Résumé: The objective of this study is to examine the dynamic behavior of a conformable fractional-order predator–prey system with a Holling-III type functional response. The fractional-order system is transformed into a discrete model through a discretization process. The fixed points are analyzed for both their existence and uniqueness. We then assess the local stability of the two fixed points using linearization techniques, and use the center manifold theorem and bifurcation theory to identify conditions for period-doubling and Neimark-Sacker bifurcations. To verify our findings, we perform numerical simulations and compute the maximum Lyapunov exponents to confirm the presence of chaotic behavior.

Résumé: Difference equations play a crucial role in modeling discrete dynamical systems across a wide range of scientific disciplines, including biology, economics, engineering, and physics. In this article, we examine a nonlinear system of rational difference equations of order four, defined by    xn+1 = ynyn−3 axn−2 −byn−3 , yn+1 = xnxn−3 cyn−2 −dyn−3 , n ∈ N0, where a,b, c, and d are real parameters. The system is initialized with nonzero real values for x−3, x−2, x−1, x0, y−3, y−2, y−1, y0. The objectives of this study are to derive explicit closed-form solutions, determine conditions for the existence and uniqueness of solutions, and analyze the qualitative behavior of the system, including stability and asymptotic properties. The results offer new insights into the intricate dependence of the system’s dynamics on both the parameters and the initial conditions. This contributes to a deeper understanding of high-order nonlinear difference systems and their complex behaviors

Résumé: In SIR epidemic models, fractional-order differential equations have recently been employed to describe long-term memory effects and genetic characteristics that are frequently observed in environments but are not adequately represented by traditional integer-order systems. In this study, we analyze the dynamics of a generalized conformable discrete-time SIR epidemic model. We establish the conditions required for the existence and local stability of both the disease-free equilibrium (DFE) and the endemic equilibrium (EE). The analysis of the SIR epidemic model reveals that the disease-free equilibrium is asymptotically stable when the basic reproduction number R0 is less than one, while the endemic equilibrium becomes asymptotically stable when R0 is greater than one. The dynamical behavior of the proposed SIR epidemic model is explored numerically for both commensurate and incommensurate fractional orders, using phase portraits, bifurcation diagrams, the maximum Lyapunov exponent, chaos control, and the 0 − 1 test. The model exhibits more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. Compared to classical integer-order structures, our fractional formulation offers a more explanation representation of population oscillations and stability transitions. In addition to providing deeper insights into the dynamics of the considered model, these findings highlight the significance of fractional calculus in expanding our understanding of a generalized conformable incommensurate fractional-order modified SIR epidemic model. Finally, MATLAB simulations are conducted to validate the presented results.

Résumé: In this paper, we study precise and exact traveling wave solutions of the conformable differential nonlinear Schrödinger equation. Then, we transform the given equation into an integer order differential equation by utilizing the wave transformation and the characteristics of the conformable derivative. To extract optical soliton solutions, we divide the wave profile into amplitude and phase components. Further, we introduce a new extension of a modified Jacobi elliptic functions method to the conformable differential nonlinear Schrödinger equation with group velocity dispersion and coefficients of second-order spatiotemporal dispersion.

Résumé: In this paper, we study precise and exact traveling wave solutions of the conformable differential nonlinear Schrödinger equation. Then, we transform the given equation into an integer order differential equation by utilizing the wave transformation and the characteristics of the conformable derivative. To extract optical soliton solutions, we divide the wave profile into amplitude and phase components. Further, we introduce a new extension of a modified Jacobi elliptic functions method to the conformable differential nonlinear Schrödinger equation with group velocity dispersion and coefficients of second-order spatiotemporal dispersion.

Résumé: In this article, a prey–predator system is considered in Caputo-conformable fractional-order derivatives. First, a discretization process, making use of the piecewise constant approximation, is performed to secure discrete-time versions of the two fractional-order systems. Local dynamic behaviors of the two discretized fractional-order systems are investigated. Numerical simulations are executed to assert the outcome of the current work. Finally, a discussion is conducted to compare the impacts of the Caputo and conformable fractional derivatives on the discretized model.

Résumé: The classic Lotka-Volterra model is a two-dimensional system of differential equations used to model population dynamics among two-species: a predator and its prey. In this analysis, we consider a modified three-dimensional fractional-order Lotka-Volterra system that models population dynamics among three-species: a predator, an omnivore and their mutual prey. Biologically speaking, population models with a discrete and continuous structure often provide richer dynamics than either discrete or continuous models, so we first discretize the model while keeping one time-continuous dependent variable in each equation. Then, we analyze the stability and bifurcation near the equilibria. The results demonstrated that the dynamic behaviors of the discretized model are sensitive to the fractional-order parameter and discretization parameter. Finally, numerical simulations are performed to explain and validate the findings, and the maximum Lyapunov exponents is computed to confirm the presence of chaotic behavior in the studied model.

Résumé: The current study aims at discussing the dynamic behaviors for a discrete prey-predator model with a fractional-order derivative. Firstly, a discretization process is carried out to obtain a discrete version of the fractional-order system. Based on the stability theorems, the stability of the equilibrium points of the discretized fractional-order system is investigated. It has been found that the dynamic behavior of the discrete model is sensitive to the fractional-order parameter and the discretization parameter. Using the intrinsic birth rate of prey, as the bifurcation parameter, it is shown that the discrete system undergoes Neimark-Sacker bifurcation around the positive equilibrium point. Finally, numerical simulations are performed to validate our theoretical findings. Moreover, the maximum Lyapunov exponents is computed to confirm the presence of chaotic behavior in the discrete model.