In this paper, we derived and studied the discrete version of a continuous fractional-order predator-prey system with constant-yield prey harvest and a monotonically increasing function response, as well as a fear effect of the predator on the bait. After studying in detail the existence and local stability of the fixed points of the system, some favorable conditions for the occurrence of its Neimark–Sacker bifurcation and period-doubling bifurcation were obtained by using the central manifold theorem and bifurcation theory. Finally, numerical simulations carried out using Matlab software illustrate the theoretical results previously obtained and reveal its some new dynamics–the chaotic occurrence.

We study in this paper the global attractivity for a higher order rational difference equation. As application, our results not only include and generalize many known ones, but also formulate some new results for several conjectures presented by Camouzis and Ladas, et al.

In this paper, we utilize the semi-discretization method to construct a discrete model from a continuous predator-prey model with herd behaviour and group defense. Specifically, some new results for the transcritical bifurcation, the period-doubling bifurcation, and the Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory.Novelty includes a smooth transition from individual behaviour (low number of prey) to herd behaviour (large number of prey). Our results not only formulate simpler forms for the existence conditions of these bifurcations, but also clearly present the conditions for the direction and stability of the bifurcated closed orbits. Numerical simulations are also given to illustrate the existence of the derived Neimark-Sacker bifurcation.

Using the forward Euler method, we derive a discrete predator-prey system of Gause type with constant-yield prey harvesting and a monotonically increasing functional response in this paper. First of all, a detailed study for the existence and local stability of fixed points of the system is obtained by invoking an important lemma. Mainly, by utilizing the center manifold theorem and the bifurcation theory some sufficient conditions are obtained for the saddle-node bifurcation and the flip bifurcation of this system to occur. Finally, with the use of Matlab software, numerical simulations are carried out to illustrate the theoretical results obtained and reveal some new dynamics of the system-chaos occuring.

In this paper, a discrete predator-prey model with double Allee effect is discussed. We first simplify the corresponding continuous predator-prey model, and use the semidiscretization method to obtain a new discrete model. Next, the existence and local stability of nonnegative fixed points of the new discrete model are studied by using a key lemma. Then, by using the center manifold theorem and bifurcation theory, the sufficient conditions for the occurrences of transcritical bifurcation and Neimark-Sacker bifurcation and the stability of closed orbit bifurcated are obtained. Finally, the numerical simulations are presented, which not only verify the existence of Neimark-Sacker bifurcation but also reveal some new dynamic phenomena of this model.

In this paper, a discrete predator-prey model incorporating Allee effect and cannibalism is derived from its continuous version by semidiscretization method. Not only the existence and local stability of fixed points of the discret system are investigated, but more important, the sufficient conditions for the occurrence of its period-doubling bifurcation and Neimark-Sacker bifurcation are obtained using the center manifold theorem and local bifurcation theory. Finally some numerical simulations are given to illustrate the existence of Neimark-Sacker bifurcation. The outcome of the study reveals that this discrete system undergoes various bifurcations including period-doubling bifurcation and Neimark-Sacker bifurcation.