In this paper, we study the dynamics and bifurcation of a two-dimensional discrete-time predator-prey model. The existence and local stability of the equilibrium points of the model are analyzed algebraically. It is shown that the model can undergo a transcritical bifurcation at equilibrium point on the $x$-axis and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium point. Some numerical simulations are presented to illustrate our theoretical results.  

In this paper, we investigate an open-access fishery model which is used to examine the dynamics of the resource and industry and to explain the current economic status of the anchovy fishery. We consider the local character of the interior and boundary equilibrium points. Also, we show that the considered system of difference equations exhibits Neimark-Sacker bifurcation under certain conditions. The existence of the repelling curve and invariant curve is demonstrated. We show that in a certain parameter region the corresponding map of the considered system is an area-preserving map, so the positive equilibrium point in that case is stable. Also, we produce numerical simulations to support our findings.

This paper investigates the dynamics of non-autonomous cooperative systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to evolutionary population cooperation models. We use two methods to extend the global attractivity results for autonomous cooperative systems to related non-autonomous cooperative systems which appear in recent problems in evolutionary dynamics.

This paper investigates the rate of convergence of a certain mixed monotone rational second-order difference equation with quadratic terms. More precisely we give the precise rate of convergence for all attractors of the difference equation $x_{n+1}=\frac{Ax_{n}^{2}+Ex_{n-1}}{x_{n}^{2}+f}$, where all parameters are positive and initial conditions are non-negative.The mentioned methods are illustrated in several characteristic examples. 2020 Mathematics Subject Classification. 39A10, 39A20, 65L20.

We consider the following system of rational difference equations in the plane: $$\left\{\begin{aligned}%{rcl}x_{n+1} &= \frac{\alpha_1}{A_1+B_1 x_n+ C_1y_n} \\[0.2cm]y_{n+1} &= \frac{\alpha_2}{A_2+B_2 x_n+ C_2y_n}\end{aligned}\right. \, , \quad n=0,1,2,\ldots $$ where the parameters $\alpha_1, \alpha_2, A_1, A_2, B_1, B_2, C_1, C_2$ are positive numbers and initial conditions $x_0$ and $y_0$ are nonnegative numbers. We prove that the unique positive equilibrium of this system is globally asymptotically stable. Also, we determine the rate of convergence of a solution that converges to the equilibrium $E=(\bar{x},\bar{y})$ of this systems.   2000 Mathematics Subject Classification. 39A10, 39A11, 39A20

This paper investigates an autonomous discrete-time glycolytic oscillator model with a unique positive equilibrium point which exhibits chaos in the sense of Li–Yorke in a certain region of the parameters. We use Marotto’s theorem to prove the existence of chaos by finding a snap-back repeller. The illustration of the results is presented by using numerical simulations.

This paper investigates an autonomous predator-prey system of difference equations with three equilibrium points and exhibits chaos in the sense of Li-Yorke in the positive equilibrium point. Numerical simulations are presented to illustrate our results.

We investigate a discrete counterpart of planar dynamical system of nonlinear differential equations induced by kinetic differential equations for a two-species chemical reaction. Chemical reactions exhibit a wide range of dynamical behavior. We show how the theoretical analysis provides insight into the potential behavior of chemical reaction systems, determining the areas of parametric space which indicate scenarios for local stability, then for one type of bifurcation co-dimension one and one type of bifurcation co-dimension two. Precisely, we prove the existence of period-doubling bifurcation and 1:2 resonance bifurcation also, by using the center manifold theorem and the technique of normal forms. All mathematical investigations are illustrated with numerical examples, bifurcation diagrams, Lyapunov exponents and phase portraits.

This paper investigates the dynamics of non-autonomous competitive systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to the evolutionary population of competition models of two species.