## 1 Introduction  
The scientific management of crowd-gathering scenarios is a significant issue regarding public safety. Using physics, transportation, and mathematical tools to describe the characteristics of a crowd flow, crowd flow dynamics has been a well-known academic hotspot for several decades. It comprises the processes of establishing a mathematical model of a crowd flow and then carrying out numerical simulations and solving them. There were some similarities between crowd flows and continuous fluids. Since Swiss mathematician and mechanics scientist [PERSON] proposed a classical assumption for fluid continuity [1], the irregular thermal motion of molecules in a fluid were simplified as the regular motion of the molecules; i.e., the fluid was considered as a continuous medium. Then, fluids were mainly studied based on a macro-mechanical motion theory of a continuous medium. In general, a crowd flow consists of pedestrians, with physical gaps between them. From a micro point of view, crowd flows are not continuously distributed, but if the distance between pedestrians and the surrounding environment is defined as a pedestrian area, it is feasible to assume a crowd flow as a continuous pedestrian fluid. Therefore, like the continuum hypothesis in fluid dynamics, the macroscopic quantities of crowd flows are the velocities, densities, and flows, which also conform to the physical laws of mass conservation, momentum conservation, and energy conservation. [PERSON] first proposed a macroscopic model for pedestrian flow [2; 3]:  
It was shown that the distribution functions for crowd densities and velocities were consistent with the results from the Maxwell-Boltzmann theory, except for significant deviations near the frequency mode of each distribution [3]. [PERSON] further proposed a hydrodynamic model of a pedestrian crowd, in which a particle collision was used to describe the interactions between macroscopic pedestrian groups [3]. [PERSON] considered pedestrian flow as a continuous fluid medium and used fluid dynamics equations to describe the relationships between the speed, density, and flow rate of pedestrians' According to references [4; 5], a typical macro dynamic model for the field of crowd flows is established in the form of Equation (1). \[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v_{h})}{\partial x}+\frac{ \partial(\rho v_{l})}{\partial y}=0 \tag{1}\]  
where \(v_{h}\) represents the horizontal velocity, and \(v_{l}\) represents the lateral velocity. Most studies on macro crowd flow used this crowd flow model in Equation (1) as their primary theory base. This model assumed that the behaviors of the crowds were homogeneous, i.e., that crowds make approximately the same responses under the same stimulus; however, this was not sufficiently realistic in practice. In real pedestrian-gathering scenarios, the panic behavior characteristics of crowd flows were relatively complex. Due to the subjective initiatives of the crowds and external environments, once internal panic disturbances due to emergencies such as pedestrian falls, stampedes, shootings, terrorist attacks, explosions, and fires occur, the stop-and-go phenomena of crowds can cause a chaotic phenomenon of the sudden acceleration and deceleration of pedestrians. In this context, more complex crowd trampling and other accidents often easily occur [5]. It was significant to keep a pedestrian crowd stable considering that a large-scale crowd (including more than 2000 pedestrians) system consisted of a large number of autonomous individual pedestrians, movement rules, and the scene environment [6]. As a continuous flow, a large-scale crowd flow also had three typical system states: stability, critical stability, and instability. Large-scale crowd trample disasters were typical phenomena of crowd instability owing to internal disturbances. Most of the existing literature about crowd stability is based on traffic flow theory and lacks systematic consideration.