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: 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.