Understanding the Behavior of Schooling Fish Through Mathematical Models  

Anne-Sophie Roy, Emma Warner, Hoi Ching Wat, Lucas Yatcyshyn 

Abstract

From sky to water, many species live and move together. Among these, shoaling and schooling fish are particularly notable for their highly coordinated behavior. Such synchronization plays a crucial role in survival, aiding in predator avoidance, navigation, and foraging. Mathematical models have shown that fish schools optimize foraging by balancing resource acquisition with minimal energy expenditure. The concept of criticality explains how schools adapt to environmental disturbances through synchronized and collective responses. These models also explain how schools form distinct shapes, enhancing hydrodynamic efficiency. Frameworks like the Aoki model, the self-propelled particle model, the Huth and Wissel model and the zonal model, give insight into the role of local interactions, such as mutual attraction, repulsion, and parallel orientation, in creating cohesive and polarized formations. Furthermore, the Rosenthal et al. model emphasizes the importance of information transfer within the school, explaining how the position of individuals influence group behaviours.  

Introduction

Even though it is only composed of some algebra and Greek letters, mathematics can further our understanding of one of the greatest structures of all time: shoals and schools of fish.  

Fish often aggregate in social groups called shoals. If fish in the shoal orient parallel to each other and maintain a fixed distance between each other, the group is considered a school as well, and fish in the school can perform complex, coordinated group movements (Fig 1) (Lopez et al., 2012; Pavlov & Kasumyan, 2000).  

Fig. 1 A school of fish (From shoals to schools: A look at fish communities).  

Schools often expand and contract in a dynamic way. Fish alternate between tighter, cohesive formations and more spread-out patterns (Fig 2) (Lopez et al., 2012; Pitcher, 1986). Schooling behavior evolved in fish primarily as a mechanism of predator defense, but also serves as a method for learning reflexes, as well as increasing hydrodynamic efficiency, and improving feeding and migration (Pavlov & Kasumyan, 2000). For instance, as an anti-predator strategy, fish imitate the signals emitted from a larger fish by moving synchronously. Moreover, by performing coordinate movements, fish can distinguish the noise generated by their movement from the environmental noises, allowing them to be more attentive to dangers  (readers are referred to Nature’s Aquatic Ballet: The Physics and Synchronization of Schooling Fish and Chemistry of Cohesion and Communication in Fish Schools for more information on the advantages of schooling, including adaptations for hydrodynamic efficiency and predator evasion) (Lopez et al., 2012).  

Fig. 2 Different school structures: (a) is a school with low polarization, (b) is a highly polarized moving school, (c) is a milling structure and (d) is a bait ball structure (Lopez et al., 2012).  

This begs the question: How do fish coordinate and self-arrange into these highly organized structures, and how do they remain cohesive and synchronized when responding to different stimuli?  

Many mathematical models and simulations have been developed, building on each other over time, with the with the aim of achieving a better understanding of the mechanisms of schooling, as well as the complex interactions between individuals in schools. In this paper, we attempt to summarize some of the mathematical models most used to describe schooling, and how they have contributed to the current understanding of schools of fish. 

Foraging Model

Foraging models combine aspects of foraging behavior and foraging theory to help explain one of the more advantageous, schooling-related adaptations in which individual fish share information and coordinated movements to optimize resources acquisition and lower individual energy costs. Foraging behaviour describes the ways organisms find, eat, retrieve, and store food for energy and nutrients within a community. Foraging theory predicts how animals decide where and how to find food, based on resource availability, competition, and risk of predators (Koy & Plotnick, 2007). 

The purpose of foraging is to ensure an organism gains more energy than it spends, not only allowing it to survive, but also to grow and reproduce. Optimal Foraging Theory suggests that organisms maximize their energy intake while minimizing the energy they expend. This means they select food with the highest energy return (E) per unit of time (t), aiming to maximize the E/t ratio.  

Foraging behavior is divided into three stages: search, assessment, and exploitation. Organisms gather information during these stages and use it to make future foraging decisions (Koy & Plotnick, 2007). During the search phase, fish identify and evaluate the quality and location of resources, which includes detecting and traveling to more distant resources. The search cost per unit time represents the energy used in this phase, specifically for locating new resources (Koy & Plotnick, 2007). 

When a fish meets a resource patch or prey item, it enters the assessment phase, where it evaluates the potential net energy gain. The fish uses past foraging experiences to decide whether to exploit the patch or continue searching. This decision involves several factors: patch encounter rate, the average net energy gained from previous encounters, the energy cost of searching, and the time needed to consume prey. Patch encounter rate is the probability of encountering a resource patch per unit time. Higher values represent a greater density of patches, thus less travel distance between patches. If the current patch offers minimal energy gain and search cost are low, a fish might choose to continue searching for a new patch. However, when encountering rates or average energy gains are low or search costs are high, it is often more efficient to exploit the current patch (Koy & Plotnick, 2007). 

If a fish decides to stay at a particular patch, the fish enters the exploitation phase and focusses on consuming resources within the current patch while continuously assessing resource availability. A fish decides to leave the current patch based on the quitting harvest rate (H), which is compared to the sum of its metabolic cost (C), predation cost (P), and missed opportunity cost (MOC). Exploitation should continue until the energy gain rate falls to or below the combined costs: H ≤ C + P + MOC. At this point, if the patch’s resources no longer exceed those of the surrounding area, the fish will move to a new patch (Koy & Plotnick, 2007). 

The foraging model highlights the costs and benefits of resource exploitation for individual fish, though these benefits are even more significant in the context of schooling. By foraging in a school, fish minimize individual metabolic, predation, and missed opportunity costs through collective resource discovery and feeding. For example, in low-light environments, foraging in groups is particularly advantageous as fish in school can also rely on each other for the discovery of new food patches. Schools also benefit from extended feeding time and more efficient sampling. Some species, such as the stone loach, even create water turbulence at the front of the school to help others in the school access the food more easily (Pitcher, 1986). By understanding the factors that influence individual decisions—particularly regarding feeding—we can gain deeper insight into overall schooling behaviour. 

Operating at criticality 

The need for resources acquisition such as through foraging is balanced with rapid, coordinated responses to environmental disturbances, such as predation by another organism, by operating at criticality.  

Poecilia sulphuraria, a type of shoaling fish endemic to a sulfidic stream system near the city of Teapa in southern Mexico, operate at criticality in the unique way that they respond to environmental disturbances. These shoals may reach up to 3000 individuals per square meter as they dwell near the oxygen-rich air-water interface, making them high frequency targets for predators. However, these fish have evolved an intriguing, synchronized diving behavior in response to disturbances (Gómez-Nava et al., 2023). 

To initiate a dive, a fish touches the water surface with their tails, spreading through the shoal much like how a human wave propagates through a stadium. This behavior acts as a deterrent which leads to less successful attacks by aerial predators. However, this diving act has also been observed in the absence of bird attacks. One hypothesis is that to maximally respond to external stimuli and effectively propagate information, it is most efficient to operate in a system at criticality. In other words, they operate between two phases: one of low and another of high alertness, known as surface-wave activity (Gómez-Nava et al., 2023). 

Based on videos of spontaneous diving behavior in absence of a predatory disturbance, the different stages of the dive of individual fish as they responded to small-scale perturbations were analyzed, then modelled by Gómez-Nava and colleagues in 2023. The different stages of each member of the shoal were identified using a computer vision processing pipeline, which sorted each pixel as either active (diving state) or non-active (surface and underwater states) (Fig 3) (Gómez-Nava et al., 2023). 

Fig. 3 The conversion of the original video into active pixels depicted in white (Gómez-Nava et al., 2023). 

The surface-activity signal measured the fraction of active pixels in a video, which represented the number of diving fish at a given moment. Peaks of activity, known as spikes, corresponded to waves spreading through the system. In this study of Poecilia sulphuraria, it was found that both the probability distribution of inter-spike time intervals and the probability distribution of spike-duration times have exponentially decaying tails.  

The way that a single wave propagated throughout the shoal was determined by defining an activity cluster as the number of active pixels corresponding to a single wave. This data was consistent with a power-law distribution with an exponent of 2.3. Therefore, the power-law distribution in this case was defined as P(x) ∝ x^-2.3. Since the wave size distribution was consistent with a power law, it suggested that the system indeed operated at criticality and demonstrated surface-wave activity (Fig. 4) (Gómez-Nava et al., 2023). 

Fig. 4 The cluster areas follow a power-law distribution, indicating that the shoals operate at a point of criticality (Gómez-Nava et al., 2023). 

To further confirm that fish tend to operate at a critical point, a generic model for the surface-activity waves was developed, simulating the dynamics and creating parameters that best represented the observed behavior. If the fish indeed operated at a critical point, then the parameters best fitting the data would also be located at a phase transition, known as a critical point. 

Model for criticality 

This model of criticality was extended to the shoal system, by subdividing the surface of the shoal into special cells representing the dynamics of each subset of fish. Each of these cells were represented as one of the states: surface, dive, or underwater. The free parameters adjusted to model the shoal were two-time constraints controlling how long the cells remained in the diving and underwater state, two parameters controlling the stochasticity of spontaneous events, and a coupling parameter representing an activation threshold (how many neighboring cells must be activated to cause a given cell to become active). First, the two-time constraints were matched to published data. Then, a systematic parameter search was performed to identify the best fitting values for the remaining three parameters (Gómez-Nava et al., 2023). This strategy confirmed the hypothesis that the best fit of the data was located at a critical point as these variables occurred at phase transitions. A summary of the methods used to create this model for the Poecilia sulphuraria shoals is shown in Figure 5. 

Fig. 5 (a) It is an infographic describing how the criticality model categorized the behavior of individual fish into swimming near the surface, diving, and hovering underwater. (b)Shows a comparison between the video data of the fish and the numerical simulations generated. (c) Representation of the surface-activity signal, describing the general actions of the shoal. (d) illustrates the inter-spike intervals, indicating the duration of each diving phase in the model. (e) similarly describes this but for spike-duration times. (f) indicates the distribution of cluster areas, following the power-law and indicating criticality (adapted from Gómez-Nava et al., 2023). 

Collective behaviour and motion 

To understand how fish coordinate and synchronize their foraging behavior environmental responses, it is important to explore the mechanisms that underlie their collective motion. The individual decision-making processes described by the foraging model and the synchronized diving behaviors of Poecilia suphuraria highlight the importance of individual actions in initiating group responses. By examining models for collective behavior and motion, one can better understand the ways in which information, such as food location or the present of a predator, propagates through a school, and how these coordinated behaviors increase the school’s ability to adapt and thrive in dynamic environments. 

Imagine walking down a busy street crowded with people during the holidays. This is analogous to the reality that many fish are living in. Fish are not only able to navigate such crowded conditions but are also able to coordinate with conspecifics in the crowd to form synchronized groups, or schools. These highly organized structures are a result on intricate interactions between neighbouring fish and can be described by collective behaviour. 

Collective behavior describes a system in which many similar, individual units interact within a space or in an underlying network. The actions of individuals are largely influenced by other units in the system such that they behave differently in the system than when on their own (Vicsek & Zafeiris, 2012). The combined actions of individual units in the system and influences that units have on each other result in unique behavioral patterns, such as those observed in schooling fish (Kawashima, 2023). 

Collective motion (also referred to as flocking) is displayed by nearly every living system containing more than 12 individuals (Vicsek & Zafeiris, 2012). For fish, groups containing more than 3 individuals exhibit collective motion (Partidge, 1982). Collective motion describes how units that exhibit collective behavior move with the same absolute velocity (Vicsek & Zafeiris, 2012). 

There are many ways that collective motion in schooling fish can be measured, including by looking at patterns and shapes formed by the school and the different interactions between individuals (Vicsek & Zafeiris, 2012). 

Patterns and shapes 

Observing the pattens and structures of the group provides context for both individual and collective responses to stimuli as well as the local interactions that inform group behavior. These formations are not random; rather, they are influenced by factors such as alignment and spacing, which facilitate coordination among individuals. Alignment refers to the synchronization of movements in a group to form polarized structures, while spacing helps maintain stability, cohesion, and overall coordination. Furthermore, the synchronization of tail movement, either in-phase or out-of-phase, affects the energy costs and flow dynamics, demonstrating hydrodynamic advantages that result from schooling.  

Diamond configuration  

The diamond configuration is a commonly observed spatial arrangement in fish schools, where each fish positions itself at a slight diagonal distance to its neighbor, forming a series of staggered rows in a diamond-like pattern (Fig. 6). This configuration, typically seen in species such as sardines and mackerel, allows fish to to keep their neighbours in view while maintaining the distance needed for efficient movement. By swimming in an offset alignment, each fish can take advantage of the vortices generated by the fish in front. This arrangement enables fish to travel long distances with minimal fatigue (Weihs, 1973) (readers are referred to Nature’s Aquatic Ballet: The Physics and Synchronization of Schooling Fish for more information on hydrodynamic efficiency and anti-predator strategies).  

Fig. 6 View fish school in a diamond configuration. The dotted line shows the “diamond” pattern, while the arrows represent the vortex created by the fish (Weihs, 1973) 

In 1973, Weihs hypothesized that, to maximize hydrodynamic efficiency, fish swimming in a diamond shape should maintain 0.4 body lengths and synchronize their tailbeats in anti-phase, while fish in the following row should align themselves between leaders at a distance of 5 body lengths. To maximize vortex advantages, lateral neighbors should reduce the spacing to 0.3 body lengths, with schools becoming more compact as swimming speed increases (Pitcher, 1986).  

Rectangular configuration  

The rectangle formation in fish schools is a structured arrangement where individuals align in a rectangular pattern, swimming in either in-phase or anti-phase modes (Fig. 7). This formation also improves hydrodynamic performance, increasing both swimming speed and energy efficiency in comparison to individual fish swimming alone. The lateral spacing between fish plays an important role in determining the stability and overall effectiveness of the formation. While the in-phase mode is excellent for energy conservation, the anti-phase mode prioritizes speed, but at a greater energy cost (Wei & al., 2023).  

Fig. 7 Representation of two fish school swimming in a rectangle configuration. The top one has in-phase tail movements, while the bottom one has anti-phase tail movements (Wei & al., 2023). 

As in the diamond configuration, a lateral spacing of 0.4 body lengths is a critical threshold affecting the hydrodynamic performance of fish schools in the rectangular configuration. When spacing falls below this value, fish swimming in-phase experience a 20% reduction in speed. In contrast, fish swimming in anti-phase at the same spacing benefit from a 17% speed increase, due to favorable vortices generated by leading fish. However, when lateral spacing exceeds 0.4 body lengths, both swimming modes show improved efficiency. Trailing fish can achieve up to a 15% reduction in energy expenditure, suggesting that more spacious arrangements offer hydrodynamic advantages (Wei et al., 2023) 

Interactions 

Different shapes and configurations of schools such as the diamond and rectangular configurations, as well as the group responses to stimuli, are a direct result of the interactions between individual fish and their neighbours. These interactions can be modelled by the metric distance model or the topological distance model (Fig. 8).  

Fig. 8 Metric distance and topological distance are two mathematical models used to describe the local interactions of a fish with its neighbors (Shoaling and schooling, 2024). 

Moreover, a series of parameters can be used to quantify the interactions and decisions of fish in a school. These parameters are described at two levels. The first level uses absolute features to describe patterns in observable phenomena, such as changes in velocity, heading, the degree of polarization of school. Absolute features describe the relationships between an individual and one other neighbor, including the metric or Euclidean distance between them (scale-dependent measurement) and their relative angular position. The second level aims to describe implicit information regarding multiple neighbours within sight, including the topological distance, ranked angular areas (scale-independent measurement), and relative metric distance, to reveal the nature of information transfer between individuals, and the relative influence and susceptibility of individuals to the behaviour of their neighbors (Rosenthal et al., 2015; Kawashima, 2023). 

 The main parameters and measurements used to describe schooling behavior including, the velocity and heading of individuals, polarization & cohesion of the school, decision making, position and centrality, and influence and susceptibility of individuals will be further analyzed. 

Basic behavioural interactions 

There are three basic behaviours observed in the interactions between individuals in a school which form a basis for current models of schooling: mutual attraction, repulsion or collision avoidance, and parallel orientation (Aoki, 1982); attraction being the most fundamental of the three behaviours (Kawashima, 2023). 

In models, these behaviours translate to rules which defining the individual responses of fish within a school to each other based on certain angles and distances between them. Mutual attraction describes the tendency of a fish to move towards a neighbour that is further than the individual’s preferred distance to its neighbors (Huth & Wissel, 1994; Partridge, 1982). Repulsion describes the tendency of a fish to move away from a neighbor that is closer to the individual than its preferred distance to avoid collision (the distance is less than the minimum approach distance). Parallel orientation describes the tendency of fish to match the velocity and direction of motion (heading) of its neighbours for neighbours that are within the range of preferred distance (Huth & Wissel, 1994).  

All fish share similar values for repulsion, parallel orientation, and attraction (measured in body lengths (BL)), which are 0.5 BL, 2 BL, and 5 BL, respectively. The range of preferred distance of a fish to its nearest neighbor is between the values for repulsion and parallel orientation (0.5-2 BL) (Huth & Wissel, 1994). 

All three behavioural interactions are essential for cohesive group motion, and the ability of individual fish to exhibit these interactions determines their ability to survive within a school (Aoki, 1982). Approach movement (mutual attraction) is necessary to allow an aggregation, and therefore a school, to form, and parallel orientation movement is necessary for the school to remain cohesive. Differences in parallel orientation exist between schooling species resulting from adaptations to their respective lifestyles and environments. For example, pelagic fish, due to the absence of visual landmarks in the open ocean, generally demonstrate stronger parallel orientation instincts, contributing to higher school cohesion than in some other fish. A combination of approach, avoidance, and parallel orientation movements are therefore essential for the compactness and coordination of a school (Aoki, 1982). 

Velocity and heading 

Since the positions and orientation of fish change with respect to time, models that measure the individual velocity and direction of movement (heading) are more useful in analyzing the interactions between fish. 

Networks commonly used to model animal behaviour are often based on the “instantaneous configuration of the individual” which includes their relative position, orientation, and velocity, as well as whether another organism is within the individual’s field of view. In terms of prediction, a model based on the relative influence of individuals on others is particularly useful when analyzing the velocity and heading of the organism.  

In these models, an individual’s velocity—an absolute feature—is it shaped both by the influence of neighboring individuals and by the organism’s own desired direction—an implicit feature. 

The relative influence of neighbours differs such that only a limited number of neighbors may influence the individual (Kawashima, 2023; Aoki, 1982). 

A variety of models have been developed to describe how individual velocity and heading, as well as neighbor influence, contribute to the collective motion observed in schools of animals.  

Aoki model 

A numerical, 2-dimensional, stochastic simulation—one based on random probability distributions—was developed to model schooling behaviour in fish, based on the three behavioural interactions: approach (mutual attraction), avoidance, and parallel orientation. This simplistic simulation highlighted the importance of mutual attraction and parallel orientation movement in fish schools and provided a framework for describing these interactions in future models (Aoki, 1982).  

The Aoki model was built on several assumptions. It was assumed that there was a unit time interval (quantized time) over which movement decisions were made. The decisions of individuals at each time interval were independent of the decision made at the previous time interval. It was further assumed that speed and direction of movement were mutually independent, stochastic variables characterized by probability distributions—the moved distance and direction at each time step of individuals (resulting in group movement) could be simulated by generating random numbers. To simplify the model, interactions between individuals were limited to directional components (Aoki, 1982). 

The velocity distribution was modeled using a probability distribution described by a gamma function, 

where is the velocity (), and are constant parameters () based on observations, and is a gamma function such that .

The direction of movement of an individual was given by a probability density,  

where is a weighting factor which scales the influence of the j≤4 neighbours within RC and with heading closest to the individual. 

The relative influence of neighbors was defined as 

and

where is a constant such that

Self-propelled particle model 

The self-propelled particle (SPP) model builds upon the Aoki model to include self-propulsion and internal forces in addition to local interactions between fish. 

Like the Aoki model, the SPP model emphasizes the attraction and repulsion between fish and made several of the same assumptions. It was assumed that decisions were made at discrete time intervals rather than continuously, and by using the attraction and repulsion forces, fish could be modelled as simple SPPs. Furthermore, the SPP model also includes formulas for the heading and the position of each particle in the system. 

The change of the direction of a particle, representing a fish, over time is expressed by the following formula (Strömbom, 2011): 

where is the direction vector of a particle at a time , represents the intertia of the particle,   is the unit vector of , c represents the attraction force towards the center of mass () of the neighbours, and the last term represents the directional error. This formula predicts the direction that a fish will move based on its tendency to continue its movement and be attracted to its conspecifics.  

Based on the heading of the particle, the position of the particle after a certain time is calculated using the following formula: 

where is the position after t+1 time, and δ represents the speed of the particle and the ζ_{i, t} models the variation in speed.  

The SPP model can predict more complex schooling patterns than the Aoki model, including swarms, undirected mills and aligned moving groups. If the attraction “force” is larger than the “inertial force” of the fish, then, fish will form a swarms; otherwise, they swim parallel to each other (Fig 9) (Strömbom, 2011).  

Fig. 9 Diagram showing different simulated structure: parallel motion, mill and swarm (from left to right) according to the ratio of the inertia force d and the attraction force c dependent on the interaction radius R and the average speed of particle 𝛿 (Strömbom, 2011).  

While the Aoki and SPP models contribute a fundamental understanding of how local interactions and behaviors such as attraction and repulsion, in combination with self-propulsion, contribute to schooling, these models are not sufficiently complex to describe observed schooling phenomena precisely and accurately.  

Huth and Wissel model 

The Huth and Wissel model added an additional level of complexity to schooling models. It considered attraction and repulsion behaviours but unlike the Aoki and SPP models, placed greater emphasis on parallel orientation. The Huth and Wissel model demonstrated how group movement is maintained by interactions between an individual and its neighbours, as well as why no more than four nearest neighbors influence a fish’s velocity and heading at a time. It was also one of the first models for freely swimming schools, which described the movement of a school that did not experience external influence (Huth & Wissel, 1994).  

After one time step, each fish in the model changed its velocity and swimming direction, which depended on the position, orientation, and velocities of a fixed number of neighbouring fish. The relative influence of a neighbor was based on the neighbor’s position with respect to the individual. For example, the new velocity of a fish was determined by calculating the average velocity of its neighbors, and the heading was based on the influence of each neighbor, calculated by averaging the potential turning angles (Huth & Wissel, 1994). 

It was determined through the model that a maximum of four neighbors influence a fish at a time, the relative influence is determined by a front priority rule by which neighbours in front of a fish within a certain distance influence the fish, and others do not (Huth & Wissel, 1994). 

It further demonstrated how schooling fish control their movement with respect to the average of a certain number of neighbors; there is low correlation between the velocity of an individual fish and that of its nearest neighbors, but a high correlation between the velocity of an individual fish and the weighted mean velocity of the entire school (Huth & Wissel, 1994).  

Increasing the number of neighbors that could influence a fish to more than 3 or 4 had no effect on the polarization of the school. Therefore, the mean velocity of four neighbors is sufficient for a polarized school, demonstrating the “economic advantage of a cooperative behaviour” (Huth & Wissel, 1994). 

Furthermore, in comparison to other models, the field of vision of fish is much lower, which demonstrates how synchronized schooling motion can be achieved based solely on the relationship between a fish and its nearest neighbours. It also helps to explain why species that rely on vision for schooling can do so even when vision is reduced (eg. at night or in turbid water) (Huth & Wissel, 1994). 

Zonal model 

While some models are made based on the number of neighbors around a fish, it has been observed that the fish adjust their behaviors according to their repulsion or attraction zones. As a neighbor is within the repulsion zone behind the fish, the fish accelerates forwards. If the distance between a fish and the neighbor in front is too large, the fish demonstrates attraction towards the neighbor and accelerate forwards as well (Herbert-Read et al., 2011).  

These observations led to the development of the zonal model, a metric distance model which describes, in greater detail, the movement of fish with respect to their neighbours based on their different attraction or repulsion zones (Fig 10) (Lukeman et al., 2010; Herbert-Read et al., 2011; Couzin et al., 2002).  

Fig. 10 The different zones in a zonal model: zor is the zone of repulsion, zoo is the zone of orientation, zoa is the zone of attraction. The volume of space that the fish cannot see is represented by the cone with an angle of (360-α)°, where α is the field of perception of the fish (Couzin et al., 2002).  

Within a given zone, the movement of fish is described by a directional vector (d_r) which indicates the desired direction of movement (Couzin et al., 2002): 

Where r_ij is the unit vector pointing in the direction of a neighbor fish j.  

The variability in the response of fish to their neighbors is modelled by the following formula (Herbert-Read et al., 2011): 

Where F_AR is an auto-regressive function, a function that takes into account the past five responses of the fish to make more reliable prediction (Ullrich, 2021). F_W is a nonlinear function on the relative position of the fish. F_Ni is a nonlinear function on the relative position of the three nearest neighbors with respect to the focal fish. ε is the stochastic variable, also called a random variable: a variable that can take any value in a given set with a certain probability (Cercignani, 2005).  

Using the zonal model, it has been determined that the single nearest neighbor has the greatest influence on an individual fish. For multiple neighbors in the attraction zone, the fish accelerates five times faster towards the first nearest neighbor than towards the second nearest neighbor. Thus, the movement of fish is determined by only a few of its neighbors. Using this model, it was hypothesized that fish process information in a way similar to humans in that they also use selectional blindness: they pay attention to and prioritize the strongest stimuli—in the case of schooling fish, the strongest stimuli is the first nearest neighbor (Herbert-Read et al., 2011).  

The findings of each of the zonal and topological velocity and heading models emphasize the role of local interactions in determining the behaviour of individuals. These behaviours can be built upon to re-examine global school structure from a perspective of collective organization. 

Polarization and cohesion 

Polarization and cohesion are key metrics for examining how individual behavior affects global school structure. 

Polarization describes the intensity of parallel orientation in the school. Cohesion is a measure of the average distance between every fish and its nearest neighbor. 

The Huth and Wissel model measured polarization and cohesion using the the polarization p, and the nearest neighbor distance nnd, respectively. Polarization was found by taking the average of the angle of deviation of each fish to the mean heading of fish in the school. Optimal polarization occurred when p=0^o and the least polarization, or maximal confusion, when p=90^o (Huth & Wissel, 1994). The nearest neighbor distance, ranged on average, between 0.5-2 body lengths (BL) of the fish (Huth & Wissel, 1994). 

Depending on the age and species of a school of fish, the mean polarization, nearest neighbor distance, and behavioral range parameters differ. For example, mackerel, which form highly polarized and cohesive school schools, have a parallel orientation p = 8-12 ^o and cohesion nnd = 0.5 BL. The repulsion area (or minimum approach distance as described in the basic rules of behavioral interactions) r1 ranges from 0.3 to 0.5 BL, and r1 – r2 = 1 to 1.5 BL (where r2 is the maximal distance that a fish will show parallel orientation behavior with respect to a neighbor) (Huth & Wissel, 1994). 

Fish have evolved such that the region in which they align their orientation with neighbors—the parallel orientation zone—is approximately uniform across different species. This value (r1 – r2 = 1 to 2 BL) is the optimal compromise between strong cohesion and strong polarization, both of which are essential for coordinated movement. The cohesion or polarization of schools significantly decreases when the parallel orientation is outside of this range (Huth & Wissel, 1994). 

The optimization of the relationship between polarization and cohesion in schools (parallel orientation area) indicates the evolutionary advantage of both aspects of schooling. For example, there is an increased ability of schooling fish to find food due to higher mobility associated with high parallel orientation, and higher rates of survival under predation in more cohesive schools (Huth & Wissel, 1994). 

Decision-making and information transfer 

Throughout the 20th century hypotheses surrounding the information transfer between animals exhibiting collective motion have evolved, from suggestions of telepathy to propositions of rapid and cascading information transfer which propagates like a wave amongst individuals in a group (Rosenthal et al., 2015). 

Previous studies have been unable to determine the social component of rapid collective responses, most likely due to incorrect underlying assumptions. While models may be able to qualitatively represent information transfer as, for example, wave propagation, more complex models may be required to describe the actual phenomenon. The self-propelled particle (SPP) model which, inspired by physical collective processes, describes interactions between particles, typically within a fixed metric (e.g. distance) or topological (e.g. number of neighbors), range using social “forces”. While mathematically convenient, many assumptions made by current models may not necessarily represent the actual perception and decision-making processes of individuals in a group (Rosenthal et al., 2015). 

It is known that information in a school is spread locally, with individuals responding primarily to the behavior of nearby neighbors (Sumpter et al., 2008). Built on this understanding of local information spread, numerous additional schooling models have been developed to address the limitations of absolute feature-based models, offering greater insight on the nuances of information transfer and decision-making in schooling fish. 

Rosenthal et al. model 

Based on the nature of evasion, Rosenthal and colleagues, in 2015, determined some of the cues which inform individual decisions to reconstruct quantitative interaction networks that could be used to better understand the cascading transfer of information in groups (Rosenthal et al., 2015). 

Information transmission in fish schools is best represented by a fractional contagion process, where the probability of response of an individual is based on the fraction of observed active neighbors. The process model has the highest correlation with the clustering coefficient (measures the interconnectedness of an individual with its neighbors) of different contagion models in that the most influential nodes are those with the highest clustering coefficient. The model also accounts for the effect position has on influence. Simple and numerical contagion processes do not match the “experimental relationship between network properties and social influence” (Rosenthal et al., 2015). 

The transmission of information described by this contagion model was observed in silverside fish in the presence of a frightening stimulus (Fig 11).  

Fig 11 The z shaped device on the right-hand side represents a fright stimulus. The spread in the headings of fish was observed over time (Sumpter et al., 2008).  

Position and centrality, influence and susceptibility 

Whether or not information is transmitted between fish in a school is dependent on the relative influence and susceptibility of the individuals. Conversely, the social influence and susceptibility of an individual in a school is primarily determined by the structural properties of the interaction network (as opposed to correlated variables such as the number of visible neighbors and local density) (Rosenthal et al., 2015). 

Individuals at the center of a school (ie. high centrality) are more inhibited from responding as they tend to have a high number of nonresponding neighbors and thus individuals with high centrality have little social influence with respect to the school (Rosenthal et al., 2015). 

Individuals at the front and side periphery of the school are the most susceptible to social cues and are the most likely to initiate behavioral cascades. Since these individuals have access to the most visual information outside of the school, they play a key role in the school’s response to predators or to attractive stimuli such as food (Rosenthal et al., 2015). 

There is a strong correlation between the clustering coefficient and influence. Redundant paths amplify behavioral change by reinforcing behaviors in the individual, increasing the activation or inhibiting the individual (in the case of many nonresponding neighbors). Individuals in a highly interconnected neighborhood (determined by the local, weighted clustering coefficient) have the greatest ability to influence behavioral change throughout a group. Similarly, these individuals are the most susceptible to changes in the group (Rosenthal et al., 2015).  

The shortest paths between individuals are more probable paths for the propagation of behavioral change (logistic regression). The probability that the behavior will propagate beyond the first responder depends on the interconnectedness of that individual’s neighbors (high clustering coefficient) (Rosenthal et al., 2015). 

These notions of position, influence and susceptibility provide context for other non-schooling models, including the previously discussed foraging model. Since fish at the periphery of a school have access to more visual information, they are also the fish that are most active in the search phase. During the assessment phase, if a fish near the periphery of the school decides to move towards a food source, there is a high likelihood that the behavior will cascade to neighboring fish since the fish has high influence, resulting in the entire school transitioning to feeding (exploitation phase). If neighbors at the periphery do not make the same assessment, or the attraction between neighbors and the fish is not sufficiently strong, the individual, since it is highly susceptible to influence of its neighbors, will return to the school (Rosenthal et al., 2015; Koy & Plotnick, 2007; Pavlov & Kasumyan, 2000).  

Conclusion 

Mathematical models offer valuable understanding of the formation and behavior of fish schools. Foraging behavior within schools is strongly influenced by interactions between fish, enabling fish to locate food efficiently while minimizing energy expenditure. Moreover, the concept of criticality shows how shoals can adapt to external disturbances, such as predator threats, through synchronized collective responses. 

Spatial configurations such as diamond and rectangular formation improve hydrodynamic efficiency and energy conservation. These structures also occur from local interactions. Fish experience attraction and repulsion forces towards their neighbors. Using these forces, simple models such as Aoki model and self-propelled particle model have been developed to predict basic schooling structures.  

By incorporating more complex parameters, such as parallel orientation of fish and the distances in which fish exhibit certain responses to their neighbors, models such as the Huth and Wissel model and zonal models have emerged. These models enable a more detailed understanding of schooling mechanisms by capturing both local interactions and global properties of the school, such as polarization, cohesion, and information transfer dynamics.   

In summary, mathematical models help explain the complex dynamics of fish schools, highlighting how local interactions lead to coordinated group behavior and enhance survival. Collectively, they offer a comprehensive view of schooling.  

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