###### Abstract  
In this integrative review paper, we explore the parallels between the physical models of gyrotrons and some equations used in diverse and broad scientific fields. These include [PERSON]'s famous equation, [PERSON] equation, the Lotka-Volterra equations and the Kuramoto model. The paper is written in the form of a pedagogical discourse and aims to provide additional insights into gyrotron physics through analogies and parallels to theoretical approaches used in other fields of research. For the first time, reachability analysis is used in the context of gyrotron physics as a modern tool for understanding the behavior of nonlinear dynamical systems.  
gyrotron; theory of gyro-devices; Adler's equation; [PERSON]'s equation; Lotka-Volterra equations; Kuramoto model, reachability analysis 2025 15 7920 April 202  
achieve the desired system performance. This makesreachability analysis an important concept in oscillator theory, particularly concerning the behavior of oscillator systems under various conditions.  
[PERSON]. [PERSON] was the first to point out that mode interaction, especially the mode competition in the resonant cavity of a gyrotron, is similar to competition between various species in ecology, e.g., the predator-prey dynamics described by the Lotka-Volterra equation [5]. These analogies were then commented on in [6]. In the present paper, we discuss in more detail the parallels between the theoretical models of gyrotron physics and some prominent equations used in other sciences.  
Gyro-devices such as gyrotron, gyro-backward oscillator (Gyro-BWO), gyro-traveling wave tube (Gyro-TWT), cyclotron auto-resonance maser (CARM), etc., are powerful sources of coherent electromagnetic radiation in a wide frequency range from millimeter to sub-millimeter wavelengths, which has recently been extended to the terahertz frequencies. They are used in numerous areas of fundamental physics research and in various advanced technologies. Various applications of gyrotrons are described in numerous recent review articles (see e.g., [7; 8; 9; 10; 11; 12; 13; 14] and the references therein). The physics of their operation is presented in a large number of publications and monographs (see, e.g., [15; 16; 17; 18; 19; 20; 21; 22]). The theory of gyrotrons comprises a rich hierarchy of physical models, beginning with simple analytical, linear, semi-self-consistent models and culminating in those formulated from first principles ([PERSON]'s equation for the electromagnetic field combined with relativistic equations of electron motion). These models are implemented in a great number of dedicated problem-oriented, adequate numerical particle-in-cell (PIC) codes (see, for instance, [23; 24; 25; 26; 27; 28; 29; 30; 31]) that are used for modeling, simulation, computer-aided design (CAD), and optimization of gyrotrons for various applications. The evolution of these models, however, can be traced back to their genesis in some famous equations, which are explored in this paper.  
The rest of the paper is organized as follows. In Section 2, we consider [PERSON]'s equation in the context of phase locking and synchronization of gyrotrons. Next, in Section 3, we discuss the Van der [PERSON] equation and its use for analyzing gyrotron oscillators. In Section 4, we explore some parallels between the Kuramoto model of synchronization and some physical mechanisms of gyrotron operation. The parallels between the models of mode interaction in gyrotrons and the models of population dynamics in ecological systems are pointed out in Section 5. Finally, the paper ends with a conclusion in Section 6.