A survey of mathematical modeling based on flocking system

2.2.1. Euler method

Euler is a holistic approach that ignores the characteristics of the individual, and views them as “whole” to study its density. Based on the theory of the classical diffusion theory of Feck, the theory of diffusion is as follows:

where $\rho$ is the change in the concentration of the particles per moment in the unit area, which is equal to the amount of displacement of the particle on the $X$-axis, $D$ is the particle diffusivity. Skellam J. G. thought it was not scientific only considering the spread of the whole itself, and ignoring the environment or from the internal resistance effect. He proposed to increase a convection factor on the $X$ axis. There are two variables are on $X$ direction, and u is the cluster of translational convection, then the equation can be written as:

Mogilner A and collaborators worked on flow problem, a force made a careful study of the particles; For the choice of diffusivity $D$, Alt W. Degenerate and Ikeda T. found that the function of $D$ in different clusters were different.

Euler’s study of the large-scale flocking that scientists have observed has worked, it uses partial differential equations to analyze the diffusion, and the analysis process does not require environmental considerations [2]. The Euler method is able to study the general movement properties of the cluster, but it ignores the individual characteristics and cannot be modeled from the individual point of view, which is not conducive to the control of the application.

2.2.2. Lagrange method

Lagrange method is based on individual microscopic modeling analysis, which using Newton equations of motion to describe the force of the object, the equation is as follows:

where $m$ is the quality, $x$ is the position, and $F$ is the resultant force of objects.

Reder C. M firstly proposed a clustering model using the force function and applied it to the behavior of the fish group; Warburton K. and his collaborators studied the effects of net forces on biological influences; Beni G. and his collaborators analyzed the stability of the model and validated the conclusion that it was stable when the environment was disturbed; The team of Grain V. and Wang extended this force to $N$-dimensional space and studied its stability.

The Lagrange method analyzes on the individual, and its modeling is based on the individual’s intrinsic factor, the individual is affected by the external environment and Individual interaction rules. It’s a natural intuitive modeling approach, it has influenced many subsequent research ideas, which is an important way of improving overall efficiency through individual control.

2.3.1. Boid model

In 1986, Reynolds used a computer to simulate the world's first cluster campaign, which was called the Boid model. This was based on mapping the actual flock of birds to the computer, by using dots on the computer screen to represent the bird, setting the coordinates, the speed direction, and so on for each point. Reynolda was obtained by the model experiment of three simple rules: attract-synchronous-rejection, which become the cornerstone of the foundation of the clustering model for subsequent model is established based on.

Wang and his partners considered on the basis of the model to increase the actual situation, environment variables, such as outside groups within the population variables and the individual character itself, which replacing the original individual consistency hypothesis in the model [3], They simulated birds, fish and insects and studied the differences in their populations.

2.3.2. Vicsek model

In 1995 the physicist T. Vicsek and his collaborators proposed particle synchronization model (Vicsek model), which range from the perspective of statistical mechanics of cluster motion, according to the individual view within the scope of the average particle velocity adjust itself, the speed and the direction of flocking synchronization in a certain space, randomly distributed $N$ particles. The position and movement direction the particle updating equations are as follows:

where $x$, $v$ and $\theta$ are the values of the position, velocity, and angle of the particle $i$ at time $t$; $\mathrm{\Delta }t$ is set to 1; the Angle at time $t+1$ is determined by the average velocity of all the particles in the area of the center of the $i$ particle; $\mathrm{\Delta }\theta$ is the noise interference that obeys the uniform distribution; In the paper [4], the study showed that when the flocking density is large and the noise is small, the individual will gradually become synchronized; Increase the noise and the particles will become confused. In Vicsek model, the state of the individual is only related to density and noise, and the noise levels interfere with the particles and make different changes. This kind of change can be compared to the three state changes of matter. The model will control the connection of science and physics, and it becomes the relevant theoretical research.

The Vicsek model is widely studied for its simple but also is a complex system. Chen and others put forward individual update rules, they thought the priority should be given to the movement of neighborhood Angle deviation of the two biggest particles, including value instead of the original model of average speed [5], and the simulation showed that this method not only reduces the calculation quantity, but also improves the convergence speed of the model. Gao and others believed that the convergence of the model is related to the number of particles in the field of vision, and introduces the degree of “degree” to describe it [6]; Li was simply adding the individual’s speed to the update rules, making the overall adjustment more accurate [7]. Tian had studied the model from a biologically-driven point of view [8].

2.3.3. Three-Circle model

In 2002 I. D. Couzin proposed the Three-Circle model [9], he incorporated the Reynolds movement three principles into the model, building the individual perception range, which consists of exclusion zones, synchronous zones. This was a model controlled by the individual of the distance parameter. Each individual had its own range of inductance, and the individual had no influence over its range; When other individuals within the scope of the induction and particles from the far, attract trend particle was governed by the rules, to be near individuals, an area known as the attract area; As the individual approaches a certain distance, it moved into the sync area of the particle, allowing the speed of the two to synchronize. If the individual was less than the distance of the particle, the particle produces an exclusion trend away from the individual, which was the exclusion zone of the particle.

The individuals around the particle will give priority to the individuals who reject the minimum distance in the area, following the following rules:

where $c$ is the position of the individual, $d$ is the direction vector, the equation shows that when there are other individuals in the exclusion zone, the particle moves in a sparse direction; If there is no other individuals in the exclusion zone, the state of the particle will be affected by the individual in the other two regions, the velocity of the particle is affected by the speed of the synchronous region, and the position is closer to the attractor. The formula is as follows:

2.3.4. Social force model

In 1995, Helbing et al proposed a social force model to simulate pedestrian flow under normal conditions [10]. The force model was based on Newton’s mechanical modeling method. The individual speed is changed by force driving:

where $m$ and $v$ are the quality and speed of the individual; $f_{ij}$ and $f_{iw}$ are individual and individual interactions with the environment.

Helbing and others worked on the basis of the model considering joining panic environment in 2000 [11], which used to study stress conditions of the reaction of the crowd, and found some behavior characteristics of the crowd in a tense; Hu and others believed that people would automatically avoid the crowded areas, and proposed an improved model of a dynamic blocking area to study the relationship between human flux and density [12].

The social force model applies to the layout of various large public places and the retreat of disasters, the model simulates the fire in the dormitory of the university [13].

2.3.5. Dynamics model

Similar to the social force model, it is the analysis of the force situation of the individual, which assumes that $N$ individuals constitute a cluster, and each individual’s dynamic equation is:

where $r_i={\left(x_i,y_i\right)}^T$ is the position vector of individual $i$, $v_i={\left(x_i,y_i\right)}^T$ is the velocity vector of the individual, $u_i={\left(x_i,y_i\right)}^T$ is the control input, which is:

where ${\alpha }_i$ is the speed control vector, so that the individual speed matching; $a_i$ is a distance vector control for avoidance; individual position under the model with adjacency graph $G=\left(V,e\right)$, $V=\left\{1,\dots N\right\}$, is the set of all the individual form of points, $\epsilon =v\times v$ is the communication between the individual under the condition of edge set.

The dynamic model is one of the main modeling methods of the cluster control model. The force acting on the individual is connected with the environment change, which is convenient for efficient and precise operation control.