In this paper, we construct and formulate the solutions and periodicity character of the following nonlinear rational systems of difference equations:

$$S_{n+1}={\displaystyle \frac{T_n{S}_{n-2}}{S_{n-2}+T_{n-1}}},T_{n+1}={\displaystyle \frac{S_n{T}_{n-2}}{\pm T_{n-2}\pm S_{n-1}}},n=0,1,2,....,(0.1)$$

where the initial conditions $s_{-2},s_{-1},s_0,t_{-2},t_{-1},t_0$ are positive real numbers. Moreover, some mathematical programs are used to support our theoretical results of each system in this paper.

In this paper, we suggest a fractional-order susceptible, exposed, infectious, and latent (SEIL) epidemic model with discrete time delays that provide for biological factors and memory effects in the spread of disease. Caputo derivatives and fixed-point theory are implemented, and we identify the existence and uniqueness of solutions, as well as the basic reproduction number for both endemic and disease-free equilibria. Local stability is analyzed through characteristic equations and linearization, while numerical simulations confirm theoretical results and illustrate the influence of fractional order, delays, and parameters. The findings show that fractional-delay models provide a more flexible and effective framework for studying disease dynamics and control.

In this paper, the dynamics of a discrete-time chemostat model were investigated. The discretization was obtained using the piecewise constant argument method. An analysis was performed to determine the existence and stability of fixed points. In addition, we have shown that the model experiences transcritical and period-doubling bifurcations. Two chaos control techniques, feedback control and hybrid control, were employed to control bifurcation and chaos in the model. Moreover, we provided numerical simulations to substantiate our theoretical results. This study illustrates that the piecewise constant argument method is more dynamically consistent than the forward Euler method.