This paper examines the dynamics of a three-dimensional system of difference equations through mathematical transformations and computational analysis. By transforming the original system into a bilinear form, we were able to simplify its structure and gain deeper insights into its behavior. This transformation also allowed us to study an equivalent two-dimensional system. The analysis revealed that the system possesses closed-form solutions under specific conditions, particularly when examining the discriminant of the quadratic polynomial associated with the system. We examined both cases of repeated and distinct characteristic roots, uncovering varying dynamical behaviors such as oscillations, stability, and growth, depending on the parameters involved in the analyzed examples. The model demonstrated its ability to capture various behaviors through extensive simulations, suggesting its potential applicability in real-world systems, including neural networks and other complex dynamic interactions. The findings highlight the model's robustness in various scenarios, making it a valuable tool for further theoretical and practical applications.

This paper presented a comprehensive study of a three-dimensional nonlinear system of difference equations, which can be reduced to a two-dimensional bilinear system. The system monitored the evolution of three sequences $\left(P_m\right),$ $\left(Q_m\right),$ $\left(R_m\right)$, governed by recursive relations. We investigated the solvability of this system and provided general closed-form solutions for various parameter conditions. Furthermore, the simulations provided valuable insights into the dynamic behavior of animals, modeled using recursive difference equations. The model encapsulated essential behavioral metrics, represented by the variables $P$, $Q$, and $R$, which corresponded to individual actions, social interactions, and environmental stressors, respectively. These variables adapted dynamically in response to internal and external influences, illustrating the system's sensitivity to various behavioral and environmental conditions.