Discretization, Bifurcation and Control for a Class of Predator–Prey Interactions

The present study focuses on the dynamical aspects of a discrete-time Leslie–Gower predator–prey model accompanied by a Holling type III functional response. Discretization is conducted by applying a piecewise constant argument method of differential equations. Moreover, boundedness, existence, uniqueness, and a local stability analysis of biologically feasible equilibria were investigated. By implementing the center manifold theorem and bifurcation theory, our study reveals that the given system undergoes period-doubling and Neimark–Sacker bifurcation around the interior equilibrium point. By contrast, chaotic attractors ensure chaos. To avoid these unpredictable situations, we establish a feedback-control strategy to control the chaos created under the influence of bifurcation. The fractal dimensions of the proposed model are calculated. The maximum Lyapunov exponents and phase portraits are depicted to further confirm the complexity and chaotic behavior. Finally, numerical simulations are presented to confirm the theoretical and analytical findings.

Spatiotemporal Dynamics Induced by Nonlocal Competition in a Diffusion Predator-Prey System with Habitat Complexity

In this paper, we study a delayed diffusive predator-prey model with nonlocal competition in prey and habitat complexity. The local stability of coexisting equilibrium are studied by analyzing the eigenvalue spectrum. Time delay inducing Hopf bifurcation is investigated by using time delay as bifurcation parameter. We give some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution by utilizing the normal form method and center manifold theorem. Our results suggest that only nonlocal competition and diffusion together can induce stably spatial inhomogeneous bifurcating periodic solutions.

Mutual interference and its effects on searching efficiency between predator–prey interaction with bifurcation analysis and chaos control

In this paper, we study the dynamic of the predator–prey model based on mutual interference and its effects on searching efficiency. The parametric conditions, existence, and stability for trivial and boundary equilibrium points are studied. Also, it has shown that by applying the center manifold theorem and bifurcation theory, system undergoes Neimark–Sacker bifurcation across the neighborhood of a positive fixed point. Moreover, due to the bifurcation and chaos which objectively exist in a system, three chaos control strategies are designed and used. Moreover, to validate our theoretical and analytical discussions, numerical simulations are applied to show complex and chaotic behavior. Finally, theoretical discussions are validated with experimental field data.

Stability and Hopf Bifurcation of a Stage-Structured Cannibalism Model with Two Delays

In this work, we consider a stage-structured cannibalism model with two delays. One delay characterizes the lag effect of negative feedback of the prey species, the other has the effect of gestation of the adult predator population. Firstly, criteria for the local stability of feasible equilibria are established. Meanwhile, by choosing delay as a bifurcation parameter, the criteria on the existence of Hopf bifurcation are established. Furthermore, by the normal form theory and center manifold theorem, we derive the explicit formulas determining the properties of periodic solutions. Finally, the theoretical results are illustrated by numerical simulations, from which we can see that the predator’s gestation time delay can make the chaotic phenomenon disappear and maintain periodic oscillation, and that a large feedback time delay of prey can make predators extinct and prey form a periodic solution.

Hopf Bifurcation Analysis of a Two-Delay HIV-1 Virus Model with Delay-Dependent Parameters

In this paper, a two-delay HIV-1 virus model with delay-dependent parameters is considered. The model includes both virus-to-cell and cell-to-cell transmissions. Firstly, immune-inactivated reproduction rate R 0 and immune-activated reproduction rate R 1 are deduced. When R 1 > 1 , the system has the unique positive equilibrium E ∗ . The local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the characteristic equation at the positive equilibrium with the time delay as the bifurcation parameter and four different cases. Besides, we obtain the direction and stability of the Hopf bifurcation by using the center manifold theorem and the normal form theory. Finally, the theoretical results are validated by numerical simulation.

Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

Dynamical Behavior Analysis of a Time-Delay SIRS-L Model in Rechargeable Wireless Sensor Networks

The traditional SIRS virus propagation model is used to analyze the malware propagation behavior of wireless rechargeable sensor networks (WRSNs) by adding a new concept: the low-energy status nodes. The SIRS-L model has been developed in this article. Furthermore, the influence of time delay during the charging behavior of the low-energy status nodes needs to be considered. Hopf bifurcation is studied by discussing the time delay that is chosen as the bifurcation parameter. Finally, the properties of the Hopf bifurcation are explored by applying the normal form theory and the center manifold theorem.

Stability and Bifurcation for a Delayed Diffusive Two-Zooplankton One-Phytoplankton Model with Two Different Functions

In this paper, a diffusion two-phytoplankton one-zooplankton model with time delay, Beddington–DeAnglis functional response, and Holling II functional response is proposed. First, the existence and local stability of all equilibria of such model are studied. Then, the existence of Hopf bifurcation of the corresponding model without diffusion is given by taking time delay as the bifurcation parameter. Next, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and center manifold theorem. Furthermore, due to the local bifurcation theory of partial functional differential equations, Hopf bifurcation of the model is investigated by considering time delay as the bifurcation parameter. The explicit formulas to determine the properties of Hopf bifurcation are given by the method of the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are performed to check out our theoretical results.

Bifurcation Analysis of a Bounded Rational Duopoly Game with Consumer Surplus

The nonlinear dynamical behaviors of economic models have been extensively examined and still represented a great challenge for economists in recent and future years. A proposed boundedly rational game incorporating consumer surplus is introduced. This paper aims at studying stability and bifurcation types of the presented model. The flip and Neimark–Sacker bifurcations are analyzed via applying the normal form theory and the center manifold theorem. This study helps determine an appropriate choice of decision parameters which have significant influences on the behavior of the game. The duopoly game that is formed by considering bounded rationality and consumer surplus is more realistic than the ordinary duopoly game which only has profit maximization. And then, some numerical simulations are provided to verify the theoretical analysis. Finally, we compare the dynamical behaviors of the built model with that of Bischi–Naimzada model so as to better understand the performance of the duopoly game with consumer surplus.