Exact Solutions and Periodic Dynamics of a Three-Dimensional Nonlinear Difference System with Delayed Cyclic Interactions

This paper investigates a nonlinear three-dimensional system of difference equations describing the interaction among three mutually dependent sequences evolving over discrete time. The proposed model accounts for nonlinear coupling effects as well as feedback structures that govern the system’s dynamics. We first establish the conditions ensuring the well-definedness and solvability of the system, followed by the construction of closed-form expressions of the solutions under appropriate assumptions on the initial data and parameter settings. To support the theoretical findings, numerical experiments are carried out, accompanied by graphical illustrations that reveal the influence of parameter variations on the qualitative dynamics of the system. As an application, we demonstrate how the proposed three-dimensional nonlinear system can be interpreted in the context of delayed cyclic competition among three interacting populations. This application illustrates the relevance of the developed framework to ecological systems exhibiting nonlinear feedback mechanisms and delayed interactions.