Active Particle Models for Animal Behavior
based on Effective Nonreciprocal Forces and Potentials
Amir Haluts1, Dan Gorbonos2, and Nir S. Gov1

1Department of Chemical and Biological Physics, Weizmann Institute of Science,
Rehovot 7610001, Israel

2Department of Collective Behaviour, Max Planck Institute of Animal Behavior,
Konstanz 78464, Germany

## Abstract

Modelling animal behavior using active-particle models is a major current
challenge. The unique properties of animals mean that such models require the
use of new types of effective interactions between the particles, and
specifically effective forces that do not obey the usual conservation laws of
Newtonian mechanics. These include nonreciprocal forces that break
conservation of energy and momentum. We demonstrate here two very different
animal behavior systems where such nonreciprocal effective forces naturally
arise: the first is when animals form contests, as many animals do, such as
fighting over some localized resource. The second system is of cohesive
swarms, that are maintained by long-range adaptive attraction. These examples
show that theoretical modelling in terms of active particles, interacting with
effective nonreciprocal forces and potentials, expands the scope of active-
particle research as well as helps to explain complex phenomena in animal
behaviour.

## Introduction

Active particle systems are composed of interacting, self-propelled particles.
In theoretical models, implemented using computer simulations, the
interactions between the particles can take a variety of forms [1]. These
interactions, can be in the form of reciprocal forces that arise from inter-
particle potentials, such as of the Lennard-Johns-type. Other interactions can
be non-physical, such as those leading to velocity alignment between
neighboring particles, such as in Vicsek-type models. The resulting behavior
of these non-equilibrium systems is determined by the self-propulsion
properties, the type of interactions, and noise.

Since animals are self-propelled, modelling the movements of animals and
animal groups is naturally approached by active particle models [2]. In these
systems the interactions between the animals, which control their movements,
are not easy to extract from experimental data. These interactions arise from
animal communication, which can involve a complex set of specific signalling,
in addition to each animal’s interpretation of the movements, intentions and
properties of the others. The mapping between the biological system and an
active particle system is therefore non-trivial. Here we demonstrate, using
two examples, models of interacting particles that describe the dynamics and
movements of animals. These examples show that the effective interactions
arising within the context of animals can have distinct properties that do not
appear in physical systems (where forces are usually reciprocal and derived
from potentials): forces between animals can be nonreciprocal, and adaptive.
Nonreciprocal interactions, motivated by interactions between biological and
active entities, have become a focus of growing recent interest for research
[3, 4, 5, 6, 7, 8].

Nonreciprocal interactions between two animals arise naturally in the context
of animal contests (fights). This is described in the first section, where we
develop a general effective-potential model for describing the escalation
process that brings two competing animals into close contact (defined as the
contest). This model was originally developed to describe a system of male
spider contestants which compete on the two-dimensional orb-web of the female
[9], and subsequently extended to describe contests between animals in
general. The dynamics of these agonistic interactions is driven by each
contestant’s assessment of its own and its rival’s ’strength’, and its
response to these assessments [10]. The asymmetric scenario where the animals
have different ’strengths’ therefore naturally gives rise to nonreciprocal
interactions, which can be described by each contestant responding to a
different effective potential landscape. The most noticeable effect of these
nonreciprocal forces during contests is the transition from bounded dynamics
(of closely matched rivals) to chase dynamics (between highly asymmetric
contestants).

In the second section we demonstrate the role of ’adaptivity’ in the effective
interactions between animals. Adaptivity is the property of animals to respond
to the relative fold-change in a signal that reaches its sensory organs [11].
This property appears in biology from bacteria to humans, and affects all
sensory systems, such as visual and audial. This property means that a
constant change in the background signal does not affect the behavior, as the
sensory modality adapts to this background, and only changes relative to this
background elicit a response. We demonstrate modelling adaptive interactions
in the context of midge swarms, which form cohesive yet disordered swarms of
flying insects [12]. The model explores long-range attractive interactions
between the particles, which are adaptive, resulting in an ’adaptive gravity’
form [13, 14, 15, 16]. The adaptivity introduces a non-equilibrium property,
by violating the conservation of energy and momentum of the particles, and
giving rise to nonreciprocal interactions.

In both sections we demonstrate the novel theoretical aspects of the
biologically-inspired interactions between the particles, as well as compare
to experimental data. This chapter emphasizes that by modelling animal
systems, we are provided with motivation to introduce and explore unique types
of interactions, that expand the spectrum of interacting active particle
systems.

##  1 Nonreciprocally interacting particles model of animal contests

It is commonplace for animals to engage in agonistic behavior to obtain and
defend food, mates, and territory [17]. These fights over resources, widely
termed ’animal contests’, inherently entail the resolution of an interplay
between risk and reward. Animal contests have therefore attracted extensive
research from ecologists and biologists who sought to understand the
evolutionary logic behind such non-trivial trade-offs [18, 19, 20, 21, 22, 23,
24, 25, 26]. From a theoretical standpoint, these risk-reward trade-offs have
also made animal contests particularly amenable to the modeling tools of game
theory, which, ever since the seminal works of the 1970s [27, 28, 29, 30, 31],
has been the central analytical framework used to model and understand them.
But while game-theoretic contest models make predictions that can, in
principle, be tested empirically [32], mapping these models to real contest
scenarios have proven difficult, and often lead to ambiguous conclusions [33,
32, 34, 35]. An intrinsic hurdle for such mapping has been the omission of
within-contest dynamics by game-theoretic models, which typically meet the
observable dynamics of contest behavior only in their endpoint predictions
(such as who won and how long was the contest), and disregard the spatio-
temporal intricacies of the contestants’ trajectories.

While detailed tracking of spatio-temporal trajectories was out of reach for
the original contest theorists, it is now a standard practice in animal
behavior research [36, 37]. This motivated the development of a new
theoretical framework for the dynamics of animal contests, which describes
them based on their spatio-temporal properties [9, 10]. In this physics-
inspired framework, the real-space dynamics of contestants is explicitly
described in terms of (active) ’contestant particles’, which interact through
effective interaction potentials. The adversarial nature of the inter-
contestant interactions makes their properties distinct from those of the
typical pairwise potentials that govern the interactions between physical
particles. Notably, since each contestant’s behavior is ultimately driven by
the assessment of its own and its rival’s ’strength’, contest interactions are
directional and nonreciprocal. This is handled by assigning the dynamics of
each contestant with the influence of a distinct potential landscape that
cannot be simply deduced from a global potential. In the following, we discuss
the motivation behind this framework, its construction, and some of its
implications—both in general and in a specific experimental system.

###  1.1 Dynamics of contestant particles

Our theoretical treatment of contestant dynamics assumes that contest behavior
can be effectively described by some archetypal interaction potentials, and
that these potentials predominantly depend on contestant position. Treating
each contestant as an active Brownian particle that moves due to the influence
of deterministic (interaction-induced) and stochastic effective forces, the
spatio-temporal dynamics of a ’contestant particle’ $i$ (where
$i\in\\{1,\,2,\,...,\,N\\}$ in a system of $N$ contestants) is modeled to be
governed by the following (overdampled) Langevin equation

$\dot{\mathbf{r}}_{i}=-\eta\nabla V_{\mathrm{tot}\rightarrow
i}(\mathbf{r}_{i},\,\\{\mathbf{r}_{j}\\})+\sqrt{2D}\,\bm{\xi}_{i}+v_{0}\,\mathbf{p}(\phi_{i}).$
(1)

On the right-hand side of Eq. (1), the first term accounts for the
deterministic driving force felt by contestant $i$ at position
$\mathbf{r}_{i}$ due to the (total) effective potential
$V_{\mathrm{tot}\rightarrow i}$, where $\eta$ is the contestant’s ’mobility’.
$V_{\mathrm{tot}\rightarrow i}$ generally depends on $\mathbf{r}_{i}$ and on
the positions $\\{\mathbf{r}_{j}\\}$ of all other rival contestants, where
these positions are defined with respect to a single resource at the origin
(as illustrated in Fig. 1A). The construction of $V_{\mathrm{tot}\rightarrow
i}$, which combines all the fundamental building blocks of the model, is
detailed in the next section; The second term is the usual translational noise
term, where $D$ is an effective translational diffusion coefficient, and
$\bm{\xi}_{i}$ is a vector of mutually uncorrelated sources of standard
Gaussian white noise for each of the spatial dimensions (i.e.,
$\bm{\xi}_{i}=(\xi_{x_{i}},\,\xi_{y_{i}})$, with
$\xi_{x_{i}},\,\xi_{y_{i}}\sim\mathcal{N}(0,1)$, in two dimensions); The third
term accounts for the internal (active) directional persistence of the
contestant, where $v_{0}$ is the persistence speed, $\mathbf{p}$ is the
persistence direction vector, and $\phi_{i}$ is the persistence angle (i.e.,
$\mathbf{p}=(\cos\phi_{i},\,\sin\phi_{i})$ in two dimensions). $\phi_{i}$
itself is commonly governed by its own Langevin equation,
$\dot{\phi}_{i}=\sqrt{2D_{R}}\,\xi_{\phi_{i}}$, where $D_{R}$ is an effective
rotational diffusion coefficient and $\xi_{\phi_{i}}\sim\mathcal{N}(0,1)$ is
the corresponding stochastic source. See [38] for a comprehensive review of
such models for active particles dynamics.

###  1.2 Effective interaction potentials and nonreciprocity

The total effective potential felt by contestant $i$ combines the influence of
a resource and and of all other contestants $\\{j\\}$ (which compete over this
particular resource) on the motion of contestant $i$. Observations suggest
that when contestants become strongly engaged in a contest interaction, their
attention is predominantly given to their rival(s). This can be modeled by an
’attention switch’—where the contestants’ motion is affected by their
interaction with the resource (which brings them into contest range) only when
they are relatively far apart, and is otherwise dominated by their
interactions with each other. Therefore, we write $V_{\mathrm{tot}\rightarrow
i}$ as follows

$V_{\mathrm{tot}\rightarrow
i}(\mathbf{r}_{i},\,\\{\mathbf{r}_{j}\\})=\sum_{j\,\neq\,i}V_{j\rightarrow
i}(\mathbf{r}_{i},\,\mathbf{r}_{j})+\theta(\mathrm{x}_{ij}-\mathrm{x}_{ij}^{\wedge})\,\,V_{\mathrm{res}\rightarrow
i}(\mathbf{r}_{i})$ (2)

where the $V_{j\rightarrow i}$’s are the ’contestant interaction
potentials’—each encoding the influence of a rival contestant $j$ on the
motion of contestant $i$, $V_{\mathrm{res}\rightarrow i}$ is the effective
’resource potential’ felt by contestant $i$—which encodes the influence of the
resource on the motion of contestant $i$, and $\theta(x)$ is a Heaviside step
function—where
$\mathrm{x}_{ij}\equiv|\mathbf{r}_{i}-\mathbf{r}_{j}|/\mathrm{x}_{0}$ is the
dimensionless (scaled) distance between contestants $i$ and $j$
($\mathrm{x}_{0}$ is a typical length scale), and $\mathrm{x}_{ij}^{\wedge}$
is the ’contest onset’ distance between $i$ and $j$ (which is defined in
section 1.3). For now, we will treat these effective potentials as time-
independent, but note that they could also vary with (interaction)
time—notably due to accumulation of fighting costs and learning—as discussed
briefly in section 1.6.

Before discussing explicit functional forms for $V_{\mathrm{res}\rightarrow
i}$ and $V_{j\rightarrow i}$, note that Eqs. (1) and (2) can be used to
describe the interactions between any number of contestants that compete over
a single resource (where the motion of each contestant is governed by its own
Langevin equation), and that one can further generalize Eq. (2) to include the
influence of more than one resource (such that the contestants move in a
’field’ of resources). Nevertheless, although many-contestant and/or many-
resource scenarios exist in nature (a specific many-contestant scenario is
discussed in section 1.5), the simplest scenario of a dyadic contest, where
two contestants fight over a resource, is by far the most common and well
studied [32, 26]. For this reason, and since a ’contest’ (in our context) is
inherently a pairwise concept, we will mostly deal with the dyadic version of
Eq. (2),

$V_{\mathrm{tot}\rightarrow
i}(\mathbf{r}_{i},\,\mathbf{r}_{j})=V_{j\rightarrow
i}(\mathbf{r}_{i},\,\mathbf{r}_{j})+\theta(\mathrm{x}_{ij}-\mathrm{x}_{ij}^{\wedge})\,\,V_{\mathrm{res}\rightarrow
i}(\mathbf{r}_{i})$ (3)

where $i,\,j=1,\,2$ are now the only two contestants. We proceed by describing
the spatio-temporal features common to contest dynamics in many animal
species, and map these generic rules into the shapes of the effective
potentials $V_{\mathrm{res}\rightarrow i}$ and $V_{j\rightarrow i}$. The finer
details that these potentials have in specific animals and setups can be
extracted empirically from contestant trajectories, as described in [9] and
[10]. In the following, we often visualize and apply the model in a two-
dimensional space, but note that the model is equally applicable in three
dimensions.

A typical dyadic contest is initiated when two animals seek access to a
spatially localized resource, commonly a territory or a mate [39, 40, 41]. In
terms of their physical influence, these resources act as attractors that
drive contestants towards each other until conflict becomes inevitable, and
can thus be thought of as the effective potential landscape on which (and due
to which) contests take place. The shape of this potential landscape depends
on the properties of the resource. For most resources, an adequate model might
be a simple radially-symmetrical ’sink’, for example

$V_{\mathrm{res}\rightarrow
i}(\mathbf{r}_{i})=p_{i}\ln(|\mathbf{r}_{i}|+\epsilon),$ (4)

where $p_{i},\,\epsilon>0$ are parameters ($p_{i}$ sets the attractiveness of
the resource, as perceived by contestant $i$). The potential landscape created
by Eq. (4) is visualized in Fig. 1A. For some animate resources, however,
behavior and setup could render the shape of the resource potential’s
landscape less trivial. For example, in the natural contest arena that is the
orb-web of a female Trichonephila clavipes spider, the resource landscape on
which male contestants are moving comprises of the female and the architecture
of its web. In this system, males are observed to approach the female from the
back and avoid its front—were the potentially cannibalistic female is most
reactive [9]. This resource should therefore be modeled by a non-isotropic
effective potential. Specifically, in [9], the influence of this resource is
described by a potential with a landscape as in Fig. 1B. Regardless of these
system-specific details, the global qualitative effect of resource landscapes
is generic: to bring the contestants into contest range due to their mutual
attraction to the resource. This generic property implies that effective
resource potentials have a global minimum, as in Fig. 1 A and B.

Figure 1: Effective resource potentials. (A) The landscape of an effective
’resource potential’ with a simple radially-symmetrical form, as in Eq. (4).
(B) The landscape of a non-isotropic effective resource potential, which
encodes the influence of a female T. clavipes spider and its orb-web on the
motion of male spider contestants [9]. Adapted from refs. [10] and [9].

The inter-contestant interactions, including the ’contest’ itself, are encoded
by the contestant interaction potential $V_{j\rightarrow i}$—the model’s most
essential building block. The archetypal shape of this potential can be
constructed based on the following generic (and in a sense minimal) features
of contest behavior, which are stated in terms of effective attraction or
repulsion between the contestants depending on the inter-contestant distance:
(1) Long-range repulsion due to mutual avoidance (which can be surmounted due
to the attracting resource), (2) Medium- to short-range attraction when the
contestants reach a separation distance in which conflict escalation is
inevitable (and hence move towards each other), (3) Strong repulsion at
contact, and (4) The strength of the interaction decays to zero when the
contestants are far apart. Importantly, conflict escalation is associated with
the tendency to decrease the inter-contestant distance (effective attraction),
while conflict de-escalation is associated with the tendency to increase this
distance (effective repulsion).

The above rules amount to a contestant interaction potential with a
qualitative shape as in Fig. 2 A and B, which can be satisfied by various
functional forms. One such particular potential is a combination of a
logarithmic repulsion and an attractive Gaussian well,

$V_{j\rightarrow i}(\mathbf{r}_{i},\,\mathbf{r}_{j})=-\alpha_{j\rightarrow
i}\exp(-\beta{\mathrm{x}_{ij}}^{2})-\delta_{j\,\rightarrow\,i}\ln(\mathrm{x}_{ij}),\qquad\mathrm{x}_{ij}=\frac{|\mathbf{r}_{i}-\mathbf{r}_{j}|}{\mathrm{x}_{0}}$
(5)

where $\alpha_{j\rightarrow i}$ and $\delta_{j\rightarrow i}$ are positive
interaction parameters that set the strengths of effective attraction and
repulsion (experienced by contestant $i$ when interacting with a rival $j$),
$\beta>0$ determines the range of effective attraction (assumed to be the same
for both contestants), and $\mathrm{x}_{0}$ is a typical length scale. This
potential has two _distinct_ local extrema (as in Fig. 2B), at

$\mathrm{x}^{\mathrm{min}}_{ij}=\sqrt{-\frac{1}{\beta}W_{0}(\Gamma_{j\rightarrow
i})}\quad\text{and}\quad\mathrm{x}^{\mathrm{max}}_{ij}=\sqrt{-\frac{1}{\beta}W_{-1}(\Gamma_{j\rightarrow
i})},\quad\text{with}\quad\Gamma_{j\rightarrow i}=-\frac{\delta_{j\rightarrow
i}}{2\alpha_{j\rightarrow i}},$ (6)

(where $W_{-1}(x)$ ($-1/e\leq x<0$) and $W_{0}(x)$ ($-1/e\leq x$) are the two
real branches of the Lambert $W$ function [42]) provided that

$\frac{\alpha_{j\rightarrow i}}{\delta_{j\rightarrow i}}>\frac{e}{2},$ (7)

where $e$ is Euler’s number. The values of $\alpha_{j\rightarrow i}$ and
$\delta_{j\rightarrow i}$ are related to contestant $i$’s assessment of itself
and of its rival, and therefore depend on the ’strength’, or (henceforth)
effective ’size’, of each contestant. In the animal contest literature, this
effective size is typically termed ’resource holding potential’ (RHP) [18, 19,
20, 21, 22, 23, 24, 25, 26]. Not to be confused with our model’s effective
interaction potentials, the RHP is a species-dependent measure for a
contestant’s ability to win fights, and is commonly correlated (but not
strictly interchangeable) with size-related features, such as claw size in
crabs or antler size in deer [26, 32]. In our case, $\alpha_{j\rightarrow i}$
reflects the motivation of contestant $i$ to escalate the interaction (that
is, how aggressive $i$ is) and is therefore associated with the (absolute or
relative) effective size of $i$, while $\delta_{j\rightarrow i}$ reflects how
intimidating (repulsive) the rival $j$ is perceived by $i$, and is therefore
associated with the (absolute or relative) effective size of $j$. Under this
interpretation of $\alpha_{j\rightarrow i}$ and $\delta_{j\rightarrow i}$, the
case where Eq. (7) is not satisfied corresponds to contestant $i$ being ’not
motivated enough’, or equivalently, ’too intimidated’, to escalate the
interaction with the rival $j$. In section 1.4, we discuss explicit models for
the dependence of $\alpha_{j\rightarrow i}$ and $\delta_{j\rightarrow i}$ on
the contestants’ effective sizes and the implications of these models in the
context of nonreciprocal interactions.

The interaction potential of Eq. (5) describes a rather minimal contest
escalation scheme, in the sense that it features a single escalation barrier
(local maximum, see Fig. 2B) towards a short-range interaction. Proposing this
functional form as the ’archetypal’ representation of contest interactions is
motivated by the premise that contestant dynamics in the vicinity of this
ultimate escalation stage account for the qualitative essence of contests in
many animal species. Nevertheless, system-specific details that diverge from
this minimal description can be included in $V_{j\rightarrow i}$, as long as
they correspond to escalation stages that occur at typical inter-contestant
distances. For example, such stages exist in contests that escalate through an
intermediate evaluation- or display stage [25, 43, 44, 45], where this stage
precedes the ultimate escalation into a short-range interaction. These two-
stage contest escalation schemes can be encoded by a contestant interaction
potential with a qualitative shape as in Fig. 2 C and D. In ref. [9], the
interaction between male spider contestants—which mostly interact through a
single modality of vibratory cues transmitted through the web—was modeled by a
similar two-stage potential.

Completing our basic pairwise treatment of contests, a respective interaction
potential $V_{i\rightarrow j}$ encodes the influence of contestant $i$ on the
motion of the rival $j$. Importantly, since these potentials manifest the
(behavioral and perceptual) asymmetries between the contestants, the
adversarial interactions encoded by $V_{j\rightarrow i}$ and $V_{i\rightarrow
j}$ are nonreciprocal (they violate Newton’s third law). This means that, for
a general pair of contestants $i$ and $j$, $V_{j\rightarrow i}\neq
V_{i\rightarrow j}$ (as illustrated in Fig. 3A). The influence of
$V_{j\rightarrow i}$ and $V_{i\rightarrow j}$ on the motion of the contestants
is governed by the nonreciprocal effective forces that they generate,

$\mathbf{F}_{j\rightarrow i}(\mathrm{x}_{ij})=-\frac{dV_{j\rightarrow
i}(\mathrm{x}_{ij})}{d\mathrm{x}_{ij}}\hat{\mathbf{x}}_{ij}\quad\text{and}\quad\mathbf{F}_{i\rightarrow
j}(\mathrm{x}_{ij})=-\frac{dV_{i\rightarrow
j}(\mathrm{x}_{ij})}{d\mathrm{x}_{ij}}(-\hat{\mathbf{x}}_{ij}),$ (8)

where
$\hat{\mathbf{x}}_{ij}=(\mathbf{r}_{i}-\mathbf{r}_{j})/|\mathbf{r}_{i}-\mathbf{r}_{j}|$
denotes the unit vector in the $j\rightarrow i$ direction, and again, in
general, $\mathbf{F}_{j\rightarrow i}\neq-\mathbf{F}_{i\rightarrow j}$.

Figure 2: Effective contestant interaction potentials. (A) The landscape of an
effective ’contestant interaction potential’ $V_{j\rightarrow i}$, as given in
Eq. (5). (B) The profiles of $V_{j\rightarrow i}$ and its corresponding force
$F_{j\rightarrow i}=-dV_{j\rightarrow i}/d\mathrm{x}_{ij}$ as a function of
the inter-contestant distance $\mathrm{x}_{ij}$ (Eq. (5) with
$\alpha_{j\rightarrow i}=7$, $\delta_{j\rightarrow i}=3$, and $\beta=1$). (C,
D) A more elaborate $V_{j\rightarrow i}$ that encodes a two-stage contest
escalation scheme. Adapted from ref. [10].

###  1.3 Definition and duration of a pairwise contest

While the adversarial inter-contestant interactions that govern contest
dynamics are nonreciprocal, to clearly define a ’contest’ in our model (in a
sense that simultaneously applies to both contestants) it is necessary to
define a pairwise contest onset between our model’s contestant particles. For
this purpose we consider the combined contributions of the effective forces
$\mathbf{F}_{j\rightarrow i}$ and $\mathbf{F}_{i\rightarrow j}$ to the
relative motion between $i$ and $j$ along the inter-contestant axis. Taking
the position of contestant $j$ as a fixed point of reference, contestant $i$
appears to be driven along $\hat{\mathbf{x}}_{ij}$ by a relative ’contest
force’ $\mathbf{F}_{\mathrm{contest}}$, given by

$\mathbf{F}_{\mathrm{contest}}(\mathrm{x}_{ij})=\mathbf{F}_{j\rightarrow
i}(\mathrm{x}_{ij})-\mathbf{F}_{i\rightarrow
j}(\mathrm{x}_{ij})=-\frac{d}{d\mathrm{x}_{ij}}\left[V_{j\rightarrow
i}(\mathrm{x}_{ij})+V_{i\rightarrow
j}(\mathrm{x}_{ij})\right]\hat{\mathbf{x}}_{ij}$ (9)

where note that $\mathbf{F}_{i\rightarrow j}$, which is applied by $i$ on $j$
in the $-\hat{\mathbf{x}}_{ij}$ direction, appears to drive $i$ relative to
$j$ in the opposing $+\hat{\mathbf{x}}_{ij}$ direction. Eq. (9), which
describes an effectively reciprocal force (it acts on $i$ and $j$ with the
same magnitude and in opposing directions), motivates the definition of a
corresponding relative ’contest potential’ $V_{\mathrm{contest}}$ as the sum
of the individual interaction potentials,

$V_{\mathrm{contest}}(\mathrm{x}_{ij})=V_{j\rightarrow
i}(\mathrm{x}_{ij})+V_{i\rightarrow j}(\mathrm{x}_{ij})$ (10)

such that $F_{\mathrm{contest}}=-dV_{\mathrm{contest}}/d\mathrm{x}_{ij}$. Note
that according to the particular contestant interaction potential of Eq. (5),
the relative contest potential is given by

$V_{\mathrm{contest}}(\mathrm{x}_{ij})=-(\alpha_{j\rightarrow
i}+\alpha_{i\rightarrow
j})\exp(-\beta{\mathrm{x}_{ij}}^{2})-(\delta_{j\rightarrow
i}+\delta_{i\rightarrow j})\ln(\mathrm{x}_{ij}),$ (11)

and its two local extrema are at

$\mathrm{x}_{ij}^{\vee}=\sqrt{-\frac{1}{\beta}W_{0}(\Gamma_{ij})}\quad\text{and}\quad\mathrm{x}_{ij}^{\wedge}=\sqrt{-\frac{1}{\beta}W_{-1}(\Gamma_{ij})},\quad\text{with}\quad\Gamma_{ij}=-\frac{\delta_{j\rightarrow
i}+\delta_{i\rightarrow j}}{2\left(\alpha_{j\rightarrow
i}+\alpha_{i\rightarrow j}\right)},$ (12)

where $\mathrm{x}_{ij}^{\vee}$ and $\mathrm{x}_{ij}^{\wedge}$ denote the
locations of the minimum and maximum, respectively, as indicated in Fig. 3B.
The contest potential of Eq. (11) has these two _distinct_ extrema when

$\frac{\alpha_{j\rightarrow i}+\alpha_{i\rightarrow j}}{\delta_{j\rightarrow
i}+\delta_{i\rightarrow j}}>\frac{e}{2},$ (13)

and becomes repulsive for all $\mathrm{x}_{ij}$ when this inequality is not
satisfied.

Figure 3: Nonreciprocity and definition of a contest. (A) The potentials
$V_{j\rightarrow i}$ and $V_{i\rightarrow j}$ encode nonreciprocal
interactions that manifest asymmetries between contestants (in general,
$V_{j\rightarrow i}\neq V_{i\rightarrow j}$). Here, these asymmetries are
described by variations in the interaction parameters of Eq. (5), where
$V_{j\rightarrow i}$ is shown with $\alpha_{j\rightarrow i}=7$ and
$\delta_{j\rightarrow i}=3$, and $V_{i\rightarrow j}$ with
$\alpha_{j\rightarrow i}=8$ and $\delta_{j\rightarrow i}=4$ ($\beta=1$ for
both). (B) The relative ’contest potential’
$V_{\mathrm{contest}}=V_{j\rightarrow i}+V_{i\rightarrow j}$ governs the
relative motion between the contestants along the inter-contestant axis
(arrows). The distance $\mathrm{x}_{ij}=\mathrm{x}_{ij}^{\wedge}$ is defined
as the contest onset. (C) Typical simulated trajectories of two identical
contestants ($V_{j\rightarrow i}=V_{i\rightarrow j}$) in the vicinity of a
resource. The dynamics of each contestant particle was simulated based on Eq.
(1) with $\eta=1$, $D=0.5$, $v_{0}=0$, and $V_{j\rightarrow i}$ as in A (see
ref. [10] for further details). Their mutual attraction to the resource brings
the contestants into contest range. Segments of the trajectories in which the
contestants were engaged in a contest
($\mathrm{x}_{ij}<\mathrm{x}_{ij}^{\wedge}$) are shown in red. (D) The
distance between the contestants throughout the simulation. As evident in the
close-up view, the contestants spend the majority of the contest near the
minimum of $V_{\mathrm{contest}}$. Adapted from ref. [10].

Regardless of the particular functional form of Eq. (11), but assuming that
$V_{\mathrm{contest}}$ has a qualitative shape as in Fig. 3B (with
corresponding local extrema at some $\mathrm{x}_{ij}^{\vee}$ and
$\mathrm{x}_{ij}^{\wedge}$), we now define the onset of a contest according to
the direction of relative motion between the contestants—governed by the sign
of $F_{\mathrm{contest}}$ (as illustrated in Fig. 3B). When the contestants
reach a separation $\mathrm{x}_{ij}$ that is shorter than
$\mathrm{x}_{ij}^{\wedge}$ (driven by their mutual attraction to the resource
and the stochastic components of their motion), the relative force becomes
attractive ($F_{\mathrm{contest}}<0$) towards $\mathrm{x}_{ij}^{\vee}$, and
the inter-contestant dynamics is characterized by a transient bounded state.
In Fig. 3 C and D, the dynamics of two contestant particles getting in and out
of this transient bounded state is illustrated by simulated contestant
trajectories. We identify this bounded state with the ultimate escalation into
a short-range contest, in which the contestants are completely engaged with
each other, and define $\mathrm{x}_{ij}^{\wedge}$ as the contest onset
distance—such that two of our model’s contestant particles are considered to
be ’engaged in a contest’ when $\mathrm{x}_{ij}<\mathrm{x}_{ij}^{\wedge}$ (in
Eqs. (2) and (3), this is the regime where $\theta=0$). Note that even if the
contest potential has a more elaborate shape than in Fig. 3B (as in Fig. 2D
and ref. [9]), the above principles can still be used to define an analogous
pairwise contest onset—notably by taking the onset of the most short-range
potential well to play the role of $\mathrm{x}_{ij}^{\wedge}$.

From our definition of a contest, it immediately follows that contest duration
($t_{\mathrm{c}}$)—a common measurable of interest in studies of animal
contests [32]—is equivalent to the escape time of our contestant particles
from their transient bounded state within the potential well of
$V_{\mathrm{contest}}$ (as illustrated in Fig. 3D). Then, with an effective
’contest bounding energy’ of

$U=V_{\mathrm{contest}}(\mathrm{x}_{ij}^{\wedge})-V_{\mathrm{contest}}(\mathrm{x}_{ij}^{\vee}),$
(14)

the mean contest duration is analogous to the Karmer’s escape time [46], and
thus follows an Arrhenius’ law

$\langle
t_{\mathrm{c}}\rangle\simeq\Pi\,\exp\left(\frac{U}{T_{\mathrm{eff}}}\right),$
(15)

where the pre-exponential factor $\Pi$ and the effective temperature
$T_{\mathrm{eff}}$ generally depend on the shape of the potential and on the
properties of the active particles [46]. Eq. (15) assumes that, after they
first cross the contest onset $\mathrm{x}_{ij}^{\wedge}$, the contestants
reach the minimum $\mathrm{x}_{ij}^{\vee}$ quickly—within an average time that
is significantly shorter than $t_{\mathrm{c}}$—and spend the rest of the
contest near $\mathrm{x}_{ij}^{\vee}$ until they escape, as in Fig. 3D.
Deriving analytical expressions for $\Pi$ and $T_{\mathrm{eff}}$ in the
context of our active particles setup is possible in some limits (see [47] for
a rather instructive derivation for active particles obeying the same Langevin
Equation as ours). But importantly, one fundamental assumption in Kramer’s
analytical derivation [46], that the potential well is very deep compared to
the stochastic fluctuations, is generally not satisfied for our contestant
particles—as they have to describe the fast engage-and-disengage dynamics of
typical animal contests [17, 9]. Nevertheless, even without exact expressions
for $\Pi$ and $T_{\mathrm{eff}}$, Eq. (15) can be used to predict the
qualitative trends of $\langle t_{\mathrm{c}}\rangle$ with respect to
variations in the parameters of $V_{j\rightarrow i}$ and $V_{i\rightarrow j}$
(which determine $U$), as shown in ref. [10].

###  1.4 Assessment strategies and interaction asymmetries

The type of RHP-related information used by contestants in their decision-
making, commonly termed the ’assessment strategy’, has been extensively
studied in game-theoretic contest models—especially in order to describe
asymmetric contests [48, 26, 30] [recall the definition of RHP, given after
Eq. (7)]. In [10], we proposed that different assessment strategies can be
expressed in our framework in terms of an ’assessment function’
$\mathbf{A}_{j\rightarrow i}$ (of contestant $i$ with respect to a rival $j$),
which defines the functional relations between the interaction parameters of
$V_{j\rightarrow i}$ and $V_{i\rightarrow j}$ and the contestants’ effective
sizes—where the effective size of contestant $i$ is defined as a dimensionless
proxy for its RHP,

$m_{i}=\frac{\text{RHP of contestant $i$}}{\text{RHP of reference}},$ (16)

with the ’RHP of reference’ being, for example, the population average. For
the contestant interaction potential $V_{j\rightarrow i}$ of Eq. (5), with
interaction parameters $\alpha_{j\rightarrow i}$ and $\delta_{j\rightarrow
i}$, we write $\mathbf{A}_{j\rightarrow i}$ as

$\mathbf{A}_{j\rightarrow
i}(m_{i},\,m_{j})\equiv\begin{pmatrix}\alpha_{j\rightarrow
i}(m_{i},\,m_{j})\\\ \delta_{j\rightarrow
i}(m_{i},\,m_{j})\end{pmatrix}=\begin{pmatrix}\alpha_{0}\,{m_{i}}^{s}(m_{i}/m_{j})^{s_{Q}}\\\
\delta_{0}\,{m_{j}}^{r}(m_{j}/m_{i})^{r_{Q}}\end{pmatrix},$ (17)

where $\alpha_{0}$ and $\delta_{0}$ are scaling parameters, and the different
power laws represent: ${m_{i}}^{s}$—absolute self-assessment;
${m_{j}}^{r}$—absolute rival-assessment; $(m_{i}/m_{j})^{s_{Q}}$ and
$(m_{j}/m_{i})^{r_{Q}}$—relative self- and rival-assessment, respectively.
These relations are best understood by recalling that $\alpha_{j\rightarrow
i}$ reflects the motivation of contestant $i$ to escalate the interaction with
the rival $j$, and therefore increases with $m_{i}$ but decreases with
$m_{j}$, while $\delta_{j\rightarrow i}$ reflects how intimidating the rival
$j$ is perceived by $i$, and therefore decreases with $m_{i}$ but increases
with $m_{j}$. Note that the rival’s assessment function,
$\mathbf{A}_{i\rightarrow j}=\left(\alpha_{i\rightarrow
j},\,\delta_{i\rightarrow j}\right)^{T}$, is obtained by swapping the
functional roles of $m_{i}$ and $m_{j}$ in Eq. (17) [that is,
$\mathbf{A}_{i\rightarrow j}(m_{i},\,m_{j})=\mathbf{A}_{j\rightarrow
i}(m_{j},\,m_{i})$]. The assessment strategy, which is now defined in terms of
how $\alpha_{j\rightarrow i}$ and $\delta_{j\rightarrow i}$ vary with $m_{i}$
and $m_{j}$, is represented by the (non-negative) values of the exponents $s$,
$r$, $s_{Q}$, $r_{Q}$. In particular, with $s>0$ and $r,\,s_{Q},\,r_{Q}=0$, we
obtain our model’s representation of the so-called ’pure self-assessment’
[32],

$\mathbf{A}_{j\rightarrow
i}(m_{i},\,m_{j})=\begin{pmatrix}\alpha_{0}\,{m_{i}}^{s}\\\
\delta_{0}\end{pmatrix}\quad\text{for pure self-assessment},$ (18)

where contestants only assess their own (absolute) effective size and
disregard their rival’s effective size, and with $s,\,r=0$ and
$s_{Q},\,r_{Q}>0$, we obtain the representation for pure ’mutual assessment’
[32],

$\mathbf{A}_{j\rightarrow
i}(m_{i},\,m_{j})=\begin{pmatrix}\alpha_{0}\,(m_{i}/m_{j})^{s_{Q}}\\\
\delta_{0}\,(m_{j}/m_{i})^{r_{Q}}\end{pmatrix}\quad\text{for pure mutual
assessment},$ (19)

where the contestants conduct a purely relative assessment of their own
effective size with respect to their rival’s effective size. Note that, by
construction, for $m_{i}=m_{j}=1$ ($\text{'RHP of contestant $i$'}=\text{'RHP
of reference'}$ in (16)), any mode of assessment described by Eq. (17) yields
the same interaction potentials.

Fig. 4 illustrates the differences between the assessment functions of Eqs.
(18) and (19) in terms of how $V_{i\rightarrow j}$ and $V_{j\rightarrow i}$
vary with $m_{i}$ and $m_{j}$; It shows these potentials, based on Eq. (5),
under pure self-assessment (Eq. (18) with $s=1$) and pure mutual assessment
(Eq. (19) with $s_{Q}=1$ and $r_{Q}=2$), for the interactions of small size-
matched contestants, a small contestant with a large contestant, and large
size-matched contestants—where ’small’ and ’large’ refer to the contestants’
effective sizes. Notably, compared to pure self-assessment—which detects
absolute changes in $m_{i}$ and $m_{j}$ (compare Fig. 4 A, B, and C), pure
mutual assessment is scale-invariant, as Eq. (19) satisfies
$\mathbf{A}_{i\rightarrow j}(c\,m_{i},\,c\,m_{j})=\mathbf{A}_{i\rightarrow
j}(m_{i},\,m_{j})$ for any scalar $c>0$. This property is illustrated in Fig.
4 D and F by the fact that, under pure mutual assessment, the interaction
potentials of two size-matched small contestants and two size-matched large
contestants—are identical. In addition, detection of asymmetries between
$m_{i}$ and $m_{j}$ is much stronger in mutual assessment, as evident by the
pronounced broken symmetry between $V_{j\rightarrow i}$ and $V_{i\rightarrow
j}$ for the interaction of unmatched contestants (compare Fig. 4 B and E).

Figure 4: Assessment strategies in terms of contestant interaction potentials.
Under pure self-assessment (Eq. (18) with $s=1$) and pure mutual assessment
(Eq. (19) with $s_{Q}=1$ and $r_{Q}=2$), the interaction potentials
$V_{i\rightarrow j}$ and $V_{j\rightarrow i}$ are shown for the interaction of
(A, D) small size-matched contestants ($m_{j}=m_{i}=0.8$), (B, E) a small
contestant with a large contestant ($m_{j}=0.8,\,m_{i}=1.2$ for pure self-
assessment, and $m_{j}=0.9,\,m_{i}=1.1$ for pure mutual assessment [different
values merely for graphing purposes]), and (C, F) large size-matched
contestants ($m_{j}=m_{i}=1.2$). For reference, each graph also shows the
potentials of medium size-matched contestants ($m_{j}=m_{i}=1$) in grey. Eqs.
(18) and (19) were used with $\alpha_{0}=7$ and $\delta_{0}=3$. Adapted from
ref. [10].

It should be noted that even a highly asymmetric inter-contestant interaction
as in Fig. 4E (where notably, $V_{i\rightarrow j}$ is strictly repulsive while
$V_{j\rightarrow i}$ has a deep attractive well with a lowered escalation
barrier), can still describe a pairwise ’contest’ as defined in section 1.3—as
long as the potential well of $V_{\mathrm{contest}}=V_{j\rightarrow
i}+V_{i\rightarrow j}$ exists (that is, as long as $V_{\mathrm{contest}}$
supports a transient bounded state for some
$\mathrm{x}_{ij}<\mathrm{x}_{ij}^{\wedge}$). This is illustrated in Fig. 5 by
very different contest interactions (in terms of the asymmetry between
$V_{j\rightarrow i}$ and $V_{i\rightarrow j}$) that were chosen to yield the
same contest potential. As discussed in section 1.5, the qualitative
properties of contest trajectories with symmetric interactions (as in Fig. 5A)
will clearly differ from those of contests with strongly asymmetric
interactions (as in Fig. 5B). But interestingly, the mean contest duration
predicted by Eqs. (14) and (15) is the same for both of these interactions (in
[10], this property is shown in simulations), as they share the same contest
potential of Fig. 5C. From a behavioral perspective, this can be understood
through the heuristic that both interactions have the same ’total’ amounts of
motivation ($\alpha_{j\rightarrow i}+\alpha_{i\rightarrow j}$) and
intimidation ($\delta_{j\rightarrow i}+\delta_{i\rightarrow j}$), and
therefore persist for the same amount of time, although in Fig. 5A, these are
split equally between the contestants, while in Fig. 5B, contestant $i$ has
most of the motivation and contestant $j$ has most of the intimidation.

Figure 5: Very different contest interactions with the same contest potential.
(A) The contestant interaction potentials of a symmetric interaction between
two identical contestants ($V_{j\rightarrow i}=V_{i\rightarrow j}$, with
$\alpha_{j\rightarrow i}=\alpha_{i\rightarrow j}=7.14$, $\delta_{j\rightarrow
i}=\delta_{i\rightarrow j}=3.25$, and $\beta=1$). (B) The contestant
interaction potentials of a strongly asymmetric interaction
($\alpha_{j\rightarrow i}=8.55$, $\alpha_{i\rightarrow j}=5.73$,
$\delta_{j\rightarrow i}=2.02$, $\delta_{i\rightarrow j}=4.48$, and
$\beta=1$), in which $V_{i\rightarrow j}$ is strictly repulsive. (C) These
very different interactions have the same contest potential
$V_{\mathrm{contest}}=V_{j\rightarrow i}+V_{i\rightarrow j}$, since
$\alpha_{j\rightarrow i}+\alpha_{i\rightarrow j}$ and $\delta_{j\rightarrow
i}+\delta_{i\rightarrow j}$ (and $\beta$) are the same for both.

###  1.5 Chase dynamics emerges from interaction asymmetry

Chasing behavior is evidently a ubiquitous feature of agonistic interactions
in animals, and is intuitively associated with some form of asymmetry between
the chasing individual and the one being chased. The essence of this behavior
is captured by our model’s contestant particles, which exhibit chase dynamics
as an emergent property of broken symmetry in their nonreciprocal interaction
forces. As illustrated in Fig. 6A for strongly asymmetric contestant
interaction potentials, within the ’chase’ range the larger contestant is
attracted by a deep potential well which drives it towards its smaller rival,
while the smaller contestant is repelled in the same direction—away from its
larger rival. Together, these effects gives rise to directed chase behavior
during the contest. This ’microscopic’ chase—at the single-particle level—is
analogous, both in qualitative nature and in the mechanism that creates it, to
the macroscopic chase and traveling wave phenomena that occur between
(chemical) species in active mixtures in the presence of nonreciprocal
interactions [4, 49].

Fig. 6 B and C compare typical trajectories of strongly asymmetric and
symmetric contests under ’pure mutual assessment’ in simulations (namely, the
dependence of $\alpha_{j\rightarrow i}$ and $\delta_{j\rightarrow i}$ on the
contestants’ effective sizes is given by Eq. (19), with $s_{Q}=1$ and
$r_{Q}=2$). While symmetric contests are characterized by scrambled and
relatively localized trajectories, the trajectories of strongly asymmetric
contests feature substantial directional alignment and persistence, and
consequently much greater displacement, due to chase dynamics. The extent to
which chase dynamics governs the contestants’ trajectories can be quantified
by considering the direction correlation between the velocity of the
contestants’ midpoint—measured by $\hat{\mathbf{v}}_{\mathrm{m}}$, and the
inter-contestant direction $\hat{\mathbf{x}}_{ij}$, as defined in Fig. 6B
(Inset). The temporal mean of this ’chase correlator’,
$\langle\hat{\mathbf{v}}_{\mathrm{m}}\cdot\hat{\mathbf{x}}_{ij}\rangle$, is
zero for symmetric interactions, and
$\left|\langle\hat{\mathbf{v}}_{\mathrm{m}}\cdot\hat{\mathbf{x}}_{ij}\rangle\right|\sim
1$ if the interaction is dominated by the chase phase.

Accounting for the dynamical aspects of asymmetric contests was particularly
important in the natural system of T. clavipes male spider contestants studied
in ref. [9]—as mature specimen of these male spiders can vary greatly in size
[50, 51]. The observed coexistence of sexually-mature T. clavipes males that
differ in their weights by as much as an order of magnitude is puzzling, as
one would expect stabilizing evolution to prevent such extraordinary size
variability [52]. Fig. 6D shows the experimentally-calibrated, two-stage
interaction potentials used to model the interactions between the spider
contestants of ref. [9]. The observation-based model for assessment in this
system (if it were to be mapped to the one-stage interaction potential of Eq.
(5)) corresponds to the following assessment function

$\mathbf{A}_{j\rightarrow
i}(m_{i},\,m_{j})=\begin{pmatrix}\alpha_{0}\,m_{i}\\\
\delta_{0}\,m_{j}\end{pmatrix}.$ (20)

The way in which Eq. (20) detects effective size asymmetries is qualitatively
similar to the pure mutual assessment of Eq. (19), as evident by how
$V_{j\rightarrow i}$ and $V_{i\rightarrow j}$ vary with an increasing $m_{i}$
in Fig. 6D. Accordingly, it predicts substantial chase dynamics for large
asymmetries in the contestants’ effective sizes, as shown in Fig. 6E Left.
This prediction is in good agreement with the dynamics of real spider
contests, as shown in Fig. 6E Right.

Figure 6: Chase dynamics emerges from interaction asymmetry. (A) Strongly
asymmetric interaction potentials illustrate how interaction asymmetry leads
to chase dynamics. Shaded rectangles mark the chase range, within which the
larger contestant is attracted by a deep minimum, while the smaller contestant
is repelled in the same direction. Arrows indicate the direction of motion for
each contestant, as dictated by the signs of the forces $F_{j\rightarrow i}$
and $F_{i\rightarrow j}$. (B) Typical trajectories of strongly asymmetric and
symmetric contests in simulations. Chase dynamics is quantified by the
direction correlation between the velocity of the contestants’
midpoint—measured by $\hat{\mathbf{v}}_{\mathrm{m}}$, and the inter-contestant
direction $\hat{\mathbf{x}}_{ij}$ (see Inset). (C) Trajectories of the
contestants’ midpoint during contests under pure mutual assessment (the
dependence of $\alpha_{j\rightarrow i}$ and $\delta_{j\rightarrow i}$ on
$m_{i}$ and $m_{j}$ is given by Eq. (19), with $s_{Q}=1$ and $r_{Q}=2$) for
different effective size ratios. These trajectories, of equal durations,
demonstrate the effect of chase dynamics on the contestants’ displacement
during the contest. In B and C, ’X’ marks the contestants’ midpoint at contest
onset and circles mark the final positions. (D) Increasing asymmetry in the
contestant interaction potentials of ref. [9] (used to model the interaction
between male spider contestants) as a function of increasing $m_{i}\geq m_{j}$
(at constant $m_{j}$). (E) Profiles of
$\langle\hat{\mathbf{v}}_{\mathrm{m}}\cdot\hat{\mathbf{x}}_{ij}\rangle$ in
contests between two male spiders (from ref. [9]), in simulations and
experiments, for symmetric and asymmetric contests. In experiments, $m_{i}$
and $m_{j}$ denote the actual body weights of the spiders. Error bars show
SEM. Adapted from refs. [10] and [9]. See these works for further details.

The ability of the model in ref. [9] to account for these details motivated
its application to a many-contestant scenario taking place in this system. It
was found that local male density, defined as the number of males occupying a
single female’s web, is highly variable in wild populations of T. clavipes
[52]. Some studies suggested that this could resolve the puzzle of high size
variability, if large males have a significant reproductive advantage when
male density is high, but this advantage is diminished when male density is
low—allowing for a wide distribution of sizes to be stable [52]. Using
simulations, we provided in ref. [9] a mechanistic explanation for the
hypothesized competitive advantage of large males at high male densities. We
have shown that the repulsive effective interaction potentials inflicted by
large males on smaller males enable them to resolve the interactions with
their smaller rivals swiftly and at a distance—and thereby reach the female-
resource faster and retain it better, while small males form long-lasting
contests with other small males (which are more abundant, [52]), and therefore
suffer disproportionately from a high density of rivals, in which the
frequency of contests is high. Conversely, when male density is low, we have
found in simulations that large males have a less significant advantage at
reaching and retaining the female-resource [9]. On such female webs, which
coexist in nature with high-density webs [52], the disadvantages of being
large (e.g. higher detectability by the potentially cannibalistic female or by
other predators), may outweigh its benefits.

###  1.6 Time dependence of contest interactions

Until now we have treated inter-contestant interactions as if they do not
(explicitly) depend on time: the effective potentials that describe these
interactions were time-independent, and so the only ’time dependence’ in our
framework was due to the motion of the contestant particles (driven by their
governing Langevin equations), which continuously alters the potential
landscapes that they experience. However, contestant interaction potentials
reflect the contestants’ current effective sizes, and these could vary during
(and due to) the interaction—notably as fighting costs accumulate (and
decrease the effective sizes compared to their initial states) and as the
contestants update their perception of their own and of their rival’s status
(learning). The dependence of $m_{i}$ and $m_{j}$ on interaction time due to
such effects introduce into $V_{j\rightarrow i}$ and $V_{i\rightarrow j}$ an
explicit time dependence.

As an illustrative example for the notable consequences of such effects on the
properties of the inter-contestant interaction, consider the simple model from
ref. [10] for the dependence of the effective sizes on the accumulated contest
time $t_{\sigma}$, given by

$m_{i}(t_{\sigma})=\frac{\mu_{i}}{1+\left(K_{\mathrm{self}}+K_{ij}\,\dfrac{\mu_{j}}{\mu_{i}}\right)t_{\sigma}}\qquad\text{and}\qquad
m_{j}(t_{\sigma})=\frac{\mu_{j}}{1+\left(K_{\mathrm{self}}+K_{ij}\,\dfrac{\mu_{i}}{\mu_{j}}\right)t_{\sigma}},$
(21)

where $\mu_{i}$ and $\mu_{j}$ are the effective sizes of contestants $i$ and
$j$ when the contest starts, $K_{\mathrm{self}}$ is the rate at which ’self-
inflicted’ costs are accrued (e.g., average rate of energy expenditure),
$K_{ij}\,\mu_{j}/\mu_{i}$ is the rate at which ’rival-inflicted’ costs are
accrued (e.g., average rate of injuries), and $t_{\sigma}$ is accumulated only
during the contest (that is, when $\mathrm{x}_{ij}<\mathrm{x}_{ij}^{\wedge}$).
Eq. (21) naively assumes that costs are accrued continuously and
deterministically, and that the rates of accruing them are independent of the
instantaneous values of $m_{i}$ and $m_{j}$. Nevertheless, it includes a
simple form of feedback, since the rate of incurring costs from the rival is
proportional to the initial effective size ratio—such that the larger
contestant inflicts costs faster but incurs them slower. Note that according
to Eq. (21), while both $m_{i}$ and $m_{j}$ decrease monotonically with
$t_{\sigma}$, their ratio $m_{i}/m_{j}$ increases with $t_{\sigma}$ when
$\mu_{i}>\mu_{j}$ and $K_{ij}>0$. It approaches a constant as
$t_{\sigma}\rightarrow\displaystyle\infty$, where

$\lim_{t_{\sigma}\,\to\,\infty}\frac{m_{i}}{m_{j}}=\frac{\mu_{i}}{\mu_{j}}\cdot\frac{K_{\mathrm{self}}\,\mu_{j}+K_{ij}\,{\mu_{i}}^{2}}{K_{\mathrm{self}}\,\mu_{i}+K_{ij}\,{\mu_{j}}^{2}}.$
(22)

The dynamics of $m_{i}$, $m_{j}$, and $m_{i}/m_{j}$ according to Eq. (21) is
shown in Fig. 7A for a case where $\mu_{i}>\mu_{j}$.

Now, through the assessment function of Eq. (17), $V_{j\rightarrow i}$ and
$V_{i\rightarrow j}$ also depend on $t_{\sigma}$. In Fig. 7B, this dependence
is visualized under pure mutual assessment (Eq. (19) with $s_{Q}=1$ and
$r_{Q}=2$). The increasing asymmetry (in terms of the growing ratio
$m_{i}/m_{j}$) is manifested by an increasingly asymmetric inter-contestant
interaction—as evident by the opposing trends of the potentials with
$t_{\sigma}$ in Fig. 7B. An interesting consequence of this interaction-
induced amplification of asymmetry is that it offers a generic mechanism for
the emergence of strong chase dynamics during the contest, even if the initial
asymmetry is small. Moreover, note that under Eq. (19) with $s_{Q}=1$ and
$r_{Q}=2$, the bounding well that defines the contest regime becomes shallower
with $m_{i}/m_{j}$, and therefore with $t_{\sigma}$ (as shown in Fig.
7C)—which in turn reduces the mean contest duration (recall Eq. (15)). This
trend can eventually lead to a cost-driven contest termination when the
bounding well completely disappears and $V_{\mathrm{contest}}$ becomes
strictly repulsive (recall the this happens when Eq. (13) is not satisfied),
as explored in detail in ref. [10]. These effects, we propose, could model a
ubiquitous mechanism for contest resolution in real animal contests.

Figure 7: Time dependence of contest interactions. (A) The dynamics of the
effective sizes $m_{i}$ and $m_{j}$, and of their ratio $m_{i}/m_{j}$,
according to Eq. (21) with $K_{\mathrm{self}}=0.01$, $K_{ij}=0.1$, and with
initial effective sizes $\mu_{i}=1.1$ and $\mu_{j}=0.9$. (B) The dependence of
$m_{i}$ and $m_{j}$ on the accumulated contest duration $t_{\sigma}$ due to
costs means that $V_{i\rightarrow j}$ and $V_{j\rightarrow i}$ themselves
depend on $t_{\sigma}$. Here, these potentials vary with the values of $m_{i}$
and $m_{j}$ in A as $t_{\sigma}$ increases, according to pure mutual
assessment (Eq. (19) with $s_{Q}=1$ and $r_{Q}=2$). (C) Under this assessment
strategy, the bounding well that defines the contest regime becomes shallower
with $m_{i}/m_{j}$, and therefore with $t_{\sigma}$.

##  2 Effective gravity model of swarming midges

A special type of collective animal behavior is the formation of cohesive
swarms [53, 12]. These swarms are characterized by having the individual
animals moving in an uncoordinated manner, unlike flocks, which are
characterized by strong directional alignment, but nevertheless maintaining a
cohesive structure that does not disperse. There have been various theoretical
models proposed to describe this phenomenon, mainly based on velocity
alignment that is mitigated by large noise [54, 55].

Here we describe a theoretical model that is based on adaptive long-range
interactions [13]. For swarming midges the effective interactions are
attractive and mediated by acoustics due to the sound they emit while flying.
Male midges form large swarms at dusk, thereby attracting females for mating.
Within our model, midges form such cohesive swarms by accelerating towards
each other in proportion to the intensity of the sound received. For pure
acoustics the leading term (monopole) gives a functional form of the
acceleration which is similar to the force of gravitational attraction,
decaying as $1/r^{2}$.

An additional, and crucial, component of this model is adaptivity. Any sensory
mechanism in nature is subject to a modification due to adaptivity, which
means that the sensory mechanism adapts itself to the background stimuli – its
sensitivity is higher when the background is lower and vice versa. Exact
adaptation means that the steady state output is independent of the steady-
state level of input. This is part of a fold-change detection mechanism [56],
which is ubiquitous in nature, and involves a response whose entire shape,
including amplitude and duration, depends only on fold change and not on the
absolute levels of the input. The adaptivity is described by a scalar symmetry
of the sensory mechanism whereby multiplication of the input fields or stimuli
by a scalar does not change the reaction of the system.

Adding the two components together we reach the formulation of “adaptive
gravity” which is described below. The adaptive-gravity model was shown to
provide a good description of the observed mass and velocity profiles of
laboratory midge swarms, which differ from those produced by regular gravity
observed in star clusters [15]. Here we present some of the main theoretical
results that describe the special features of the adaptive-gravity, focusing
on the special properties that emerge due to these nonreciprocal long-range
interactions.

###  2.1 Adaptive gravity

Let us consider N identical particles, whose positions are denoted by
$\vec{x}_{i}$ ($i=1,..,N$), interacting by adaptive gravity. The contribution
of a particle at $\vec{x}_{j}$ to the (regular) gravitational potential at
$\vec{x}_{i}$ is given by

$V^{ij}(\vec{x}_{i})=\begin{cases}\frac{c_{g}}{|\vec{x}_{i}-\vec{x}_{j}|},&i\neq
j\\\ 0,&i=j.\end{cases}$ (23)

where $c_{g}$ is a positive constant with dimensions of $mass\cdot
length^{3}/time^{2}$.

The contribution of the particle at $\vec{x}_{j}$ to the gravitational force
at the point $\vec{x}_{i}$ is given by the gradient of the potential

$\vec{F}_{ij}^{g}(\vec{x}_{i})=-\nabla V^{ij}(\vec{x}_{i}),$ (24)

so that total gravitational (regular) linear force that acts on the particle
at $\vec{x}_{i}$ is

$\vec{F}_{i}^{g}(\vec{x}_{i})=\sum_{j=1}^{n}\vec{F}_{ij}^{g}(\vec{x}_{i})=-\sum_{j=1}^{n}\nabla
V^{ij}(\vec{x}_{i})$ (25)

In adaptive gravity, the regular gravitational force is “renormalized” by a
factor that depends on the total background force (scalar sum in the
denominator), resulting in a highly nonlinear form of the interactions between
the particles

$\vec{F}^{ad}(\vec{x}_{i})=-\frac{\sum_{j=1}^{n}\nabla
V^{ij}(\vec{x}_{i})}{1+(c_{g}^{-1}R_{ad}^{2})\sum_{j=1}^{n}|\nabla V^{ij}|}$
(26)

where $R_{ad}$ is the length scale over which adaptivity occurs. This way to
introduce adaptivity first appeared in the context of alignment interactions
[57, 58]. In the context of the midge swarm the effective interactions are
mediated by acoustic (and also visual) signals, such that the background force
represents the local background noise produced by the swarm.

Since $|\nabla V^{ij}|\sim|\vec{x}_{i}-\vec{x}_{j}|^{-2}$, when the distances
between pairs are large $|\vec{x}_{i}-\vec{x}_{j}|\gg\sqrt{N}R_{ad}$ the
adaptivity does not play a role, and the interaction approaches regular
gravity in this limit

$\vec{F}^{ad}(\vec{x}_{i})\sim\vec{F}^{g}(\vec{x}_{i}).$ (27)

On the other hand, when $|\vec{x}_{i}-\vec{x}_{j}|<\sqrt{N}R_{ad}$ adaptivity
is strong and then the following expression is a good approximation, which we
term “perfect adaptvity”

$\vec{F}^{ad}(\vec{x}_{i})\sim\frac{c_{g}}{R_{ad}^{2}\,N_{tot}(\vec{x}_{i})}\,\sum_{j=1}^{n}\nabla
V^{ij}(\vec{x}_{i}),$ (28)

where $N_{tot}(\vec{x}_{i})$ is a factor that represents the total background
“noise” (or the total stimuli) at the point $\vec{x}_{i}$ as a result of all
the interactions that involve the particle at this point

$N_{tot}(\vec{x}_{i})=\sum_{j=1}^{n}|\nabla V^{ij}(\vec{x}_{i})|.$ (29)

The appearance of this factor in the denominator of the expressions in Eqs.
(26) and (28) of the effective force, due to adaptivity, is responsible for
the nonlinear and nonreciprocal nature of the interactions in this model. It
is also a good example of the scalar symmetry in adaptation mechanisms [56].
We refer to $V^{ij}$ as our input fields and multiplication of the input
fields by a scalar does not change the reaction of the system $\vec{F}^{ad}$.

For two particles at positions $\vec{x}_{1}$ and $\vec{x}_{2}$ this factor is
reciprocal

$N_{tot}(\vec{x}_{1})=N_{tot}(\vec{x}_{2})=\frac{c_{g}}{|\vec{x}_{1}-\vec{x}_{2}|},$
(30)

and therefore the interaction is reciprocal. Note that the effective force
between two particles under perfect adaptivity becomes a constant, independent
of the separation between them. Also, energy is conserved and the force from
Eq. (26) can be integrated to give the effective potential between the pair of
particles

$U_{\mbox{\scriptsize
pair}}(|\vec{x}_{1}-\vec{x}_{2}|)=\frac{c_{g}}{R_{\mbox{\scriptsize
ad}}}\left[\arctan{(|\vec{x}_{1}-\vec{x}_{2}|/R_{\mbox{\scriptsize
ad}})}-\pi/2\right].$ (31)

Figure 8: The potential energy for two particles in adaptive gravity
$U_{\mbox{\scriptsize pair}}(|\vec{x}_{1}-\vec{x}_{2}|)$ calculated according
to Eq. 31 using $R_{ad}/c_{g}=10$ (blue line). The red line is regular gravity
potential for comparison with $c_{g}=1$. Adapted from [13].

Unlike regular gravity, where the potential energy diverges when the two
particles approach each other, the adaptivity prevents this and the potential
in Eq.31 approaches a finite value with a “cusp” shape, so that it is linear
in $|\vec{x}_{1}-\vec{x}_{2}|$. For $|\vec{x}_{1}-\vec{x}_{2}|\geq
R_{\mbox{\scriptsize ad}}$ the effective potential approaches the long-range
regular gravitational behavior of: $U_{\mbox{\scriptsize
pair}}=c_{g}/|\vec{x}_{1}-\vec{x}_{2}|$.

In the case of $N\geq 3$ the interactions are in general nonreciprocal as we
can see for three particles at $\vec{x}_{i},\;(i=1,..,3)$, which have
different normalization factors

$\displaystyle N_{tot}(\vec{x}_{1})$ $\displaystyle=$
$\displaystyle\frac{c_{g}}{|\vec{x}_{1}-\vec{x}_{2}|}+\frac{c_{g}}{|\vec{x}_{1}-\vec{x}_{3}|},$
(32) $\displaystyle N_{tot}(\vec{x}_{2})$ $\displaystyle=$
$\displaystyle\frac{c_{g}}{|\vec{x}_{2}-\vec{x}_{1}|}+\frac{c_{g}}{|\vec{x}_{2}-\vec{x}_{3}|}$
(33) $\displaystyle N_{tot}(\vec{x}_{3})$ $\displaystyle=$
$\displaystyle\frac{c_{g}}{|\vec{x}_{1}-\vec{x}_{3}|}+\frac{c_{g}}{|\vec{x}_{2}-\vec{x}_{3}|}$
(34)

We therefore can not write an effective potential for this system. When
Newton’s third law is not satisfied, there is no conservation of momentum. In
addition nonreciprocal interactions imply that the system is out of
equilibrium and time reversibility is broken [59], and the energy cannot be
conserved. We now explore some of the notable consequences of the adaptive
interactions on the dynamics and properties of the swarms.

###  2.2 Enhanced stability of adaptive spherical-uniform swarms

We now show that due to adaptivity a swarm of particles interacting through
adaptive gravity does not suffer from the famous Jeans instability that is
responsible for collapse of groups of particles under normal gravitational or
other attractive power-law long range forces [60, 61]. Adaptivity therefore
endows swarms with a natural mechanism for self-stabilization. The argument
that we present below is heuristic. A more rigorous derivation in terms of a
critical wave number for a wavelike periodic perturbation appeared in [14] and
reached the same expressions up to numerical constants.

For the purpose of the argument let us consider a spherical uniform swarm
whose radius is $R_{s}=\langle r\rangle$ (defined as the mean distance of a
particle from the center of the mass of the swarm, which is $R_{s}=(3/4)R$ for
a uniform density spherical swarm of radius $R$). We want to compare the
response to a local fluctuation in the density for regular vs. adaptive
gravity. In the case of a random fluctuation in the density, the random
movement of the particles could stabilize the swarm if their random velocities
are high enough. The typical time for stabilization is the typical time it
takes to move across the swarm: $t_{esc}=R_{s}/\sigma_{v}$, where
$\sigma_{v}=\sqrt{\langle v^{2}\rangle}$ is the root-mean-square velocity.
This velocity can arise from thermal motion, or from the chaotic motion due to
the attractive forces themselves (that depend on initial conditions), as well
as due to the noisy active propulsion forces of the living (active) particles.
The question is whether $\sigma_{v}$ is high enough so that the typical time
to stabilize (or equivalently the time for the density fluctuation to escape)
$t_{esc}$ is shorter than the typical time to collapse $t_{col}$, which is
given by the typical time a particle falls to the center of mass due to the
overall gravitational attraction: $t_{col}=2\,\pi/\sqrt{K}$, where $K$ is the
effective spring constant for the linear restoring force of the form
$\vec{F}=-K\vec{r}$ that acts towards the center of the swarm. This linear
force appears for a uniform density spherical swarm, and is calculated in the
appendix for both regular gravity $K^{g}$ (Eq. (67)) and adaptive gravity (Eq.
(71)).

The criterion for instability is therefore $t_{esc}>t_{col}$. For regular
gravity we get from it the following inequality

$\rho>\frac{3\,\pi\,\langle v^{2}\rangle}{c_{g}\,R_{s}^{2}},$ (35)

which gives the famous critical Jeans density for collapse in regular gravity
[60]

$\rho_{Jeans}=\frac{3\,\pi\,\langle v^{2}\rangle}{c_{g}\,R_{s}^{2}},$ (36)

which means that for any density higher then this value the swarm will be
unstable and collapse as a result of any arbitrarily small density
fluctuation.

In the case of adaptive gravity we get the following inequality:

$\langle v^{2}\rangle<\frac{c_{g}R_{s}}{16\,\pi^{2}\,R_{ad}^{2}},$ (37)

which shows that the collapse does not depend on the density, and there is a
threshold on the value of the thermal velocity fluctuations below which the
swarm will collapse. This means that any swarm that is ”hot” enough is
perfectly stable against density fluctuations. The adaptivity makes the mutual
interactions weaker in the dense regions and as a result the dependence of
stability on the density disappears, and this way it contributes to the
stability of the swarm.

The calculation given above for the stability condition of a uniform density
swarm applies also for a real swarm (as well as a galaxy or a star-cluster)
even though its density is not uniform [15].

It was found in [14] that for attractive forces with higher order power laws
and adaptivity the stabilization is even stronger. Instead of a minimal
density for collapse, there is a maximal density for collapse, and above it
the swarm becomes stable, which therefore protects the swarms from collapse.

###  2.3 Generalized virial relation for active systems

We now develop virial equations that relate the kinetic energy to a
generalized analogue of potential energy when the system is stationary, based
only on the conservation of mass. The equations are of second order since they
relate second-order moments. In their derivation we follow closely
Chandrasekhar [62] that derived them for the gravitational potential and we
write them in a more general form for any force assuming only mass
conservation. We derive here the scalar virial equation, while the more
general tensorial version appeared in [13].

Let us start from the following version of Newton’s second law governing the
velocity $\vec{v}$ of an element of the swarm whose density is $\rho$:

$\rho\,\frac{d\vec{v}}{dt}=-\nabla p+\rho\,\vec{f},$ (38)

where $p(\vec{r},t)$ is an isotropic pressure whose gradient creates a force
and $\vec{f}$ stands for the rest of the forces per unit mass. We multiply the
equation by the position vector $\vec{r}$ (scalar product) and integrate over
the entire volume $V$. Then from the left hand side of Eq. (38) we obtain from
mass conservation

$\int\rho\vec{r}\cdot\frac{d\vec{v}}{dt}\,dV=\frac{d^{2}I}{dt^{2}}-2\,K,$ (39)

where $I$ is the scalar moment of inertia defined as

$I\equiv\frac{1}{2}\int\rho\vec{r}^{2}\,dV,$ (40)

and

$K\equiv\frac{1}{2}\int\rho\vec{v}^{2}\,dV.$ (41)

The first term on the right hand side of Eq. (38) gives us:

$\int\vec{r}\cdot\nabla p\,dV=\int\nabla\cdot(\vec{r}p)\,dV-3\int p\,dV.$ (42)

Then using Gauss’s theorem for a volume $V$ and its boundary $\partial V$ we
get a surface term for the pressure:

$\int\vec{r}\cdot\nabla p\,dV=S-3\Pi,$ (43)

where the surface term is

$S\equiv\oint_{\partial V}p\,\vec{r}\cdot d\vec{s},$ (44)

and we define the total pressure over the volume of the swarm:

$\Pi\equiv\int p\,dV.$ (45)

The second term in the right hand side of Eq. (38) gives the analogue of
potential energy of the swarm:

$W\equiv\int\rho\vec{r}\cdot\vec{f}\,dV.$ (46)

Then combining Eqs. (38),(39) and (43) we obtain the following equation:

$\frac{d^{2}I}{dt^{2}}=2\,K-S+3\,\Pi+W.$ (47)

Assuming that the system is stationary and defining the total analogue of
kinetic energy $T$ as a sum of the kinematic motion and the internal thermal
motions in the form of isotropic pressure:

$T\equiv K+\frac{3}{2}\,\Pi,$ (48)

we arrive to a generalized virial theorem that relates (with a surface term)
the kinetic energy with the “potential energy” W:

$2\,T-S+W=0.$ (49)

This relation is very general for any force $\vec{f}$ that acts on the
particles without a requirement for any conservation laws except for the mass
conservation.

Comparing this virial expression to data of midge swarms the following mean
values per midge were found [13]:

$\displaystyle\langle T\rangle=(3.42\pm 0.08)\cdot 10^{2}\quad cm^{2}/s^{2},$
$\displaystyle\langle-W/2\rangle=(2.80\pm 0.08)\cdot 10^{2}\quad
cm^{2}/s^{2}.$ (50)

which were found to be approximately constant as a function of the swarm size
$R_{s}$ (which increases with the number of midges in the swarm). Then
according to Eq. (49), the difference between the two gives the mean surface
pressure in the swarm:

$S=-(1.24\pm 0.14)\cdot 10^{2}\quad cm^{2}/s^{2}.$ (51)

This pressure is negative, indicating that the swarm is experiencing a
stabilizing inwards effective pressure on its surface. The origin of this
pressure could arise from interactions of the swarm midges with midges outside
the swarm or the environment. Such external stabilizing pressures are commonly
found in astrophysical stellar systems, such as globular clusters [63].

For a spherical swarm of uniform density we can calculate the “potential
energy” $W$, find its scaling with the swarm size $R_{s}$, and compare with
the experimental data. We expect that near the center of the swarm the
adaptivity will be the strongest and the “perfect adaptivity” regime will be
the best approximation for the effective force between the particles. The
“potential energy” (Eq. 46) in a uniform spherical symmetric swarm is:

$W=\frac{27}{64\,R_{s}^{3}}\int_{0}^{\frac{4}{3}R_{s}}r^{3}\,f(r)\,dr.$ (52)

If it were a regular gravity then $f(r)=F^{g}(r)$ according to Eq.
$(\ref{New_grav})$ and

$W^{g}=-\frac{9\,c_{g}}{20\,R_{s}}\propto-\frac{1}{R_{s}},$ (53)

However, in adaptive gravity in the “perfect adaptivity” regime we have to
divide $F^{g}$ by $N_{tot}(\vec{r})$ according to the continuum limit of Eq.
(28)

$f(r)=\frac{c_{g}\,F^{g}(r)}{R_{ad}^{2}\,N_{tot}(r)}.$ (54)

Substituting into Eq. (52) and taking the expression for $N_{tot}(r)$ from Eq.
(68) and $F^{g}(r)$ from Eq. $(\ref{New_grav})$ we get:

$W^{ad}=-\frac{16\,c_{g}\,R_{s}}{3\,R_{ad}^{2}}\int_{0}^{1}\frac{x^{5}\,dx}{2x+(x^{2}-1)\ln{\left(\frac{1-x}{1+x}\right)}}\sim-0.4\frac{c_{g}}{R_{ad}^{2}}\,R_{s}\propto-
R_{s},$ (55)

and this linear scaling with $R_{s}$ is indeed observed in the swarm data when
we constrain the data closer to the center of the swarm (where the density is
rather uniform and the effect of adaptivity is strongest), as we see from the
slopes in Fig. 9.

Figure 9: The “potential energy” W in adaptive gravity Log-Log plot for the
“potential energy” W in the center of the swarm with a linear fit, constrained
for $r<R_{s}/2,R_{s},\infty$ (yellow, purple and blue respectively). The
power-law slopes are (yellow and purple respectively): $1.16\pm 0.12,0.71\pm
0.05$. Adapted from [13].

Another way to look at Eq. (47) (when the system is stationary) is as a
definition of the pressure:

$\Pi=-\frac{1}{3}\left(2\,K+W\right)+\frac{S}{3}.$ (56)

It was used to characterize the phases of midge swarms [64] and to write the
equation of state for the swarms [65].

###  2.4 Spontaneous formation of bounded pairs in adaptive-gravity swarms

Besides the quantitative modifications as a result of adaptivity, which gives
rise to modified global swarm behavior and scaling as shown above, there is a
striking new local phenomenon of pair formation that can be qualitatively
explained to arise from adaptivity. It has been observed that midges form
transient pairs with synchronized relative motion while moving through the
swarm [66]. An example from real data is shown in Fig. 10)(a,b) and from
simulations in Fig. 10)(c,d). The definition of a pair in the data is
according to the frequency of their relative movement and in the simulation
the pair is defined by the energy ratio that forms a bound pair. It can be
shown [16] that the two definitions are consistent with each other.

Within the adaptive-gravity model these pairing events occur whenever two
midges happen to move together from the center of the swarm (where the total
background noise is high) towards the swarm periphery (where the total
background noise is low). As a result of adaptivity, during this movement, the
attraction in the pair increases as the total noise decreases, thereby forming
a bound state (see Fig. 11A). The full analysis and comparison with
experimental data appeared in [16]. Here we give a simple analytical sketch of
the formation of a bound pair as a result of adaptivity.

Let us consider the interaction of two midges, that happen by chance to come
very close to each other, in the background of the rest. We assume that the
separation between the two midges is small compared to their average distance
to the rest of the midges in the swarm, and therefore the interactions with
the rest of the swarm will be negligible except for a contribution to the
total noise in Eq. (26), so that the effective force felt by one member of the
pair is

$\vec{F}^{ad}(\vec{x}_{1})=-\frac{\nabla
V^{12}(\vec{x}_{1})}{1+c_{g}^{-1}R_{ad}^{2}\,I_{background}+c_{g}^{-1}R_{ad}^{2}|\nabla
V^{12}|},$ (57)

where

$I_{background}\equiv\sum_{i=3}^{n}|\nabla V^{1i}|\sim
I_{bgd}+\mathcal{O}.(|\vec{x}_{1}-\vec{x}_{i}|)$ (58)

The leading order of the expansion at infinity gives us a constant background
contribution. Integrating this force we get the effective two-body potential
(compare with Eq. (31))

$U_{\mbox{\scriptsize pair}}(|\vec{x}_{1}-\vec{x}_{2}|)=\frac{c_{g}}{\gamma
R_{\mbox{\scriptsize
ad}}}\left[\arctan{(\gamma\,|\vec{x}_{1}-\vec{x}_{2}|/R_{\mbox{\scriptsize
ad}})}-\pi/2\right],$ (59)

where

$\gamma\equiv\sqrt{1+c_{g}^{-1}R_{ad}^{2}\,I_{bgd}}$ (60)

is the background sound parameter. Hence we effectively have two-body motion
under the influence of a mutual central force. In the case of two bodies, not
only the conservation laws are valid but also the additivity of the effective
force, so we can use all the conservation laws of a central force such as
energy and angular momentum conservation. In particular, this two-body system
can be reduced to an equivalent one-dimensional motion in the effective
potential

$U_{\mbox{eff},12}(|\vec{x}_{1}-\vec{x}_{2}|)=\frac{\tilde{l}^{2}}{2\,|\vec{x}_{1}-\vec{x}_{2}|^{2}}+\frac{c_{g}}{\gamma\,R_{\mbox{\scriptsize
ad}}}\left(\arctan\left(\frac{\gamma\,|\vec{x}_{1}-\vec{x}_{2}|}{R_{\mbox{\scriptsize
ad}}}\right)-\frac{\pi}{2}\right),$ (61)

where $\tilde{l}$ is the angular momentum per unit mass (and we take the
reduced mass of the pair).

Figure 10: Pair formation in laboratory observations of midge swarms  (A)
Trajectories of two midges in a laboratory swarm that exhibited pairing
(identified as being above a threshold value for the oscillation frequency,
see [16, 67]). The midges were identified as belonging to a pair between the
blue and black points. Paired parts of the trajectories are colored in red,
while unpaired parts are in grey. Distances are in mm. (B) Distance between
the members of the laboratory pair in (A) as a function of time. Distances are
normalized by the swarm size $R_{s}$. The time is normalized by the typical
orbit time around the center of mass $T_{com}$. (C) Pairing event in the
simulation (defined as a pair in a bound orbit). Distances are in simulation
unit lengths. Here the segments between the blue and the black points are only
a part of a longer pairing event. The red and the grey colors are as in (A).
(D) Distance between the members of the pair in (C) (simulation) as a function
of time. Distances are normalized by the swarm size $R_{s}$. The time is
normalized by the typical orbit time around the center of mass $T_{com}$.
Adapted from [16].

When the background sound is reduced (as the pair move away from the swarm
center), the sensitivity of each pair member increases, their effective mutual
force becomes stronger and an unbound pair may become bound. Let us analyze it
for simplicity when the background sound parameter decreases as a step
function (Fig. 11):

$\gamma=\begin{cases}\gamma_{1}&t\leq t_{0}\\\ \gamma_{2}&t>t_{0},\end{cases}$
(62)

where $\gamma_{1}>\gamma_{2}$.

Figure 11: The background sound parameter (A) Illustration of the proposed
pair formation mechanism. When two interacting midges leave the dense region
of the swarm (darker dashed lines), where the background sound parameter
$\gamma$ is high, and move to a lower density region (such that
$\gamma_{1}>\gamma_{2}$), the mutual pull between them becomes stronger, their
orbit gets tighter, and they become bound. (B) The background sound parameter
$\gamma$ is taken to be a step function. For $t\leq t_{0}$ the background
sound level is higher than the background sound for $t>t_{0}$. Adapted from
[16].

When the mutual attractive force is stronger, even when we start from an
elliptical bound orbit, it becomes tighter and therefore the particles
approach closer to each other. The total energy, which is conserved, is given
by the following expression:

$E_{i}=E_{k}(|\vec{x}_{1}-\vec{x}_{2}|)+\frac{\tilde{l}^{2}}{2\,|\vec{x}_{1}-\vec{x}_{2}|^{2}}+\frac{c_{g}}{\gamma_{i}\,R_{\mbox{\scriptsize
ad}}}\left(\arctan\left(\frac{\gamma_{i}\,|\vec{x}_{1}-\vec{x}_{2}|}{R_{\mbox{\scriptsize
ad}}}\right)-\frac{\pi}{2}\right),$ (63)

where $E_{k}(|\vec{x}_{1}-\vec{x}_{2}|)$ is the kinetic energy of the radial
movement. At $t=t_{0}$ let us denote the radial distance of the reduced mass
by $r_{0}$ and the energy of the pair is changing due to the change in the
background sound. The energy difference is given by:

$\Delta E=\Delta
U_{\mbox{eff},12}=\frac{c_{g}}{\gamma_{2}\,R_{\mbox{\scriptsize
ad}}}\left(\arctan\left(\frac{\gamma_{2}\,|\vec{x}_{1}-\vec{x}_{2}|}{R_{\mbox{\scriptsize
ad}}}\right)-\frac{\pi}{2}\right)-\frac{c_{g}}{\gamma_{1}\,R_{\mbox{\scriptsize
ad}}}\left(\arctan\left(\frac{\gamma_{1}\,|\vec{x}_{1}-\vec{x}_{2}|}{R_{\mbox{\scriptsize
ad}}}\right)-\frac{\pi}{2}\right)$ (64)

When $\gamma_{2}<\gamma_{1}$ in this case $\Delta E<0$ since
$U_{\mbox{eff},12}(r)$ which is given in Eq. (61) is monotonically increasing
(becomes less negative) as a function of $\gamma$ for any positive value of
$R_{ad}$

$\partial_{\gamma}U_{\mbox{eff},12}=c_{g}\,\frac{2\,\gamma\,R_{ad}\,|\vec{x}_{1}-\vec{x}_{2}|+(R_{ad}^{2}+\gamma^{2}\,|\vec{x}_{1}-\vec{x}_{2}|^{2})(\pi-2\,\arctan\left(\frac{\gamma\,|\vec{x}_{1}-\vec{x}_{2}|}{R_{ad}}\right))}{2\,\gamma^{2}\,R_{ad}\left(R_{ad}^{2}+\gamma^{2}\,|\vec{x}_{1}-\vec{x}_{2}|^{2}\right)}>0.$
(65)

The “tightness” of an orbit can be regarded as the absolute value of the ratio
of the kinetic to potential energy $\left|E_{k}/U_{\mbox{eff},12}\right|$.
When it is smaller than one $\left|E_{k}/U_{\mbox{eff},12}\right|<1$, the
orbit is bound, and that is how a pair is defined, if the two midges move in
bound orbits (see Fig. 10). Since the effective potential energy is lowered as
a result of the reduction of the background sound and the kinetic energy
remains the same, this process can indeed produce bound orbits out of unbound
ones. In addition, it can be shown [16] that the maximal distance between the
members of the pairs is larger when the background sound parameter is higher.
The proposed mechanism for pair formation is supported by experimental data
[16], where it is shown that the background sound gradient tends to be
negative where the pairs form (upon leaving the swarm center) and tend to be
positive where they dissociate (entering the swarm center).

This phenomenon of separated particles forming bounded pairs can be explained
by the adaptive-gravity model, while in regular gravitational attraction it is
extremely unlikely due to constraints of conservation of momentum and energy
[68]. Due to momentum conservation, two particles in regular gravity can form
a bound pair only if a third particle removes the excess momentum.

Note that the pairing mechanism that we describe is general to any many-body
system with long-range adaptive interactions, and is not limited to the
gravity-like functional form that we used here.

## Discussion

Nonreciprocal interactions between active particle has become a topic of great
interest recently. What we have demonstrated in this chapter is that
nonreciprocal forces, which arise from effective potentials that do not obey
Newton’s third law, and do not allow to formulate conservation of momentum and
energy, arise naturally within the context of interactions between animals. We
presented two very different examples for such effective interactions, in the
form of (mostly) pair-wise animal contests, and in the context of the many-
body long-range interactions within a cohesive swarm. These examples show that
modelling animal behavior in terms of active particle systems expands the
range of nonreciprocal interactions that are considered by physicists and
mathematicians, by introducing unique features (such as adaptivity and
learning).

At the same time, these examples demonstrate the usefulness of active-particle
models for describing and explaining animal behavior. In the case of the
contests model, we demonstrate that extracting the effective pair-wise
interactions from experimental animal trajectories can be used to explain
observations of evolutionary traits that arise within the context of larger
groups. In the case of the swarms, the adaptivity property was shown to
explain the observed scaling with the swarm size (global property), as well as
the emergence of spontaneous formation of bound pairs (local property).

We therefore conclude that the study of active-particle systems with
unconventional interactions enriches our understanding of animal behavior,
while in turn the study of animal systems enriches and expands the scope of
both statistical mechanics and applied math.

## Appendix

##  Appendix A Explicit Calculation of the Linear Restoring Force of Adaptive
Gravity

Under the assumptions of a spherical and uniform swarm there is a linear
restoring force towards the center of the swarm of the form
$\vec{F}=-K\vec{r}$ in the leading order in the deviation from it (where
$\vec{r}=0$ is the center of the swarm). Let us calculate explicitly the
effective spring constant of the linear adaptive-gravitational field at a
point inside a spherical swarm with radius $R_{s}=(3/4)R$ (the mean distance
from the center is $3/4$ of the radius $R$) and uniform density $\rho$,
according to Eq. (26) where we take the continuum limit of the sums. We will
use cylindrical coordinates $(r,z,\varphi)$ and calculate the field at ($r=0$,
$z=z_{0}$) without loss of generality (the point $A$ in Fig. 12). The symmetry
of the problem implies that the field is along the $z$ axis. The contribution
of a point at $(r^{\prime},z^{\prime})$ to the gravitational force at
$(0,z_{0})$ is

$\frac{c_{g}}{r^{\prime 2}+(z^{\prime}-z_{0})^{2}},$

and the angle is

$\cos\varphi=\frac{z^{\prime}-z_{0}}{\sqrt{r^{\prime
2}+(z^{\prime}-z_{0})^{2}}}.$

Hence the regular gravitational force at $z_{0}$ is

$F^{g}(z_{0})=2\,\pi\,\rho\,c_{g}\int_{-R}^{R}\\!\\!\\!dz^{\prime}\int_{0}^{\sqrt{R^{2}-z^{\prime
2}}}\\!\\!\\!\\!\\!r^{\prime}dr^{\prime}\,\frac{z^{\prime}-z_{0}}{\left[r^{\prime
2}+(z^{\prime}-z_{0})^{2}\right]^{\frac{3}{2}}}=-\frac{4\,\pi\,\rho\,c_{g}}{3}\,z_{0},$
(66)

and then the effective spring constant of this restoring force is

$K^{g}=\frac{4\,\pi\,\rho\,c_{g}}{3}.$ (67)

This result can be also obtained from Gauss’s law (as it was derived in [13]).
The “total noise” factor in the denominator (also in Eq. (29)), which is given
by the summation of the absolute values of the contributions to the point
$(0,z_{0})$, reads

$N_{\mbox{\scriptsize
tot}}(z_{0})=2\,\pi\,c_{g}\,\rho\,\int_{-R}^{R}dz^{\prime}\int_{0}^{\sqrt{R^{2}-z^{\prime
2}}}\frac{r^{\prime}dr^{\prime}}{r^{\prime
2}+(z^{\prime}-z_{0})^{2}}=\pi\,c_{g}\,\rho[2R-\frac{(R^{2}-z_{0}^{2})}{z_{0}}\ln\left(\frac{R-z_{0}}{R+z_{0}}\right)].$
(68)

To leading order in $z_{0}$, we have

$N_{\mbox{\scriptsize tot}}(z_{0})=4\pi\rho\,c_{g}\,R+\mathcal{O}(z_{0}^{2}),$
(69)

which after substitution to Eq. (26) gives to leading order in $z_{0}$ the
following restoring force towards the center:

$F^{ad}(z_{0})=-\frac{4\,\pi\,\rho\,c_{g}}{3+16\,\pi\,\rho\,R_{s}\,R_{ad}^{2}}\,z_{0}.$
(70)

The number of particles $N$ in a uniform and spherical swarm is given by
$N=4\,\pi\,\rho\,R_{s}^{3}/3$ and then we can write the denominator in Eq.
(70) in the following way

$3+16\,\pi\,\rho\,R_{s}\,R_{ad}^{2}=3+16\,N\,R_{ad}^{2}/R_{s}^{2}.$

Assuming the adaptivity range is approximately the size of the swarm
$R_{ad}\sim R_{s}$ and $N\geq 1$. Then the first constant term in the
denominator is negligible compared to the second. Then within this
approximation we get the following expression for the effective spring
constant:

$K^{ad}=\frac{c_{g}}{4\,R_{s}\,R_{ad}^{2}}.$ (71)

Figure 12:  The cylindrical coordinates $(r,z)$ (and $\varphi$) that we use
for the calculation of the effective gravitational field at a point A in a
uniform-density spherical swarm.

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