualitatively similar. When the difference in instantaneous speed between clusters and isolated bacteria is either a disadvantage or too small an advantage, this difference in instantaneous speed and persistence does not affect the predation speed of bacteria *M. xanthus*. More specifically, in this case, the front is pulled: the invasion speed matches the linear propagation speed, which, for these parameter values, corresponds to the parabolic speed. However, when the speed advantage of bacterial clusters (coupled with the corresponding disadvantage of cluster persistence) becomes significant enough, the coexistence of *A* and *S* motility enhances the bacterial invasion speed. In this scenario, the front is pushed, and the impact on the predation speed is proportional to the degree of difference in instantaneous speed and persistence.

In summary, our model predicts that the coexistence of two motility systems either *does not affect predation efficiency*, in which case the invasion speed is governed solely by isolated bacteria (pulled front), or *enhances predation efficiency* (pushed front). Thus, according to our results, the coexistence of two motility systems never constitutes a disadvantage for bacterial predation.

# 5 Conclusion

In this work, we introduce a mathematical model to study the predation dynamics of the social bacterium *M. xanthus*. This model, corresponding to an extension of previous models, aims primarily to model and analyze the impact of differences between the two motility systems, *A* and *S*. Within this model, we derive an explicit formula for the linear propagation speed and demonstrate the existence of anomalous speeds arising due to the linear coupling between bacterial clusters and isolated bacteria. Regarding the nonlinear speed, our numerical simulations reveal a transition between a pulled-front regime and a pushed-front regime, where the selected speed surpasses the linear prediction.

From a modeling point of view, the numerical simulations of our model reproduce key features observed in biological experiments. In particular, our model accurately captures the spatial distribution of cluster sizes within the invasion front. It also provides an explanation for the observed differences in invasion speed among the *Wildtype*, *A+S*−, and *A−S+* strains. As reported in experiments, the *A−S+* strain had great difficulty in invading the prey. Furthermore, our model highlights that the coexistence of the two motility systems, *A* and *S*, either has a neutral effect or enhances predation efficiency. If the motility differences do not induce sufficiently large disparities in instantaneous speed and persistence, the coexistence of both systems does not influence propagation speed. Conversely, when these differences become significant, they lead to a proportional increase in the invasion speed.

To assess whether the coexistence of *A* and *S* motility has a neutral or positive impact on predation, it is crucial to precisely determine the transition between a pulled-front and a pushed-front regime. However, numerically approximating this transition with high precision remains challenging. A recent numerical method based on far-field decomposition has demonstrated remarkable accuracy in identifying this threshold for a broad class of reaction-diffusion systems [64]. It would be interesting to extend these results to hyperbolic systems.

Once this threshold is determined with high precision, an important next step would be to
<figure><img src="image_9.png" /><figcaption>Figure 7: **Evolution of the numerical critical speed of the model (3)-(4)-(5) under the constraint $1 = \tau v^2$ with $v$ a linear function.** Top. On the left, we represent a family of instantaneous speed functions $v_j$. The instantaneous speeds $v_j$ are lines connecting the point (1,1) to the point $(n, \bar{v}_j)$, with $(\bar{v}_j)$ a uniform discretization of the interval $[0.1, 5]$, represented by the color spectrum from blue ($\bar{v} = 0.1$) to red ($\bar{v} = 5$). On the right, the figure illustrates the functions $\tau_j$ associated with instantaneous speeds $v_j$ under the constraint $1 = v_j^2 \tau_j$. **Bottom.** For each couple of functions $(v_j, \tau_j)$, we numerically compute the critical speed of the model (3)-(4)-(5) and we indicate the result by a square with the corresponding colour. When $\bar{v}$ is small enough, the critical speeds are equal to the linear speed, and the front is pulled. In this case, the propagation speed is independent of the coefficient $\bar{v}$. However, when the coefficient, $\bar{v}$, becomes large enough, the front becomes pushed and predation of the bacteria is faster. The values of the other parameters are given by $n = 4$, $\epsilon = 0.9$, $\alpha_1 = \alpha_2 = \gamma = \beta = N_M = 1$ and $\delta = 3$.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 8: **Evolution of the numerical critical speed of the model (3)-(4)-(5) under the constraint $1 = \tau v^2$ with $v$ a Hill function.** Top. On the left, we represent a family of instantaneous speed functions $v_j$. The functions $v_j$ are of the form $v_j(i) = 1 + \bar{v}_j \cdot h(i)^m / (1 + h(i)^m)$, where $h$ is an affine function given by $h(i) = i \cdot h_1 + h_0$, $m$ is an integer and $(\bar{v}_j)$ a uniform discretization of the interval $[-1, 5]$. On the right, the figure illustrates the functions $\tau_j$ associated with instantaneous speeds $v_j$ under the constraint $1 = v_j^2 \tau_j$. **Bottom.** For each couple of functions $(v_j, \tau_j)$, we numerically compute the critical speed of the model (3)-(4)-(5) and we indicate the result by a square with the corresponding colour. When $\bar{v}$ is small enough, the critical speeds are equal to the linear speed, and the front is pulled. In this case, the propagation speed is independent of the coefficient $\bar{v}$. However, when the coefficient, $\bar{v}$, becomes large enough, the front becomes pushed and predation of the bacteria is faster. The values of the other parameters are given by $n = 8$, $\epsilon = 0.9$, $\alpha_1 = \alpha_2 = \gamma = \beta = N_M = 1$, $\delta = 3$, $m = 10$, $h_0 \approx 0.28$ and $h_1 \approx 0.21$.</figcaption></figure>
estimate biologically relevant parameter values to determine whether the invasion dynamics of *M. xanthus* predominantly follow a pulled or pushed regime. While some experimental data on bacterial coagulation and fragmentation can already be incorporated, crucial information is still lacking, particularly regarding division rates within clusters and the precise interactions between *M. xanthus* and its prey. Further experimental studies are needed to enable accurate model calibration.

In contrast to other bacterial predators that specialize in specific prey, *M. xanthus* is a generalist predator capable of feeding on a wide variety of bacterial species [9]. Given the vast spectrum of potential prey for *M. xanthus*, the predation process of this bacterium is likely to be extremely complex, involving various molecular mechanisms acting independently or synergistically to attack different species. It would therefore be relevant to explore, both mathematically and biologically, the variations in the predation process exerted by *M. xanthus* on different prey species. Additionally, certain bacterial prey, such as *Bacillus subtilis*, *Pseudomonas fluorescens* and *Escherichia coli*, adopt defensive strategies, including the formation of specific biofilms or the secretion of antibiotics [65, 66, 67, 68]. Investigating these prey defense mechanisms and their interactions with *M. xanthus* could further refine our understanding of bacterial predation dynamics.

In this article, we focused on the predation front of *M. xanthus*. In the wake of the invasion front, two main phenomena occur sequentially: *rippling*, followed by the formation of fruiting bodies. As the predation front advances, the presence of cellular debris enables *M. xanthus* cells to coordinate their movement by forming accordion-like undulations, similar to the undulations of water [7, 68, 69, 72]. During these undulations, known as *ripples*, each wave crest oscillates back and forth without generating a net displacement, although individual cells continuously change positions. Then, under starvation conditions, cells form aggregates, exchanging extracellular chemical signals and physical contact cues, ultimately leading to the formation of multicellular fruiting bodies [73]. Within these structures, cells undergo transformation into myxospores, highly resistant to heat [74]. Sporulation confers a survival advantage in hostile environments and enhances germination and growth rates when favorable conditions return.

Several models have been developed to describe the rippling phenomenon [72, 75, 76], including those incorporating *Run and Tumble* dynamics [46, 77, 49, 78]. Similarly, numerous models exist for fruiting body formation (see, for example, [79, 80]). To develop a model encompassing all these phenomena, it would be interesting to extend our hyperbolic model to determine if it can also capture, in the wake of the invasion front, the rippling phenomenon and then, once nutrients are depleted, the formation of fruiting bodies. The emergence of periodic solutions in space (formation of fruiting bodies) and/or in time (rippling) are phenomena that have already been extensively studied for hyperbolic equations and systems of equations [81, 82, 83, 84, 85, 86, 87].

## Acknowledgements

The author thanks Vincent Calvez and Thomas Lepoutre for several insightful discussions. In addition, the author gratefully acknowledges Antoine Le Gall and Tâm Mignot for their helpful explanations regarding the bacterium *Myxococcus xanthus*.
# A Derivation of the linear propagation speed for $n = 2$

First, we provide the definition of the term *pinched double roots* from [34, Defintion 4.2]

**Definition 2 (Pinched double root).** We say $(\nu_*, \lambda_*)$ is a *pinched double root* of the dispersion relation $D_s(\nu, \lambda)$ if

1. *Double root:* we have

$$D_s(\nu_*, \lambda_*) = 0, \quad \text{and,} \quad \partial_\nu D_s(\nu_*, \lambda_*) = 0. \tag{8}$$

2. *Pinching:* there exists a continuous curve $\lambda(\tau)$, with $\tau \in \mathbb{R}_+$ satisfying:

* $\mathrm{Re}\,\lambda(\tau)$ is strictly increasing,

* $\lambda(0) = \lambda_*$,

* and $\mathrm{Re}\,\lambda(\tau) \to \infty$ as $\tau \to \infty$,

and continuous curves of roots $\nu^\pm(\lambda(\tau))$ which satisfy

* $\nu^\pm(\lambda_*) = \nu_*$,

* and

$$\lim_{\tau \to \infty} \mathrm{Re}\,\nu^\pm(\lambda(\tau)) = \pm\infty. \tag{9}$$

In this work, we are only interested in time-stationary fronts in the moving frame reference. Consequently, we consider the values $\lambda \in \mathbb{R}^+$, and negative eigenvalues, $\nu \in \mathbb{R}^-$.

## A.1 Derivation of normal speeds

In the decoupled case, $k = 0$, the first two equations (7a)-(7b) give us the following system

$$\left\{ \partial_t p_1^+ + \frac{1}{\epsilon} \partial_x p_1^+ = \frac{1}{2\epsilon^2} (p_1^- - p_1^+) + \frac{\alpha_1}{2} p_1, \right. \tag{10a}$$

$$\left. \partial_t p_1^- - \frac{1}{\epsilon} \partial_x p_1^- = \frac{1}{2\epsilon^2} (p_1^+ - p_1^-) + \frac{\alpha_1}{2} p_1, \right. \tag{10b}$$

which can be expressed as,

$$\epsilon^2 \partial_{tt} p_1 + (1 - \alpha_1 \epsilon^2) \partial_t p_1 = \partial_{xx} p_1 + \alpha_1 p_1.$$

This linear equation has been extensively studied (See for example [23, 29]). In particular, it has been shown that the associated critical propagation speed is given by the following formula

$$\left\{ \begin{array}{ll} \frac{2\sqrt{\alpha_1}}{1 + \epsilon^2\alpha_1}, & \text{if } \epsilon \le 1/\sqrt{\alpha_1}, \quad \text{(Parabolic regime)} \\ \frac{1}{\epsilon}, & \text{otherwise.} \quad \text{(Hyperbolic regime)} \end{array} \right. \tag{11}$$
In the parabolic regime, the speed is a perturbation of the critical speed of the Fisher-KPP parabolic equation. While, for the hyperbolic regime, the critical speed corresponds to the maximum possible speed of the wave, which is equal to the coefficient of the transport term.

In the moving frame, $\xi = x - st$, we seek solutions in the exponential form $e^{\lambda t + \nu \xi} \overline{p_1}$, we obtain the following dispersion relation,

$$D_s^{(1)}(\nu, \lambda) = \nu^2 + \nu \frac{s}{1 - \epsilon^2 s^2} (1 - \alpha_1 \epsilon^2 - 2\lambda\epsilon^2) + \frac{1}{1 - \epsilon^2 s^2} [\alpha_1 - \epsilon^2 \lambda^2 - (1 - \alpha_1 \epsilon^2)],$$

with the two following associated roots,

$$\nu_1^{\pm}(\lambda, s; \alpha_1, \epsilon) = \frac{s(1 - \alpha_1 \epsilon^2) + 2s\epsilon^2\lambda \mp \sqrt{s^2(1 + \alpha_1 \epsilon^2)^2 - 4\alpha_1 + 4\lambda(1 - \alpha_1 \epsilon^2 + \lambda\epsilon^2)}}{2(\epsilon^2 s^2 - 1)}. \quad (12)$$

The dispersion relation has a double root when $\nu_1^+ = \nu_1^-$. Since the speed of $p_1^\pm$ is given by the value $1/\epsilon$, we consider only the range of plausible invasion speeds $s \in [0, 1/\epsilon]$. By performing a Taylor expansion of the eigenvalues as $\lambda \to \infty$, we obtain the following relation

$$\nu_1^{\pm}(\lambda, s) = \frac{\lambda\epsilon}{(\epsilon s \pm 1)} + o(\lambda).$$

Consequently, we have the following limit, $\nu_1^{\pm}(\lambda, s) \to \pm\infty$ as $\lambda \to +\infty$. The pinching condition is therefore satisfied. The equality $\nu_1^+ = \nu_1^-$ gives us

$$s_1^2(\lambda; \alpha_1, \epsilon) = \frac{4\alpha_1}{(1 + \alpha_1 \epsilon^2)^2} - \frac{4\lambda(1 - \alpha_1 \epsilon^2 + \lambda\epsilon^2)}{(1 + \alpha_1 \epsilon^2)^2}. \quad (13)$$

To determine the linear propagation speed, we need to minimize the function $\Lambda$ defined by

$$\Lambda(\lambda; \alpha_1, \epsilon) := \frac{4\lambda(1 - \alpha_1 \epsilon^2 + \lambda\epsilon^2)}{(1 + \alpha_1 \epsilon^2)^2}.$$

* When $\epsilon \le 1/\sqrt{\alpha_1}$, the minimum of the function $\Lambda$ is reached at $\lambda = 0$. In this case, we indeed obtain the parabolic propagation speed.

$$s_1 = s_{\text{para}} := \frac{2\sqrt{\alpha_1}}{1 + \epsilon^2 \alpha_1},$$

already demonstrated by Hadeler [23] and Fedotov [25] using completely different methods.

* When $\epsilon > 1/\sqrt{\alpha_1}$, the minimum of the function $\Lambda$ is reached at $\lambda = \lambda^*$, with $\lambda^* := -(1 - \alpha_1 \epsilon^2)/(2\epsilon^2)$. Substituting this into equation (13), we obtain the hyperbolic propagation speed $s_1 = s_{\text{hyp}} := 1/\epsilon$. In this case, the associated $\nu^*$ is null. Contrary to the pinched double root of the parabolic case, the one of the hyperbolic case is not simple relatively to $\lambda$.

Consequently, the concept of pinched double roots allows us to easily obtain the critical linear speed of the system (10a)-(10b), which was already demonstrated in the article [29] using a different method.
## A.2 Derivation of anomalous speeds

To determine a potential anomalous speed, we now consider equations (7c)-(7d). Following a similar approach as in the previous analysis, we can derive the pair of eigenvalues associated with the system of two equations (7c)-(7d). These eigenvalues, denoted as $\nu_2^-$ and $\nu_2^+$, are defined by

$$ \nu_2^{\pm} = \frac{s(1 + 2\epsilon^2\tau k) + 2\epsilon^2\tau\lambda s \mp \sqrt{4\lambda\tau v^2 (1 + 2\epsilon^2\tau k + \epsilon^2\tau\lambda) + s^2 + 4k\tau v^2(1 + \epsilon^2\tau k)}}{2\tau(\epsilon^2 s^2 - v^2)}. \quad (14) $$

The dispersion relation for the complete linear system (7a)-(7d), due to the triangular structure of the problem, is given by the following formula:

$$ D_s(\nu, \lambda) := (\nu - \nu_1^+(\lambda, s))(\nu - \nu_1^-(\lambda, s))(\nu - \nu_2^+(\lambda, s))(\nu - \nu_2^-(\lambda, s)). $$

This dispersion relation has double roots if one of the equalities $\nu_1^+ = \nu_1^-$, $\nu_2^+ = \nu_2^-$, $\nu_1^+ = \nu_2^-$ and $\nu_1^- = \nu_2^+$ is satisfied. As previously shown, the double root when $\nu_1^+ = \nu_1^-$ is pinched and leads to parabolic and hyperbolic speeds. The equality $\nu_2^+ = \nu_2^-$ is associated with a complex propagation speed. Let us now consider the equality $\nu_1^- = \nu_2^+$. When $\nu_2^+$ is real, its sign is always positive according to definition (14). Therefore, we do not consider this equality. Finally, we examine the last double root $\nu_1^+ = \nu_2^-$. This time, analyzing the signs of the eigenvalues shows that consistent triplets $(\nu, s, \lambda)$ can be obtained. As we have seen before, $\nu_1^+$ tends to $+\infty$ as $\lambda \to +\infty$. Conversely, the expansion of $\nu_2^-$ for $\lambda \to +\infty$ is given by the following relation

$$ \nu_2^-(\lambda, s) = \frac{\lambda\epsilon}{\epsilon s - v} + o(\lambda). $$

Consequently, when $s < v/\epsilon$, we have the limit $\nu_2^- \to -\infty$ as $\lambda \to +\infty$. Here, the possible wave propagation speeds, $s$, lie between 0 and $\max(1/\epsilon, v/\epsilon)$. Therefore, it is possible to have $s > v/\epsilon$ when $v < 1$. In this case, the root does not satisfy the pinching condition, and no anomalous speed is expected.

The anomalous speed associated with the equality $\nu_1^+ = \nu_2^-$ is a solution of the following polynomial equation,

$$ a_4 s^4 + a_2(\lambda)s^2 + a_0(\lambda) = 0, \quad (15) $$

with

$$ a_0(\lambda) := ((\lambda + k) [1 + \epsilon^2\tau(\lambda + k)] - \tau v^2(\lambda - \alpha_1) [1 + \epsilon^2\lambda])^2, \quad (16) $$

$$ a_2(\lambda) := [(1 - \alpha_1\epsilon^2)(1 + 2\epsilon^2\tau k) + 2\alpha_1\tau\epsilon^2 - 2k\epsilon^2(1 + \epsilon^2\tau k)] \\ \{ (\lambda + k) [1 + \epsilon^2\tau(\lambda + k)] + \tau v^2(\lambda - \alpha_1) [1 + \epsilon^2\lambda] \} \\ - \tau v^2(1 + \alpha_1\epsilon^2)^2(\lambda + k) [1 + \epsilon^2\tau(\lambda + k)] - (\lambda - \alpha_1)(1 + \lambda\epsilon^2)s, \quad (17) $$

and,

$$ a_4 := \epsilon^2(\alpha_1 + k)(k\epsilon^2 - 1)(\tau[k\epsilon^2 - 1] + 1)(1 + \epsilon^2k\tau + \alpha_1\epsilon^2\tau). \quad (18) $$

Solving equation (15) gives us two possible real positive speeds

$$ s^{\pm}(\lambda) := \sqrt{\frac{-a_2(\lambda) \pm \sqrt{a_2^2(\lambda) - 4a_0(\lambda)a_4}}{2a_4}}. \quad (19) $$
We define $\Gamma^{\pm}$ as the sets given by

$$ \Gamma^{\pm} = \{ \lambda \in \mathbb{R}^{+} : \nu(\lambda, s^{\pm}(\lambda)) \text{ is a pinched double root of } D_s \text{ with admissible } (s^{\pm}, \nu) \}. $$

**Conjecture 1.** If $\Gamma^+ \cup \Gamma^- \neq \emptyset$, then $0 \in \Gamma^+ \cup \Gamma^-$, and we have the following results:

- If $0 \in \Gamma^+$, then

$$ s^{+}(0) \geq \max \left( \max_{\lambda \in \Gamma^{+}} s^{+}(\lambda), \max_{\lambda \in \Gamma^{-}} s^{-}(\lambda) \right). $$

- If $0 \in \Gamma^-$, then

$$ s^{-}(0) \geq \max \left( \max_{\lambda \in \Gamma^{+}} s^{+}(\lambda), \max_{\lambda \in \Gamma^{-}} s^{-}(\lambda) \right). $$

In other words, when there exists a family of admissible pinched double roots associated with the equality $\nu_1^+ = \nu_2^-$ parameterized by $\lambda$, the maximum associated speed is given by the one at $\lambda = 0$.

The high complexity of the polynomial coefficients (16), (17), and (18) makes proving such a result challenging. To ensure its validity, we verified that this result holds true for $10^{15}$ different parameter values, uniformly distributed in the parameter space $(\alpha_1, \epsilon, \tau, v, k) \in [0.05, 10]^5$. For each of these tests, the maximum admissible speed was indeed located at $\lambda = 0$.

Under the assumption of the conjecture 1, we thus define the two anomalous speeds of our problem given by

$$ s_{\text{anom}}^{\pm} := s^{\pm}(0) = \sqrt{\frac{-a_2(0) \pm \sqrt{a_2^2(0) - 4a_0(0)a_4}}{2a_4}}. \qquad (20) $$

Unlike speeds for the parabolic and hyperbolic regimes, these anomalous speeds depend on the parameters $v$, $\tau$, and $k$. When $\epsilon = 0$, the term $a_4$ becomes zero, thus there is only one anomalous speed, which is given by the formula,

$$ s_{\text{anom}}^{\pm}(\epsilon = 0) = \alpha_1 \frac{\sqrt{\theta - 1}}{\sqrt{k + \alpha_1}} + \frac{\sqrt{\alpha_1 + k}}{\sqrt{\theta - 1}}, $$

with $\theta = \tau v^2$. This equality is in good agreement with the result obtained in the article [36].

To conclude, we obtain the following result:

**Proposition 1** (Linear propagation speed for $n = 2$). We assume conjecture 1. By definition 1, the critical linear propagation speed, $s_{\text{lin}}^{*}$, can take one of four possible values:

* The two normal speeds: parabolic and hyperbolic

$$ s_{\text{para}} := \frac{2\sqrt{\alpha_1}}{1 + \alpha_1\epsilon^2}, \qquad s_{\text{hyp}} := \frac{1}{\epsilon}. $$

* The two anomalous speeds:

$$ s_{\text{anom}}^{\pm} = s^{\pm}(0) = \sqrt{\frac{-a_2(0) \pm \sqrt{a_2^2(0) - 4a_0(0)a_4}}{2a_4}}, $$

with $a_0$, $a_2$, and $a_4$ defined by equations (16), (17), and (18), respectively.
### A.3 Numerical simulations

In this section, we focus on the selected wave speed for a nonlinear model sharing the same linear part of the model (6). More precisely, we introduce a nonlinear model without certain nonlinear terms such as coagulation, in order to have fronts always linearly determined. This allows us to numerically verify whether the linear speed predicted by Proposition 1 is indeed the one selected by a nonlinear model in the pulled front regime.

We consider the following nonlinear model,

$$ \partial_t p_1^+ + \frac{1}{\epsilon} \partial_x p_1^+ = \frac{1}{2\epsilon^2} (p_1^- - p_1^+) + \frac{k}{2} p_2(1-p_1) + \frac{\alpha_1}{2} p_1(1-p_1), \quad (21a) $$

$$ \partial_t p_1^- - \frac{1}{\epsilon} \partial_x p_1^- = \frac{1}{2\epsilon^2} (p_1^+ - p_1^-) + \frac{k}{2} p_2(1-p_1) + \frac{\alpha_1}{2} p_1(1-p_1), \quad (21b) $$

$$ \partial_t p_2^+ + \frac{v}{\epsilon} \partial_x p_2^+ = \frac{1}{2\tau\epsilon^2} (p_2^- - p_2^+) - k p_2^+, \quad (21c) $$

$$ \partial_t p_2^- - \frac{v}{\epsilon} \partial_x p_2^- = \frac{1}{2\tau\epsilon^2} (p_2^+ - p_2^-) - k p_2^-. \quad (21d) $$

corresponding to an hyperbolic extension of the model presented in the article [36].

In Figure 9, we plot the selected speed of the model (21a)-(21d) using numerical simulations (black points) and the theoretical linear speeds presented in Proposition 1 (colored lines) for various values of the parameter $\epsilon$. We observe that the numerical selected speed indeed corresponds to the predicted linear speed. When $\epsilon$ is small enough, the linear speed matches the parabolic speed (red line). As $\epsilon$ increases, the selected speed first becomes the minus anomalous speed (cyan line) and then the plus anomalous speed (magenta line). For other simulations not presented in this article, with different parameter values, we have also observed the possibility to select the hyperbolic speed (green line).

For the parabolic and anomalous regimes, the linear propagation speed is determined by a simple pinched double root at $\lambda = 0$. In these regimes, our numerical simulations show that the modified nonlinear model indeed selects the marginally stable front. For these initial steep conditions, this scenario is expected and corresponds to the marginal stability conjecture. (see [33, 88, 89, 34] for further details). This conjecture was recently proven in the context of higher-order scalar parabolic equations by Avery and Scheel in [90], and later extended to reaction-diffusion systems by Avery [91].

For the hyperbolic regime, the linear propagation speed is induced by a double pinched double root at $\lambda = \lambda^* > 0$. This type of situation is more complex and has been studied, for instance, in the articles [34, 92, 93]. In this situation, it is possible to overestimate pointwise growth rates and fail to correctly determine the marginally stable front [34, Remark 4.5]. According to our numerical simulations, we observe that this speed can indeed be selected by the model (21a)-(21d). For example, under the assumption $v < 1$ and $\epsilon > 1/\sqrt{\alpha_1}$, the hyperbolic speed is always selected.
<figure><img src="image_11.png" /><figcaption>Figure 9: Comparison between the numerical selected speed of the modified model (21a)-(21d) and various linear speeds for different values of $\epsilon$. The numerical selected speed of the model (21a)-(21d) is represented by black points. The theoretical linear speeds are depicted by solid colored lines: red (parabolic), green (hyperbolic), cyan/magenta (anomalous minus/plus). The selected speed of the model (21a)-(21d) aligns with the linear prediction of Proposition 1. The parameter values are given by: $k = 2$, $\alpha_1 = 2$, $v = 1.5$, $\tau = 1$.</figcaption></figure>

## B Derivation of the linear propagation speed for $n > 2$

**Linear propagation speed** Linearizing around the zero equilibrium point, we obtain the following linear system

$$ \left\{ \begin{aligned} \partial_t p_1^{\pm} \pm \frac{1}{\epsilon} \partial_x p_1^{\pm} &= \pm \frac{1}{2\epsilon^2} (p_1^{-} - p_1^{+}) + \sum_{j=2}^{n} \beta(j) \eta(j, 1) p_j + \frac{\alpha_1}{2} p_1, \\ \partial_t p_i^{\pm} \pm \frac{v(i)}{\epsilon} \partial_x p_i^{\pm} &= \pm \frac{1}{2\tau(i)\epsilon^2} (p_i^{-} - p_i^{+}) + \sum_{j=i+1}^{n} \beta(j) \eta(j, i) p_j - \beta(i) p_i^{\pm}, \end{aligned} \right. \quad (22) $$

for all $i \in \{2, \dots, n\}$.

The derivation of the linear propagation speed is similar to the case $n = 2$. Due to the triangular structure of the system (22), the dispersion relation is given by

$$ D_s(\nu, \lambda) = \prod_{i=1}^{n} (\nu - \nu_i^+(\lambda, s)) (\nu - \nu_i^-(\lambda, s)), \quad (23) $$

where $\nu_i^{\pm}$ corresponds to the pair of eigenvalues associated with the two equations for clusters of size $i$, independent of all the other equations, i.e., under the assumption $\beta(j) = 0$ for all $j > i$. Consequently, $\nu_1^{\pm}$ are the same as previously, explicitly defined by equations (12). The eigenvalues $\nu_i^{\pm}$, for all $i \in \{2, \dots, n\}$, are defined by equation (14) in the case $k = \beta(i)$, $v = v(i)$, and $\tau = \tau(i)$.
The pinched double roots of (23) associated with an admissible speed and slope can be obtained when $\nu_1^+ = \nu_1^-$ or when one of the equalities $\nu_1^+ = \nu_i^-$ with $i \in \{2, \dots, n\}$ is satisfied. In the first case, the associated speeds are the two normal speeds, while the second case leads to anomalous propagation speeds. For each cluster size $i \in \{2, \dots, n\}$, we assume the equivalent of conjecture 1, allowing us to define the two anomalous speeds for each cluster size.

$$s_{\text{anom},i}^{\pm} := s_i^{\pm}(0) = \sqrt{\frac{-a_{2,i}(0) \pm \sqrt{a_{2,i}^2(0) - 4a_{0,i}(0)a_{4,i}}}{2a_{4,i}}}, \quad \text{for } i \in \{2, \dots, n\}. \quad (24)$$

where the coefficients $a_{0,i}$, $a_{2,i}$, and $a_{4,i}$ are calculated in the same way as the coefficients $a_0$, $a_2$, and $a_4$, defined respectively by equations (16), (17), and (18).

We thus obtain the following proposition:

**Proposition 2** (Critical linear propagation speed for $n > 2$). We assume conjecture 1 for the different cluster sizes. By definition 1, the critical linear propagation speed, $s_{\text{lin}}^{*}$, can take the following possible values:

* The two normal speeds: parabolic and hyperbolic

$$s_{\text{para}} := \frac{2\sqrt{\alpha_1}}{1 + \alpha_1 \epsilon^2}, \quad s_{\text{hyp}} := \frac{1}{\epsilon}.$$

* The different anomalous speeds: $s_{\text{anom},i}^{\pm}$ for all $i \in \{2, \dots, n\}$.

As before, the linear speed selected among these possible linear speeds corresponds to the maximum of the admissible speeds.

**Numerical simulations** As previously, we introduce a nonlinear system that shares the same linear system (22), but whose nonlinear terms are chosen to have no impact on the wave speed,

$$\left\{ \begin{aligned} \partial_t p_1^{\pm} \pm \frac{1}{\epsilon} \partial_x p_1^{\pm} &= \pm \frac{1}{2\epsilon^2} (p_1^{-} - p_1^{+}) + \sum_{j=2}^{n} \beta(j) \eta(j, 1) p_j + \frac{\alpha_1}{2} p_1(1-p), \\ \partial_t p_i^{\pm} \pm \frac{v(i)}{\epsilon} \partial_x p_i^{\pm} &= \pm \frac{1}{2\tau(i)\epsilon^2} (p_i^{-} - p_i^{+}) + \sum_{j=i+1}^{n} \beta(j) \eta(j, i) p_j - \beta(i) p_i^{\pm}, \end{aligned} \right. \quad (25)$$

for all $i \in \{2, \dots, n\}$.

Case $n = 3$. In the first step, we assume that $n = 3$ to simplify the problem. There are six admissible linear speeds: the parabolic speed, the hyperbolic speed, the two anomalous speeds associated with clusters of size 2, and finally the two anomalous speeds associated with clusters of size 3.

In Figure 10, we plot the selected speed of the modified model (25) using numerical simulations (black points), as well as the admissible theoretical normal and anomalous linear speeds (colored lines) for different parameter values of $v(3)$. The anomalous speed associated with clusters of size 2, $s_{\text{anom},2}^{\pm}$, is represented in blue while the anomalous speed associated with cluster
of size 3, $s_{\text{anom},3}^{\pm}$, is in yellow. Again, for the numerical simulations, we consider only steep initial data. For this choice of parameter values, the hyperbolic speed is not admissible, and the only normal linear speed is the parabolic speed, represented by the red curve. To illustrate the phenomenon of anomalous speed selection, presented in Definition 1 and Proposition 2, we vary the value of $v(3)$ so that the admissible anomalous speed associated with clusters of three bacteria becomes greater than the admissible anomalous speed associated with clusters of two bacteria. As we can see in Figure 10, the selected speed will transition from the anomalous speed associated with clusters of two bacteria to the anomalous speed associated with clusters of size three.

<figure><img src="image_12.png" /><figcaption>Figure 10: For $n = 3$, comparison between the numerical selected speed of the modified model (25) and various linear speeds for different values of $v(3)$. The black dots correspond to the numerical selected speed of the system (25). The red curve represents the parabolic speed, the blue curve the anomalous speed associated with cluster of size 2, $s_{\text{anom},2}^{\pm}$, and finally, the yellow curve corresponds to the anomalous speed associated with cluster of size 3, $s_{\text{anom},3}^{\pm}$. For this parameter regime, the hyperbolic speed is not admissible. The parameter values are given by $\beta = 1$, $\alpha_1 = 1$, $\epsilon = 0.9$, and the functions $\tau$ and $v$ are given by $\tau(2) = \tau(3) = 1$ and $v(2) = 2$.</figcaption></figure>

Case $n > 3$. We also perform numerical simulations in the case $n > 3$. In Figure 11, we plot the numerical selected speed of the modified model (25) using black dots, as well as the theoretical linear speeds for different values of the parameter $\epsilon$ and for two different cases, $n = 8$ and $n = 12$. When $\epsilon$ is small enough, the linear speed indeed corresponds to the parabolic speed, represented by the red curve. For both figures, the results are similar. When $\epsilon$ becomes large enough, the numerical speed becomes the largest of the admissible anomalous speeds represented by the blue curve. We also illustrate the hyperbolic speed with the green curve, even though it is never selected for these parameter values. These numerical results are in agreement with the
Proposition 2.

<figure><img src="image_13.png" /><figcaption>Figure 11: Comparison between the numerical selected speed of the modified model (25) and the various possible linear speeds for different values of $\epsilon$, and for: Top, $n=8$; Bottom, $n=12$. The black dots represent the numerical selected speed of the system (25). The red (resp. green) curve corresponds to the parabolic (resp. hyperbolic) linear speed. The largest of the admissible anomalous speeds is represented by the blue curve. The numerical simulations in the two figures are in agreement with Proposition 2. The values of the other parameters are $\beta = 1$, $\alpha_1 = 1$, and the functions $\tau$ and $v$ are given by $\tau(i) = 1$ and (Top) $v(i) = 1 + 1.3 \times \mathbf{1}_{i \le 4}(i)$, (Bottom) $v(i) = 1 + 1.6 \times \mathbf{1}_{i \le 6}(i)$ for all $i \in \{2, \dots, n\}$.</figcaption></figure>
References

[1] Laura M. Faure et al. "The mechanism of force transmission at bacterial focal adhesion complexes". In: *Nature* 539.7630 (2016). DOI: 10.1038/nature20121.

[2] Romain Mercier et al. "The polar Ras-like GTPase MglA activates type IV pilus via SgmX to enable twitching motility in *Myxococcus xanthus*". In: *Proceedings of the National Academy of Sciences* 117.45 (2020). DOI: 10.1073/pnas.2002783117.

[3] Jonathan Hodgkin and Dale Kaiser. "Genetics of gliding motility in Myxococcus xanthus (Myxobacterales): Two gene systems control movement". In: *Molecular and General Genetics* MGG 171.2 (1979). DOI: 10.1007/BF00270004.

[4] Eugene Rosenberg and Mazal Varon. "Antibiotics and Lytic Enzymes". In: *Myxobacteria*. Ed. by Alexander Rich and Engene Rosenberg. Series Title: Springer Series in Molecular Biology. New York, NY: Springer New York, 1984. DOI: 10.1007/978-1-4613-8280-5_5.

[5] Andrew D. Morgan et al. "Comparative Analysis of *Myxococcus* Predation on Soil Bacteria". In: *Applied and Environmental Microbiology* 76.20 (2010). DOI: 10.1128/AEM.00414-10.

[6] José Muñoz-Dorado et al. "Myxobacteria: Moving, Killing, Feeding, and Surviving Together". In: *Frontiers in Microbiology* 7 (2016). DOI: 10.3389/fmicb.2016.00781.

[7] James E. Berleman et al. "Predataxis behavior in *Myxococcus xanthus*". In: *Proceedings of the National Academy of Sciences* 105.44 (2008). DOI: 10.1073/pnas.0804387105.

[8] Juana Pérez et al. "Rhizobial galactoglucan determines the predatory pattern of Myxococcus xanthus and protects Sinorhizobium meliloti from predation". In: *Environmental Microbiology* 16.7 (2014). DOI: 10.1111/1462-2920.12477.

[9] Susanne Thiery and Christine Kaimer. "The Predation Strategy of Myxococcus xanthus". In: *Frontiers in Microbiology* 11 (2020). DOI: 10.3389/fmicb.2020.00002.

[10] Susanne Thiery et al. "The predatory soil bacterium Myxococcus xanthus combines a Tad- and an atypical type 3-like protein secretion system to kill bacterial cells". In: *Cell Reports* 40.11 (2022). DOI: 10.1016/j CELREP.2022.111340.

[11] Wenchao Zhang et al. "Dynamics of Solitary Predation by Myxococcus xanthus on Escherichia coli Observed at the Single-Cell Level". In: *Applied and Environmental Microbiology* 86.3 (2020). Ed. by Hideaki Nojiri. DOI: 10.1128/AEM.02286-19.

[12] Sofiene Seef et al. "A Tad-like apparatus is required for contact-dependent prey killing in predatory social bacteria". In: *eLife* 10 (2021). DOI: 10.7554/eLife.72409.

[13] E Rosenberg, K H Keller, and M Dworkin. "Cell density-dependent growth of Myxococcus xanthus on casein". In: *Journal of Bacteriology* 129.2 (1977). DOI: 10.1128/jb.129.2.770-777.1977.

[14] B D Blackhart and D R Zusman. ""Frizzy" genes of Myxococcus xanthus are involved in control of frequency of reversal of gliding motility." In: *Proceedings of the National Academy of Sciences* 82.24 (1985). DOI: 10.1073/pnas.82.24.8767.

[15] M Dworkin. "Tactic behavior of Myxococcus xanthus". In: *Journal of Bacteriology* 154.1 (1983). DOI: 10.1128/jb.154.1.452-459.1983.
[16] Romain Mercier and Tâm Mignot. "Regulations governing the multicellular lifestyle of Myxococcus xanthus". In: *Current Opinion in Microbiology* 34 (2016). DOI: 10.1016/j.mib.2016.08.009.

[17] Martin Dworkin and David Eide. "*Myxococcus xanthus* Does Not Respond Chemotactically to Moderate Concentration Gradients". In: *Journal of Bacteriology* 154.1 (1983). DOI: 10.1128/jb.154.1.437-442.1983.

[18] Hélène Bloch et al. "A new modeling approach of *Myxococcus xanthus* bacteria using polarity-based reversals". In: *ESAIM: Proceedings and Surveys* 77 (2024). Ed. by Emmanuel Franck et al. DOI: 10.1051/proc/202477025.

[19] Vincent Calvez et al. "Regime switching on the propagation speed of travelling waves of some size-structured Myxobacteriapopulation models". In: *In preparation* (2023).

[20] Maxime Estavoyer and Thomas Lepoutre. "Travelling waves for a fast reaction limit of a discrete coagulation-fragmentation model with diffusion and proliferation". In: *Journal of Mathematical Biology* 89.1 (2024). DOI: 10.1007/s00285-024-02099-4.

[21] Angela Gallegos, Barbara Mazzag, and Alex Mogilner. "Two continuum models for the spreading of myxobacteria swarms". In: *Bulletin of Mathematical Biology* 68.4 (2006). DOI: 10.1007/s11538-005-9031-2.

[22] Pintu Patra et al. "Colony Expansion of Socially Motile Myxococcus xanthus Cells Is Driven by Growth, Motility, and Exopolysaccharide Production". In: *PLOS Computational Biology* 12.6 (2016). Ed. by Christopher V. Rao. DOI: 10.1371/journal.pcbi.1005010.

[23] K. P. Hadeler. "Hyperbolic travelling fronts". In: *Proceedings of the Edinburgh Mathematical Society* 31.1 (1988). DOI: 10.1017/S001309150000660X.

[24] Vicenç Méndez and Juan Camacho. "Dynamics and thermodynamics of delayed population growth". In: *Physical Review E* 55.6 (1997). DOI: 10.1103/PhysRevE.55.6476.

[25] Sergei Fedotov. *Traveling waves in reaction-diffusion system*. arXiv:cond-mat/9807352. 1998.

[26] K. P. Hadeler. "Reaction transport systems in biological modelling". In: *Mathematics In-spired by Biology*. Ed. by Vincenzo Capasso. Vol. 1714. Series Title: Lecture Notes in Math-ematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 1999. DOI: 10.1007/BFb0092376.

[27] Vicenç Méndez, Joaquim Fort, and Jordi Farjas. "Speed of wave-front solutions to hy-perbolic reaction-diffusion equations". In: *Physical Review E* 60.5 (1999). DOI: 10.1103/PhysRevE.60.5231.

[28] B.H. Gilding and R. Kersner. "Wavefront solutions of a nonlinear telegraph equation". In: *Journal of Differential Equations* 254.2 (2013). DOI: 10.1016/j.jde.2012.09.007.

[29] Emeric Bouin, Vincent Calvez, and Grégoire Nadin. "Hyperbolic traveling waves driven by growth". In: *Mathematical Models and Methods in Applied Sciences* 24.06 (2014). arXiv:1110.3242 [math]. DOI: 10.1142/S0218202513500802.

[30] Vicenç Méndez, Daniel Campos, and Werner Horsthemke. "Growth and dispersal with inertia: Hyperbolic reaction-transport systems". In: *Physical Review E* 90.4 (2014). DOI: 10.1103/PhysRevE.90.042114.
[31] E. Barbera, C. Currò, and G. Valenti. "On discontinuous travelling wave solutions for a class of hyperbolic reaction-diffusion models". In: *Physica D: Nonlinear Phenomena* 308 (2015). DOI: 10.1016/j.physd.2015.06.011.

[32] Elvira Barbera and Giovanna Valenti. "Wave features of a hyperbolic reaction-diffusion model for Chemotaxis". In: *Wave Motion* 78 (2018). DOI: 10.1016/j.wavemoti.2018.02.004.

[33] G. Dee and J. S. Langer. "Propagating Pattern Selection". In: *Physical Review Letters* 50.6 (1983). DOI: 10.1103/PhysRevLett.50.383.

[34] Matt Holzer and Arnd Scheel. "Criteria for Pointwise Growth and Their Role in Invasion Processes". In: *Journal of Nonlinear Science* 24.4 (2014). DOI: 10.1007/s00332-014-9202-0.

[35] Hans F. Weinberger, Mark A. Lewis, and Bingtuan Li. "Anomalous spreading speeds of cooperative recursion systems". In: *Journal of Mathematical Biology* 55.2 (2007). DOI: 10.1007/s00285-007-0078-6.

[36] Matt Holzer. "Anomalous spreading in a system of coupled Fisher-KPP equations". In: *Physica D: Nonlinear Phenomena* 270 (2014). DOI: 10.1016/j.physd.2013.12.003.

[37] W Van Saarloos. "Front propagation into unstable states". In: *Physics Reports* 386.2-6 (2003). DOI: 10.1016/j.physrep.2003.08.001.

[38] R. A. Fisher. "The wave of advance of advantageous genes". In: *Annals of Eugenics* 7.4 (1937). DOI: 10.1111/j.1469-1809.1937.tb02153.x.

[39] Andrei Kolmogorov. "Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique". In: *Moscow Univ. Bull. Math.* 1 (1937).

[40] Akira Okubo. "Diffusion and ecological problems: mathematical models". In: *Biomath* 10 (1980). Publisher: Springer-Verlag.

[41] Hans G. Othmer and S. Levin, eds. *Nonlinear Oscillations in Biology and Chemistry*. Vol. 66. Lecture Notes in Biomathematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. DOI: 10.1007/978-3-642-93318-9.

[42] Elizabeth Eli Holmes. "Are Diffusion Models too Simple? A Comparison with Telegraph Models of Invasion". In: *The American Naturalist* 142.5 (1993). DOI: 10.1086/285572.

[43] Thomas Hillen. "Hyperbolic models for chemosensitive movement". In: *Mathematical Models and Methods in Applied Sciences* 12.07 (2002). DOI: 10.1142/S0218202502002008.

[44] José A. Carrillo et al. "Double milling in self-propelled swarms from kinetic theory". In: *Kinetic & Related Models* 2.2 (2009). DOI: 10.3934/krm.2009.2.363.

[45] José A. Carrillo, Raluca Eftimie, and Franca K. O. Hoffmann. *Non-local kinetic and macroscopic models for self-organised animal aggregations*. Version Number: 1. 2014. DOI: 10.48550/ARXIV.1407.2099.

[46] Lutscher and Stevens. "Emerging Patterns in a Hyperbolic Model for Locally Interacting Cell Systems". In: *Journal of Nonlinear Science* 12.6 (2003). DOI: 10.1007/s00332-002-0510-4.
[47] T. Hillen and A. Stevens. "Hyperbolic models for chemotaxis in 1-D". In: *Nonlinear Analysis: Real World Applications* 1.3 (2000). DOI: 10.1016/S0362-546X(99)00284-9.

[48] J. Saragosti et al. "Directional persistence of chemotactic bacteria in a traveling concentration wave". In: *Proceedings of the National Academy of Sciences* 108.39 (2011). DOI: 10.1073/pnas.1101996108.

[49] Angelika Manhart. "Counter-propagating wave patterns in a swarm model with memory". In: *Journal of Mathematical Biology* 78.3 (2019). DOI: 10.1007/s00285-018-1287-x.

[50] Mark Kac. "A stochastic model related to the telegrapher's equation". In: *The Rocky Mountain Journal of Mathematics* 4.3 (1974). Publisher: JSTOR.

[51] David J. Aldous. "Deterministic and Stochastic Models for Coalescence (Aggregation and Coagulation): A Review of the Mean-Field Theory for Probabilists". In: *Bernoulli* 5.1 (1999). DOI: 10.2307/3318611.

[52] F. P. Da Costa. "Mathematical Aspects of Coagulation-Fragmentation Equations". In: *Mathematics of Energy and Climate Change*. Ed. by Jean-Pierre Bourguignon et al. Vol. 2. Series Title: CIM Series in Mathematical Sciences. Cham: Springer International Publishing, 2015. DOI: 10.1007/978-3-319-16121-1_5.

[53] Sara Rombouts et al. "Multi-scale dynamic imaging reveals that cooperative motility behaviors promote efficient predation in bacteria". In: *Nature Communications* 14.1 (2023). DOI: 10.1038/s41467-023-41193-x.

[54] Matt Holzer. "A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations". In: (2014). Publisher: arXiv Version Number: 2. DOI: 10.48550/ARXIV.1409.8641.

[55] Grégory Faye and Gwenaël Peltier. "Anomalous invasion speed in a system of coupled reaction-diffusion equations". In: *Communications in Mathematical Sciences* 16.2 (2018). DOI: 10.4310/CMS.2018.v16.n2.a7.

[56] Francis Filbet and S. Jin. "A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources". In: *Journal of Computational Physics* 229.20 (2010). arXiv:0905.1378 [math]. DOI: 10.1016/j.jcp.2010.06.017.

[57] Bruno Després and Frédéric Lagoutière. "Un schéma non linéaire anti-dissipatif pour l'équation d'advection linéaire". In: *Comptes Rendus de l'Académie des Sciences - Series I - Mathematics* 328.10 (1999). DOI: 10.1016/S0764-4442(99)80301-2.

[58] R.J Leveque and H.C Yee. "A study of numerical methods for hyperbolic conservation laws with stiff source terms". In: *Journal of Computational Physics* 86.1 (1990). DOI: 10.1016/0021-9991(90)90097-K.

[59] Corrado Lattanzio et al. *Analysis and numerics of the propagation speed for hyperbolic reaction-diffusion models*. arXiv:2206.09714 [cs, math]. 2022.

[60] Matt Holzer and Arnd Scheel. "A slow pushed front in a Lotka–Volterra competition model". In: *Nonlinearity* 25.7 (2012). DOI: 10.1088/0951-7715/25/7/2151.
[61] Alfred M. Spormann. "Gliding Motility in Bacteria: Insights from Studies of *Myxococcus xanthus*". In: *Microbiology and Molecular Biology Reviews* 63.3 (1999). DOI: 10.1128/MMBR.63.3.621-641.1999.

[62] John S. Mattick. "Type IV Pili and Twitching Motility". In: *Annual Review of Microbiology* 56.1 (2002). DOI: 10.1146/annurev.micro.56.012302.160938.

[63] Dale Kaiser and Cathy Crosby. "Cell movement and its coordination in swarms of myxococcus xanthus". In: *Cell Motility* 3.3 (1983). DOI: 10.1002/cm.970030304.

[64] Montie Avery, Matt Holzer, and Arnd Scheel. "Pushed-to-Pulled Front Transitions: Continuation, Speed Scalings, and Hidden Monotonicty". In: *Journal of Nonlinear Science* 33.6 (2023). DOI: 10.1007/s00332-023-09957-3.

[65] Susanne Müller et al. "Bacillaene and Sporulation Protect Bacillus subtilis from Predation by Myxococcus xanthus". In: *Applied and Environmental Microbiology* 80.18 (2014). Ed. by A. M. Spormann. DOI: 10.1128/AEM.01621-14.

[66] Susanne Müller et al. "Predation by Myxococcus xanthus Induces Bacillus subtilis To Form Spore-Filled Megastructures". In: *Applied and Environmental Microbiology* 81.1 (2015). Ed. by M. A. Elliot. DOI: 10.1128/AEM.02448-14.

[67] William H. DePas et al. "Biofilm Formation Protects Escherichia coli against Killing by Caenorhabditis elegans and Myxococcus xanthus". In: *Applied and Environmental Microbiology* 80.22 (2014). Ed. by H. Nojiri. DOI: 10.1128/AEM.02464-14.

[68] Marie Vasse et al. "Killer prey: Ecology reverses bacterial predation". In: *PLOS Biology* 22.1 (2024). Ed. by Lotte Søgaard-Andersen. DOI: 10.1371/journal.pbio.3002454.

[69] L J Shimkets and D Kaiser. "Induction of coordinated movement of Myxococcus xanthus cells". In: *Journal of Bacteriology* 152.1 (1982). DOI: 10.1128/jb.152.1.451-461.1982.

[70] B Sager and D Kaiser. "Intercellular C-signaling and the traveling waves of Myxococcus." In: *Genes & Development* 8.23 (1994). DOI: 10.1101/gad.8.23.2793.

[71] James E. Berleman et al. "Rippling Is a Predatory Behavior in *Myxococcus xanthus*". In: *Journal of Bacteriology* 188.16 (2006). DOI: 10.1128/JB.00559-06.

[72] Oleksii Sliusarenko et al. "Accordion waves in *Myxococcus xanthus*". In: *Proceedings of the National Academy of Sciences* 103.5 (2006). DOI: 10.1073/pnas.0507720103.

[73] Dale Kaiser. "Signaling in myxobacteria". In: *Annual Review of Microbiology* 58.1 (2004). DOI: 10.1146/annurev.micro.58.030603.123620.

[74] K A O'Connor and D R Zusman. "Development in Myxococcus xanthus involves differentiation into two cell types, peripheral rods and spores". In: *Journal of Bacteriology* 173.11 (1991). DOI: 10.1128/jb.173.11.3318-3333.1991.

[75] Oleg A. Igoshin et al. "Pattern formation and traveling waves in myxobacteria: Theory and modeling". In: *Proceedings of the National Academy of Sciences* 98.26 (2001). DOI: 10.1073/pnas.221579598.

[76] Oleg A. Igoshin et al. "Waves and aggregation patterns in myxobacteria". In: *Proceedings of the National Academy of Sciences* 101.12 (2004). DOI: 10.1073/pnas.0400704101.
[77] R. Eftimie, G. De Vries, and M. A. Lewis. "Complex spatial group patterns result from different animal communication mechanisms". In: *Proceedings of the National Academy of Sciences* 104.17 (2007). DOI: 10.1073/pnas.0611483104.

[78] Arnd Scheel and Angela Stevens. "Wavenumber selection in coupled transport equations". In: *Journal of Mathematical Biology* 75.5 (2017). DOI: 10.1007/s00285-017-1107-8.

[79] Olga Sozinova et al. "A three-dimensional model of myxobacterial fruiting-body formation". In: *Proceedings of the National Academy of Sciences* 103.46 (2006). DOI: 10.1073/pnas.0605555103.

[80] Oleksii Sliusarenko, David R. Zusman, and George Oster. "Aggregation during Fruiting Body Formation in *Myxococcus xanthus* Is Driven by Reducing Cell Movement". In: *Journal of Bacteriology* 189.2 (2007). DOI: 10.1128/JB.01206-06.

[81] Raluca Eftimie. "Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review". In: *Journal of Mathematical Biology* 65.1 (2012). DOI: 10.1007/s00285-011-0452-2.

[82] Vicenç Méndez, Sergej A. Fedotov, and Werner Horsthemke. *Reaction-transport systems: mesoscopic foundations, fronts, and spatial instabilities*. Springer series in synergetics. Berlin Heidelberg: Springer, 2010.

[83] Evgeny P. Zemskov and Werner Horsthemke. "Diffusive instabilities in hyperbolic reaction-diffusion equations". In: *Physical Review E* 93.3 (2016). DOI: 10.1103/PhysRevE.93.032211.

[84] Giancarlo Consolo, Carmela Currò, and Giovanna Valenti. "Pattern formation and modulation in a hyperbolic vegetation model for semiarid environments". In: *Applied Mathematical Modelling* 43 (2017). DOI: 10.1016/j.apm.2016.11.031.

[85] Giancarlo Consolo et al. "Oscillatory periodic pattern dynamics in hyperbolic reaction-advection-diffusion models". In: *Physical Review E* 105.3 (2022). DOI: 10.1103/PhysRevE.105.034206.

[86] C. Currò and G. Valenti. "Pattern formation in hyperbolic models with cross-diffusion: Theory and applications". In: *Physica D: Nonlinear Phenomena* 418 (2021). DOI: 10.1016/j.physd.2021.132846.

[87] Joshua S. Ritchie, Andrew L. Krause, and Robert A. Van Gorder. "Turing and wave instabilities in hyperbolic reaction-diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation". In: *Annals of Physics* 444 (2022). DOI: 10.1016/j.aop.2022.169033.

[88] Wim Van Saarloos. "Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection". In: *Physical Review A* 37.1 (1988). DOI: 10.1103/PhysRevA.37.211.

[89] Wim Van Saarloos. "Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence". In: *Physical Review A* 39.12 (1989). DOI: 10.1103/PhysRevA.39.6367.
[90] Montie Avery and Arnd Scheel. “Universal selection of pulled fronts”. In: *Communications of the American Mathematical Society* 2.5 (2022). DOI: 10.1090/cams/8.

[91] Montie Avery. *Front selection in reaction-diffusion systems via diffusive normal forms*. arXiv:2211.11829 [nlin]. 2023.

[92] Grégory Faye, Matt Holzer, and Arnd Scheel. “Linear spreading speeds from nonlinear resonant interaction”. In: *Nonlinearity* 30.6 (2017). DOI: 10.1088/1361-6544/aa6c74.

[93] Montie Avery et al. “Instability in large bounded domains—branched versus unbranched resonances”. In: *Nonlinearity* 34.11 (2021). DOI: 10.1088/1361-6544/ac2a15.