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http://www.aimspress.com/journal/mbe
MBE, 21(4): 5536–5555.
DOI: 10.3934/mbe.2024244
Received: 05 January 2024
Revised: 28 February 2024
Accepted: 05 March 2024
Published: 27 March 2024
Research article
Mechanistic modeling of alarm signaling in seed-harvester ants
Michael R. Lin1, Xiaohui Guo2,*, Asma Azizi3, Jennifer H. Fewell4and Fabio Milner1,5
1Simon A. Levin Mathematical, Computational and Modeling Sciences Center, Arizona State
University, Tempe 85281, USA
2Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7632706,
Israel
3Department of Mathematics, Kennesaw State University, Marietta 30062, USA
4School of Life Sciences, Arizona State University, Tempe 85287, USA
5School of Mathematical and Statistical Sciences, Arizona State University, Tempe 85287, USA
*Correspondence: Email: xiaohui.guo@weizmann.ac.il.
Abstract: Ant colonies demonstrate a finely tuned alarm response to potential threats, offering a
uniquely manageable empirical setting for exploring adaptive information diffusion within groups. To
effectively address potential dangers, a social group must swiftly communicate the threat throughout
the collective while conserving energy in the event that the threat is unfounded. Through a combination
of modeling, simulation, and empirical observations of alarm spread and damping patterns, we
identified the behavioral rules governing this adaptive response. Experimental trials involving alarmed
ant workers (Pogonomyrmex californicus) released into a tranquil group of nestmates revealed a
consistent pattern of rapid alarm propagation followed by a comparatively extended decay period [1].
The experiments in [1] showed that individual ants exhibiting alarm behavior increased their movement
speed, with variations in response to alarm stimuli, particularly during the peak of the reaction. We
used the data in [1] to investigate whether these observed characteristics alone could account for the
swift mobility increase and gradual decay of alarm excitement. Our self-propelled particle model
incorporated a switch-like mechanism for ants’ response to alarm signals and individual variations in
the intensity of speed increased after encountering these signals. This study aligned with the established
hypothesis that individual ants possess cognitive abilities to process and disseminate information,
contributing to collective cognition within the colony (see [2] and the references therein). The elements
examined in this research support this hypothesis by reproducing statistical features of the empirical
speed distribution across various parameter values.
Keywords: alarm response; collective behavior; ant colony; agent-based model; information
dissemination
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1. Introduction
The “alarm response” commonly employed by social insect colonies, serves as a vivid metaphor for
a diverse range of prototypical infection processes, including rumor spread [3], social contagion [4,5],
and disease outbreaks [6]. In the case of harvester ants, for instance, a localized disturbance, such as a
single intruder entering the nest, can prompt workers to adopt a distinct behavioral alarm state. In this
state, they exhibit more animated movement patterns and increased contact rates with nestmates [7].
Workers often transmit this heightened state to nearby peers through direct contact and associated alarm
pheromones (Ketone 4-Methyl-3-heptanone) [8–11]. Under certain conditions, this process initiates an
alarm signaling cascade, resulting in a collective behavioral shift toward more turbulent motion—the
colony alarm response. At its peak, the alarm response can trigger a qualitative shift in colony behavior,
leading to attacks on invaders or the abandonment of the nest space. However, when the threat is
unfounded, the alarm response quickly subsides to pre-alarm activity levels [1]. The physical contacts
associated with the diffusion-modulated pattern of alarm information contribute to the amplification,
flow, and dampening of the alarm response, making it a valuable model for understanding transmission
and infection processes at various scales. Examples include infectious disease epidemics (see [12] and
the references therein), the spread of viral content in online networks [13], immune responses to targets
such as viruses or tumor cells [14–18], and biochemical cascades such as apoptosis [19], where local
interactions magnify a small stimulus until systemic changes occur.
In contrast to numerous similar cases, the response to colony alarms is highly manageable from
an empirical perspective, despite receiving limited attention regarding its practicality as a real-world
experimental context for investigating information cascades. In a laboratory setting, alarm behavior can
be consistently induced by introducing pre-alarmed ants to a tranquil colony, as documented by Guo
et al. [1]. This can be achieved by confining ants to a two-dimensional surface, allowing for individual
tracking through multi-object tracking software, such as ABCtracker [20]. The primary behavioral
change observed in individual ants during an alarm event is a rapid increase in speed, coupled with
heightened interactions with colony mates. By interpreting high speed as a proxy for alarm excitement,
we can analyze the propagation of an alarm stimulus through the group by studying its impact on
speed distribution.
Using the object tracking apparatus, spatially explicit agent-based models, such as self-propelled
particles, are highly effective for investigating group-level alarm responses. In self-propelled particle
models, collective motion is emulated by representing individuals as particles navigating through space
and engaging in interactions [21–25]. Each entity adjusts its speed and heading direction based on local
interactions with others [26]. These models offer utility by enabling the specification and assessment
of motion rules at the individual level, with subsequent comparisons between simulated group-level
outcomes and empirical observations.
In this study, we introduced artificially alarmed worker ants of the species P. californicus to
inactive colonies and monitored the response of each colony by analyzing its speed distribution. Upon
encountering an alarm signal, individual ants exhibited varying responses; some displayed subtle
changes in behavior, while others transitioned into an alarmed state characterized by increased contact
with nestmates and the transmission of alarm signals. This diversity in individual responses likely
plays a crucial role in shaping the colony’s ability to amplify or mitigate its overall response to
alarms. Colonies responding to alarm events exhibited distinctive increases in speed and associated
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speed variance, followed by a gradual decline in speeds, as outlined in previous research [1]. The
modulation of collective alarm responses by physical contact between alarmed and unalarmed
individuals contributes to the differentiation of genuine alarm events from false alarms. Consequently,
we inquired whether these specific characteristics of group dynamics can be explained by a simple set
of rules governing how individuals adjust their speeds.
Drawing on empirical observations, we developed a self-propelled particle model that incorporates
contact-based mechanisms for alarm signal transmission and speed modification at the individual
level. The model enables virtual ants to move at various speeds within a circular region. When an
alarmed ant encounters an unalarmed ant, the latter responds by 1) increasing its walking speed and
direction by explicitly defined amounts and 2) gaining the ability to transmit the alarmed state to
others. Our findings indicate that this minimal model effectively replicates the observed empirical
speed dynamics across a broad range of parameters. The strong alignment between our empirical and
modeling approaches can also contribute to a more comprehensive understanding, such as in the
context of infection and signal transduction within groups.
In the subsequent sections, we elucidate the self-propelled particle model, employing empirical
distributions derived from our alarm inducement experiments. Following that, we elaborate on the
materials and methods used in these experiments. Lastly, we substantiate the validity of our model by
comparing its outputs to the corresponding quantities observed in the experimental results.
2. Mathematical model
We employed the self-propelled particle modeling framework to elucidate how ants react to alarm
signals in relation to changes in velocity. Additionally, we investigated whether these individual
responses can collectively represent an alarm response at the colony level. The model’s underlying
assumptions focused on three key aspects of the alarm transmission mechanism: 1) the influence of
receiving alarm information on movement, 2) variations in acceleration response among ants, and 3)
the process by which alarmed ants reduce their speed back to normal levels.
To address the initial inquiry, we categorized ants into a binary state: either “alarmed” or 1
“unalarmed.” When unalarmed ants come into close contact with alarmed nestmates, more than 80%
of them transition to an alarmed state and simultaneously increase their speed within 1.51 seconds, as
noted by Guo et al. [1]. This finding implies that an immediate speed surge yields more realistic
outcomes than allowing ants to accumulate speed gradually over multiple contacts and signals.
Regarding the second query, we posited that each ant can enhance its speed by an individual-specific
amount upon receiving an alarm signal. Our empirical findings suggested that while the transmission
of alarm information generally boosts speed, it does not uniformly result in a complete transition to a
qualitatively “alarmed” behavioral state for all individuals receiving the signal [1]. Thus, despite the
binary nature of the state change, the degree of response in speed should exhibit variability. Finally,
we assumed that ants decrease their speed at a rate directly proportional to their current speed,
ensuring a prioritization of energy conservation in the absence of alarm stimuli.
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2.1. Model description
We simulated a group of Nant workers navigating a two-dimensional circular arena with a radius
of M=520 pixels in continuous space (see Figure S1 in the Supplementary file). Our time scale was
limited to just a few minutes, negating the need to account for birth or death processes.
Let λ∗represent the colony’s baseline speed in the unalarmed state. Ants were consecutively
numbered from 1 to Nfor unique identification. For each ant indexed by p(1 ≤p≤N), its complete
set of attributes at time-step tis denoted by
ϕp
t=(xp
t,yp
t, θp
t,Ap
t,sp
t,Λp, λp
t),
where
•xp
tand yp
trepresent the ant’s spatial location in the arena at the end of time-step t, with Cartesian
coordinates based on the origin at the center of the arena;
•θp
tdenotes the angle of the ant’s heading during time-step t, defined as the counterclockwise
angle from the x-axis of the vector connecting two consecutive positions (see Figure S2 in the
Supplementary file);
•Ap
t∈[0,1] indicates the alarm state during time-step t;
•sp
trepresents the length of the step that ant-ptakes at time-step tdivided by the length of the time-
step (see Figure S2 in the Supplementary file). Henceforth, any s-value will be simply referred to
as “speed”.
•λp
trepresents the expected value of ant-p’s speed during time-step t;
•Λp∈[0,Λmax ] is ant-p’s speed jump parameter, indicating the amount by which λp
tincreases
when an ant becomes alarmed.
Some of an ant’s characteristic parameters exhibited values distributed as follows:
1) Change in Heading (∆θp
t):
(∆θp
t):
(∆θp
t): The alteration in heading, ∆θp
t=θp
t−θp
t−1, is randomly sampled at each
time-step tfor each ant p. Based on the empirical distribution of turning angles ∆θp
t(visualized
in Figure S3), our model assumes that the values of ∆θp
tare sampled from a Laplace distribution
(probability density function),
P(∆θp
t)=ωexp −ω|∆θp
t|
2,∆θp
t∈(−π, π),(2.1)
where ω−1signifies the mean magnitude of the turning angle, consistent for all ants across all
time-steps. The adjustment of heading influences correlated random walks in ants, as observed in
individual ant movement data collected from empirical videos (refer to Figure S3 in the
Supplementary file and experiments in [1]).
2) Alarm State: In our model, “alarm” denotes a binary state, where Ap
t=0 indicates an unalarmed
state for ant-pat time-step t, and Ap
t=1 signifies an alarmed state. The alarmed state can be
communicated from one alarmed ant to others, with conditions defining when an unalarmed ant
transitions to an alarmed state.
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In a biological context, we describe an ant in an “alarmed” state as having the function of
recognizing and alerting the colony to potential threats. This state is behaviorally distinguished by
notably elevated walking speeds, increased circular motion, and higher rates of contact compared
to the typical behavior in a normally functioning colony [1]. During an actual colony alarm
response, only a specific subset of individuals that receive the alarm signal enter this precisely
defined behavioral state, while most generally respond by increasing their speed. It is important to
note that our characterization of an “alarmed” ant deviates from the strict ethological definition.
Instead, it signifies that an ant has received information and responded with some degree of
increased speed. In our model, unalarmed ants consistently maintain a mean speed of λ∗.
Conversely, alarmed ants exhibit mean speeds greater than λ∗.
3) Speed Jump Parameter (Λp):
(Λp):
(Λp): If the ant becomes alarmed at any time t, a random number Λp
is added to her speed parameter λp
t. For ant-p, this speed jump is defined as the parameter Λp.
It is randomly sampled once (prior to the simulation) from a truncated exponential distribution
because based on anecdotal evidence, the majority of ants are less responsive to alarm information,
while a minority display heightened response, thus exponential distribution aligns well with this
observation. The distribution has a mean of a−1, and Λptakes values within the range of [0,Λmax]:
P(Λp)=aexp(−aΛp)
1−exp(−aΛmax).(2.2)
We truncated the tail at Λmax to prevent excessively large, biologically unrealistic speed jumps. The
denominator serves to normalize the probability density function. An exponential distribution was
chosen because we want to give agents a spread in their intrinsic excitability. We found anecdotally
that ants respond differently to alarm information, with some ants being more reactive than others.
While we could have used a uniform or normal distribution for speed jump, we noticed that most
ants are less responsive, but a few are very responsive. So an exponential distribution matches this
observation well.
4) Speed at Time-step t+1 (sp
t+1):
t+1 (sp
t+1):
t+1 (sp
t+1): At time-step t+1, we determined the value of sp
t+1through
random sampling from the exponential distribution with a mean of λp
t+1(see Figure S4 in the
Supplementary file):
P(sp
t+1)=
exp −sp
t+1
λp
t+1
λp
t+1
.(2.3)
5) Speed Dynamics (λp
t+1):
(λp
t+1):
(λp
t+1): The value of λp
t+1is determined by one of three mutually exclusive
conditions, depending on the alarm state of the ant at time-step t. An exception occurs when
selecting any of these conditions would lead the ant to exit the physical arena
i.e., xp
t+12+yp
t+12>5202, in such cases, resampling from an alternative distribution is done,
as detailed later.
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a) At all time-steps twhen ant- pis unalarmed (Ap
t=0), thus, λp
t=λ∗.
b) If ant- pdid not exhibit alarm at time-step t−1 but becomes alarmed at time-step t(i.e., Ap
t−1=0
and Ap
t=1), we incorporated her speed jump parameter, denoted as Λpand constrained within
the range [0,Λmax ], into the baseline speed of the colony. This addition is made to determine
the expected speed parameter for this particular ant. It is important to note that this adjustment
occurs at most once per simulation, specifically during the initial time-step when the ant enters
the alarmed state.
λp
t=λ∗+ Λp(2.4)
As Λpvaries among individual ants, a range of responses to alarm signals was expected,
reflecting the diversity observed in our empirical trials.
c) If ant- pis alarmed at both time-steps t−1 and t(i.e., Ap
t−1=Ap
t=1), the following equation
governs its expected speed during time-step t:
λp
t=λp
t−1−β(λp
t−1−λ∗)=βλ∗+(1 −β)λp
t−1,(2.5)
here, the parameter β, known as the speed decay or speed-loss factor, is an independent
parameter for each ant. The specific value of βis determined through fitting the model to
experimental data, as explained in Subsection 3.1. This updating rule is derived from our
empirical observation that ants with initially higher walking speeds experience a more
pronounced reduction in speed during an alarm. Given the higher energetic cost associated with
faster speeds, this functional form enables our synthetic ants to prioritize energy conservation,
ensuring that faster ants decrease their speed at a comparatively higher rate.
Let D={n∈N: 1 ≤n≤N}be the collection of the Nant indexes comprising the whole colony.
For each p∈D, we defined the neighbors of ant- pat time-step tas the collection of ants (using their
indexes as proxy) who are located within runits from ant-p:
Bp
t=nk∈D: (xp
t−xk
t)2+(yp
t−yk
t)2≤r2o.(2.6)
Here rrepresents the alarm signal perceptual radius of any ant. That is, ris the maximum distance at
which an ant can sense alarm signals, due to all possible sensory modalities: tactile, vision,
and/or olfaction.
We will now briefly summarize now the core model assumptions:
1) Worker-pgets alarmed in time-step t+1 whenever one of its Bp
tneighbors is alarmed;
2) There is an abrupt increase in her λp
t+1(and likewise her sampled speed sp
t+1) when she
gets alarmed;
3) The expected value of ant-p’s moving speed gradually decays towards the colony’s baseline
speed after she enters the alarmed state.
2.2. Model initialization
1) We randomly distribute Nants within a two-dimensional circular region with a radius of M=520
pixels. The initial location of ant- pis set as
xp
0=Rpcos(θp
1),(2.7)
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yp
0=Rpsin(θp
1).(2.8)
We set Rpuniformly at random from the interval [0,M], and chose θp
1uniformly at random from
the interval [0,2π].
2) Let ρ∈(0,1) denote the initial fraction of alarmed ants. We randomly selected a subset V⊂D,
comprising [ρN+0.5] ants (where square brackets indicate the integer value function) to serve as
alarm seeds. In other words, Ap
1=1 for all ants in this subset. We established the expected value
for their speed as follows:
λp
1=λal =λ∗+ Λmax .(2.9)
Subsequently, the speed sp
1of each alarm seed ant is stochastically sampled from the exponential
distribution with mean λal, where P(s)=1
λal exp −s
λal .
3) Each ant- pin the set U=D−Vexists in an unalarmed state represented by Ap
1=0. We establish
the expected speed for these ants as λp
1=λ∗, and their initial speed sp
1was randomly selected from
an exponential distribution with a mean λp
1=λ∗, expressed as P(s)=1
λ∗exp −s
λ∗.
4) During the first time-step, we relocated the ants from their initial positions to their final positions
using the following update formulas:
xp
1=xp
0+sp
1cos(θp
1),(2.10)
yp
1=yp
0+sp
1sin(θp
1).(2.11)
5) The speed jump parameter for ant- p, denoted as Λp
1= Λp, was sampled in accordance with Eq (2.2).
This defines the initial parameter set, ϕp
1, for ant-p, where 1 ≤p≤N.
2.3. Update procedure
At each time-step tfollowing the initial one (which was initialized above), we iterated through all N
ants in the order of their indices. For each ant-p, we modified all components of ϕp
t−1to ϕp
tas outlined
below. Subsequently, we proceeded to the next time-step, repeating the process until the final time-step
update was completed.
1) The speed jump parameter remained constant for each ant across all time steps, implying that
Λp
t+1= Λp
t= Λp.
2) The heading angle θp
twas updated according to the formula θp
t=θp
t−1+ ∆θp
t, where the value of ∆θp
t
was randomly chosen from the probability function P(∆θp
t+1)=ωexp(−ω|∆θp
t+1|)/2, as illustrated in
Eq (2.1).
3) The update process for the alarm state (Ap
t), expected speed (λp
t), and speed of ant-p(sp
t) to their
respective values at time t+1 (Ap
t+1,λp
t+1, and sp
t+1) is outlined below:
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•If Ap
tequals 1, ant-pis already alarmed, and it remains in that alarmed state with Ap
t+1being 1.
The ant’s anticipated speed undergoes an update from λp
tthrough discrete exponential decay,
as described in Eq (2.5):
λp
t+1=λp
t−β(λp
t−λ∗).
On the other hand, if Ap
tis 0, the alarm state Ap
t+1at time t+1 is determined by the alarm states
of its neighbors Bp
tbased on the following two mutually exclusive cases:
a) If all neighbors Bp
tof ant-pare unalarmed, we set Ap
t+1=0 and maintain the ant’s expected
speed unchanged:
λp
t+1=λ∗.
b) If at least one of ant-p’s neighbors Bp
tis alarmed, we set Ap
t+1=1. In this case, we also
increase the ant’s expected speed by her speed jump:
λp
t+1=λ∗+ Λp.
•The speed, sp
t+1, is determined by randomly sampling from the exponential distribution with a
mean of λp
t+1, as specified in Eq (2.3).
4) The position (xp
t,yp
t) undergoes an update to (xp
t+1,yp
t+1) in one of two ways, depending on whether
the proposed new position:
˜xp
t+1=xp
t+sp
t+1cos(θp
t+1),
˜yp
t+1=yp
t+sp
t+1sin(θp
t+1),
lies outside the arena. If the new position is inside or on the boundary of the arena, such that
|( ˜xp
t+1,˜yp
t+1)| ≤ M, then:
xp
t+1=˜xp
t+1,yp
t+1=˜yp
t+1.
If the new position falls outside the region defined by |( ˜xp
t+1,˜yp
t+1)|>M, the ant is redirected inward
as follows:
xp
t+1=xp
t−sp
t+1cos(θp
t+1),yp
t+1=yp
t−sp
t+1sin(θp
t+1).
Subsequently, θp
t+1is updated to account for the new orientation of the velocity vector:
θp
t+1=cos−1"xp
t+1−xp
t
sp
t+1#.
The detailed procedure for each update following initialization can be elucidated using the
schematic depicted in Figure 1.
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Figure 1. A schematic illustrating the process of updating the characteristic parameters of
an individual ant at a specific time-step. The turning angle ∆θp
t+1is selected from a Laplace
distribution with a location parameter of 0 and a scale parameter of 1
ωas described in Eq (2.1).
Additionally, the speed sp
t+1is chosen from an exponential distribution with an average of λp
t
according to Eq (2.3), and the speed jump Λpis selected from a truncated exponential distribution
with an average of a−1as outlined in Eq (2.2).
3. Model validation and results
In this section, we first employed the empirical data available in [1] to validate our model. Then, we
conducted systematic simulations to investigate the influence of crucial parameters on the dynamics of
group mean speed.
3.1. Model validation
There are three set of data for three different colonies A, B, and C in [1]. For each colony, we
adjusted our model to match the empirical data both before and after the introducing seed alarmed
ants, as explained in [1], to find parameters λ∗,r,a,and βas shown in Figure 2. All remaining
parameters were held constant, following the specifications in Table 1. To perform the fitting process,
we utilized MATLAB’s fminsearch function, employing the simplex search method developed by
Lagarias et al. [27]. This approach aimed to minimize the error between the model’s simulation and
the actual data, as detailed in Table 1.
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Figure 2. The average speed of the group, denoted as st, is depicted through blue lines in both
pre- and post-introduction videos: The red circles represent the average curve derived from 300
replicates of the model under fitted parameters, with error bands indicating standard deviation
across replicates. The model parameters utilized are detailed in Tables 1 and 2. All curves have
been smoothed for clarity. Notably, the post-introduction data is plotted immediately after the
pre-introduction data to facilitate comparison, with the 25-minute gap omitted for simplicity.
Table 1. Initialization, movement, and contact parameters: The baseline values provided were
applied consistently across all replicates and figures, unless specified otherwise.
Symbol Description Baseline value Estimated using
TTotal time-steps in simulation 4500 (150 s) Video duration
MRadius of arena 520 pxl (75.5 mm) Empirical arena
NPopulation size 60 Empirical population
ω−1Mean turning angle 21.14◦Empirical data
rRadius within which an interaction occurs 58.27 pxl (5.8 mm) Model fitting
λ∗Steady state speed parameter 1.8739 pxl/time-step (8.16 mm/s) Pre-introduction sp
t
Λmax Maximum increase in λdue to excitation 6.88 pxl/time-step (30 mm/s) Empirical data
aDecay rate of P(Λp) over Λp0.1497 Model fitting
βDecay rate of λpover time 0.00032 Model fitting
ρFraction of colony initially alarmed 0.05 Empirical data
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Table 2. Predictions for the parameters of our colonies’ model: The parameter vector
(λ∗,r,a, β) for each colony was adjusted to minimize the combined root mean square error
(RMS Epre−introduction +RMS Epost−introduction ) between experimental and model-predicted values of
st, while keeping all other parameters fixed based on Table 1. The averages of each column served
as the default parameters in Table 1.
Colony λ∗(pxls/f rame)r(pxls)aβ
A1.149 ±0.046 40 ±3 0.100 ±0.007 1.00 ×10−3±1×10−5
B0.918 ±0.045 60 ±3 0.060 ±0.006 2.5×10−4±1×10−5
C0.0459 ±0.044 61 ±3 0.250 ±0.005 2.2×10−4±1×10−5
We defined the framewise group mean speed as st=1
NPN
p=1sp
t,where Nrepresents the group size,
and time-averaged speeds as sp=1
TPT
t=1sp
t, where Tis the total number of time-steps.
The blue curves in Figure 2 depict framewise group mean speeds (st) both before and after the
introduction. Meanwhile, the red curves represent the model’s st, averaged across 300 model replicates.
These replicates, described in Section 2.2, are initialized with randomized starting positions (xp
0,yp
0),
speeds sp
0, and orientations θp
0. It is important to note that for pre-introduction replicates, the fraction of
the colony initially alarmed (ρ) was set to zero. The fitted parameters used in the model are summarized
in Table 2 and offer a rough estimate of each colony’s true latent parameters.
In Figure 2, the group mean speed, denoted as st, exhibits a tendency to rise rapidly and decline
gradually, with variations in the alarm response among different colonies. A comparison between
the empirical results and our fitted model reveals that the model exhibits the necessary flexibility to
capture this range of colony responses. In a biological context, the typical pattern of swift speed
increase and slower speed decrease can be interpreted as ants quickly getting agitated upon initial
exposure to alarm signals. This rapid response to the stimulus presumably enables the colony to react
promptly to potential threats. The decay phase in the colony’s response indicates that although ants
take a longer time to return to a calm state, they do revert to their baseline behavior within a few
minutes unless subjected to another alarm stimulus. Due to the positive linear relationship between
energetic cost of locomotion and speed [28, 29], the ability to swiftly return to baseline states allows
colonies to minimize energy expenditure on false alarms and reduce the disruption of normal colony
function caused by an alarm event.
Figure 3(a) illustrates histograms that aggregate post-introduction speed data from colonies A, B,
and C, divided into five non-overlapping, 30-second windows. On the horizontal axis, individual
framewise speeds sp
tare binned, while the vertical axis depicts the frequency of sp
twithin each bin,
aggregated over each 30-second time window. The frequencies were normalized to represent a
probability density function and plotted on a linear scale (with insets in semilog scale).
In Figure 3(b), all parameters were fixed according to Table 1. We conducted 300 model replicates
and pooled sp
tfrom all replicates. The parameters in Table 1 were carefully chosen to closely calibrate
the model to the spatial and time scales observed in the pooled post-introduction videos. Each curve
in Figure 3(b) represents data aggregated from a 30-second window (equivalent to 900 simulation
time-steps) and displays the probability distribution over sp
t. The empirical distributions in Figure 3(a)
encompass a total of 670,680 data points, while the model distributions in Figure 3(b) comprise
approximately 81 million points.
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Comparing the dynamics of speed distribution between Figure 3(a),(b) reveals that the
distributions maintained an exponential pattern throughout both post-introduction videos and model
replicates. Beyond the initial 30-second interval, the slope of the distribution progressively becomes
more negative. This suggests that ants are converging toward a specific baseline distribution
characterized by a lower mean and variance. In the context of the selected parameters (see Table 1),
the decay of the distribution occurs roughly on comparable time scales.
Figure 3. Depicted speed distributions within consecutive 30-second time intervals for (a)
combined empirical trials post-introduction and (b) 300 aggregated model replicates: Each curve
illustrates the histogram of individual framewise speeds (sp
t) within the respective 30-second
time window. Error bands, where visible, represent the standard error. Insets display the
same distributions on semilogarithmic axes, truncated to highlight the speed range of 0 to 50
mm/s. In both experimental and model scenarios, the distribution approximates an exponential
pattern with a diminishing tail over time. Refer to Table 1 for the parameters employed in the
aforementioned replicates.
3.2. Dynamical outcomes
Here, we initially investigate the influence of colony size (N), or equivalently, the spatial density of
individuals, on the dynamics of alarm spread. We gain insights into how increasing group sizes might
impact the colony-level alarm response by observing its effect on several speed-based metrics. Next,
we individually vary each of the other model parameters and observe how they affect the dynamics of
st(Figure 4). Each plot in this section is derived from averaging 300 model replicates.*
*The parameter settings for all figures align with those in Table 1. The initial conditions are uniform across all simulations, except
for randomizing starting positions (xp
0,yp
0), speeds sp
0, and orientations θp
0.
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Figure 4. The mean curve staveraged over 300 replicates: The error bands depict the standard
deviation across replicates. For details regarding the parameters employed in all replicates,
excluding the one undergoing variation, refer to Table 1.
In Figure 5, we scrutinized three metrics of the degree of alarm excitation: colony mean speed (st),
colony expected mean speed parameter (λt), and the fraction of alarmed ants. The curves depict the
dynamics at three colony sizes (N=20,60,180), with the horizontal axis representing time converted
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from simulation time-steps to seconds. For this set of results only, we introduced the alarm seed ants
at time-step t0=720 =24 sto clarify the speed growth phase. The “fraction of colony alarmed” in
Figure 5(c) represents the number of individuals in the state Ap
t=1, divided by N. According to all
three metrics, the model predicts that increased density would lead to a more rapid spread and decay
of alarm.
Figure 5. The impact of group size on the dynamics of the group mean speed (top left), the group
expected mean speed parameter (top right), and the fraction of the group alarmed (bottom): A
larger group size implies higher density, given that the size of the arena is fixed. Increased density
results in a more rapid overall propagation of alarm. The curves in the top and middle panels
represent the means of 300 replicates, with their corresponding standard deviations depicted in
shaded regions. In the bottom panel, the curves illustrate the first quartile (Q1), second quartile
(Q2), and third quartile (Q3) of 300 replicates, along with their standard deviations. Default
parameters from Table 1 were used for all plots.
The result also suggests that the group mean speed (st) should reach a higher peak in denser groups.
Table 3 lists each group’s “peak height”, defined as maxst−λ∗. Each value in the peak height column
is then normalized by the N=20 group’s peak height for ease of comparison. Increasing Nby factors
of 3 and 9 led to 1.55 times and 1.92 times increases in peak height, respectively. These peaks also
occurred at earlier times. “Time-to-peak” was computed as tpeak−t0, where tpeak is the time of maximum
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stand t0=720 time-steps is the time at which the alarmed ants were introduced. All values in the time-
to-peak column were normalized by the time-to-peak of the N=20 group. The last column lists the
fraction of ants alarmed at tpeak.
Table 3. Metrics for colony alarm levels across three colony sizes: “Peak height” denotes the
maximum group mean speed, represented as st. “Time-to-peak” measures the duration between
introducing alarmed ants (at t0) and reaching the maximum speed. “Fraction alarmed at peak”
indicates the proportion of alarmed ants at the peak time divided by N. All peak heights and
time-to-peaks were normalized based on the values for the N=20 group for ease of comparison.
NScale factor Peak height Time to peak Fraction alarmed at peak
20 1 1.000 ±0.147 1 0.590 ±0.015
60 3 1.549 ±0.162 0.403 0.845 ±0.010
180 9 1.920 ±0.170 0.108 0.915 ±0.010
Table 3 illustrates that, according to our assumptions, a more densely populated environment
expedites the rapid dissemination of the initial alarm signal within the group. The N=180 group
reached its peak excitation in only one-tenth of the time required by the N=20 group. Moreover, at
the peak time, the N=180 group exhibited an alarm response in 91.5% ±1% of ants, while the
N=20 group had only 59% ±1.5% alarmed. It is crucial to acknowledge that the model may
overestimate the actual increases in the rate of spread and the maximum excitation at higher densities.
This overestimation is attributed to the model not accounting for pauses in motion that may occur
during ant-to-ant interactions.
In Figure 4, we systematically varied each fitted parameter individually while keeping the others
constant, as outlined in Table 1. Our observations revealed that augmenting either the contact radius,
denoted as r(see panel 4), or the initially alarmed fraction, denoted as ρ(see panel 2), had analogous
effects on the dynamics of stcompared to increasing the parameter N. This manifested in a quicker
rise, a faster decay, and a higher peak of st. These outcomes align with intuitive expectations. A larger
contact radius (r) facilitates a swifter alarm spread due to more frequent contacts, while introducing a
greater fraction (ρ) of initially alarmed ants enhances the stimulus and accelerates signal transmission
within the group.
In Figure 4, panel 5, we observe that reducing the parameter a(resulting in an increased variation
in speed jumps, Λp, among individual ants) leads to higher peak heights without significant differences
in time-to-peak. Similarly, substantial increases in group baseline speed, λ∗(panel 1), do not result in
earlier peaks in excitation. An elevated decay rate of the expected speed, regulated by the parameter
β, leads to a decrease in the group’s mean speed ¯st. It attains a lower peak and subsequently declines
to the baseline of 8.16 mm/s at a faster rate. This outcome aligns with the model assumption that
as βincreases, the time-series of expected speed diminishes (refer to Eq (2.5)). Consequently, this
phenomenon signals a reduction in alarmed ant speeds, subsequently impacting the overall group mean
speed. Regardless of the chosen parameter values we explored, the model consistently exhibits a
swift rise and gradual decay of sp, aligning with observations from post-introduction videos. This
consistency in model behavior reinforces the validity of our underlying assumptions regarding alarm
signal transmission and damping.
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4. Discussion and conclusions
In formulating the model, we incorporated several crucial empirical observations regarding ant
colony alarm responses: (i) the swift onset of alarm excitation, (ii) the diversity in individual-level
responses during an alarm, and (iii) the relatively gradual decay of alarm. Since speed is the movement
feature most strongly correlated with alarm [1], we investigated the role of this behavior in regulating
alarm. Our model employed a parallel set of three individual-level speed adjustment mechanisms,
collectively designed to replicate the observed group-level speed distribution dynamics (Figure 3(a)).
The first mechanism bestowed upon the ants is a switch-like response to alarm signals: They rapidly
increase speed upon encountering alarmed nestmates, and this increase varies among individuals. An
alternative mechanism could involve ants gradually increasing their speed in proportion to the strength
or frequency of received alarm signals. Several studies examining the role of encounter rate [30–32]
and signal strength [33] on task recruitment in ant species suggest that ants have the potential for a more
modulated response. The use of quorum rules by some house-hunting ants [34, 35] implies that alarm-
susceptible ants might be aggregating data from multiple contacts as well.†While such mechanisms are
conceivable, our results underscore that an instantaneous speed jump upon close contact is sufficient to
capture the extremely rapid rise in stin the initial phase of the alarm response, as shown in Figure 2.
This rapid initial response also proved highly robust when varying each data-fitted model parameter
individually, as demonstrated in Figure 4.
The second key assumption revolved around the variability in the response (speed increase) of our
ants when alarmed. Notably, individual variations among ants within the same colony, and
even within the same task group, have been well-documented in various contexts such as task
performance [36], task specialization [37], and activity level [38]. This behavioral diversity has been
attributed to phenotypic, age-based [39], physiological (Flexible Task Allocation), and experiential
differences [40].
Our observations indicate that ants exhibit diverse responses during alarm situations (Section 3.2).
We postulated that allowing individuals to increase their speed by different amounts, denoted as Λp,
during alarms may lead to similar effects on the group-level, framewise speed distributions as observed
in post-introduction videos (Figure 3). In the initial 30 seconds (red curve, Figure 3(a)), the histogram
slope is relatively higher than at baseline (blue curve, Figure 3(a)), indicating a thicker-tailed speed
distribution with a higher mean, sp
t, and greater variance in sp
tduring the peak of alarm. Our model’s
assumptions regarding the variation in individuals’ Λp(Eq (2.2)) contribute to this initial shift in the
framewise speed distribution.
Additionally, during the decay phase of the alarm response, we discovered that allowing alarmed
ants to decrease their current average speed λp
tat a rate proportional to λp
t−λ∗was sufficient to
capture the decay of the framewise speed distribution toward its baseline (Figure 3). The gradual
decrease in the distribution’s mean stunder most parameter settings supports our assumptions about
ants’ consistent prioritization of energy conservation. Since moving at high speeds consumes more
energy, faster-moving ants tend to decrease their speed more rapidly. It is worth noting that both
empirical data in [1] and model data indicate that the individual’s speed over consecutive time-steps is
not monotonic. However, on average, the speed increases when ants are alarmed and decreases
†We tested a model version based on these rules (unpublished data) and obtained reasonable but less-accurate agreement with the
empirical videos.
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afterward. This suggests that decay of the individual excitement state mediates alarm decay at the
group level most effectively.
We contribute several unique features in the context of other self-propelled particle models. Most
notable is the mechanism that allows agents to adapt to new information. When an agent receives the
alarm signal, it modulates its behavior by changing its speed parameter λp
t. This affects the distribution
from which the agent samples its speed. This distribution then continues to shift over time, as the speed
parameter decays post-alarm. Allowing adaptive speed distributions is not common in self-propelled
particles, but was essential to capture the dynamics in our alarm experiment.
Discrepancies between empirical dynamics and model output partly stem from the model’s lack of
realism in depicting ant movement. For instance, in actuality, ants commonly pause during encounters
with nestmates, a behavior omitted in the model for the sake of simplicity. Additionally, the model
employs a random sampling approach for ant speeds at each time-step, neglecting the likely correlation
between speeds in consecutive frames. This absence of speed correlation in the model results in a
much smoother decay in stcompared to pre- and post-introduction trials (Figure 2, Supplemental
Videos). These movement characteristics constrain the accuracy of group-level dynamics, and future
research should aim to model individual-level movement more precisely. Approaches such as machine-
learning-based regression models, recurrent neural networks, or reinforcement learning methods could
be employed to train models that accurately represent the velocity vector of each agent sequentially.
The ant colony’s alarm response presents an excellent avenue for investigating adaptive
information flow and signaling mechanisms in animal groups and other complex systems. This
system is amenable to experimental manipulation, and agent-based modeling allows for the
prototyping of individual signal-response rules. Although closely linked to the existing body of work
on collective motion [21,23, 26, 41, 42] and general infection processes, this study stands out as one of
the few examinations of insect alarm behavior utilizing object tracking. Through simulating variations
in a natural colony, we aimed to offer biological insights into the latent factors influencing a colony’s
alarm response. Moreover, the proposed mechanisms of signal transmission and decay in the model
are likely applicable to other systems where transmission is instantaneous, and the group response
is adaptive.
Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
This work was supported by the NSF-DMS (Award Number 1716802); the NSF-IOS/DMS (Award
Number 1558127); the James S. McDonnell Foundation 21st Century Science Initiative in Studying
Complex Systems Scholar Award (UHC Scholar Award 220020472); and the Defense Advanced
Research Projects Agency (DARPA)-SBIR (2016.2 SB162-005). We would like to thank the
members of the Fewell lab for their assistance in ant rearing and for their feedback on the manuscript.
Mathematical Biosciences and Engineering Volume 21, Issue 4, 5536–5555.
5553
Conflict of interest
The authors declare there is no conflict of interest.
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