一类具有收获率竞争系统的稳定性及Hopf分岔

首先建立了一类具有时滞的捕获率的竞争系统,该系统具有Holling II功能.接着应用特征方程,发现当τ穿过某些数时出现了Hopf分岔,并用规范型方法和中心流形定理得到Hopf分岔和分岔周期解的稳定性的计算公式.最后举例论证.     

Hopf bifurcation of a predator-prey system with stage structure and harvesting

In this paper, a two-species predator-prey system with stage structure and harvesting is investigated. The existence of Hopf bifurcations of the system is given. And the stability and directions of Hopf bifurcations are determined by applying the normal form theory and the center manifold theorem.     

Hopf bifurcation analysis for a delayed predator-prey system with a prey refuge and selective harvesting

In this paper, a delayed predator-prey system with Holling type III functional response incorporating a prey refuge and selective harvesting is considered. By analyzing the corresponding characteristic equations, the conditions for the local stability and existence of Hopf bifurcation for the system are obtained, respectively. By utilizing normal form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations supporting the theoretical analysis are given.     

Stability and Hopf bifurcation analysis of a prey–predator system with two delays

In this paper, we have considered a prey–predator model with Beddington-DeAngelis functional response and selective harvesting of predator species. Two delays appear in this model to describe the time that juveniles take to mature. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By analyzing the associated characteristic equation, its linear stability is investigated and Hopf bifurcations are demonstrated. The stability and direction of the Hopf bifurcation are determined by applying the normal form method and the center manifold theory. Numerical simulation results are given to support the theoretical predictions.     

Hopf bifurcation and stability for a differential-algebraic biological economic system

In this paper, we analyze the stability and Hopf bifurcation of the biological economic system based on the new normal form and the Hopf bifurcation theorem. The basic model we consider is owed to a ratio-dependent predator-prey system with harvesting, compared with other researches on dynamics of predator-prey population, this system is described by differential-algebraic equations due to economic factor. Here μ as bifurcation parameter, it is found that periodic solutions arise from stable stationary states when the parameter μ increases close to a certain limit. Finally, numerical simulations illustrate the effectiveness of our results.     

Hopf bifurcation in a delayed Lokta–Volterra predator–prey system

In this paper, we consider a delayed Lotka–Volterra predator–prey system with a single delay. By regarding the delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the positive equilibrium, the linear stability of the system is investigated and Hopf bifurcations are demonstrated. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Hopf bifurcation in a diffusive predator-prey model with herd behavior and prey harvesting

In this paper, the dynamics of a diffusive delayed predator-prey model with herd behavior and prey harvesting subject to the homogeneous Neumann boundary condition is considered. Firstly, choosing the harvesting term as a bifurcation parameter, then we obtain the existence and the stability of the equilibrium by analyzing the distribution of the roots of associated characteristic equation. Secondly, time delay is regarding as a bifurcation parameter, and the use of the normal form theory and center manifold theorem, the existence, stability and direction of bifurcating periodic solutions are all demonstrated detailly. Finally, summarizing some numerical simulations to illustrate the theoretical analysis.     

Hopf bifurcation in a three-species system with delays

A kind of three-species system with Holling II functional response and two delays is introduced. Its local stability and the existence of Hopf bifurcation are demonstrated by analyzing the associated characteristic equation. By using the normal form method and center manifold theorem, explicit formulas to determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solution are also obtained. In addition, the global existence results of periodic solutions bifurcating from Hopf bifurcations are established by using a global Hopf bifurcation result. Numerical simulation results are also given to support our theoretical predictions.     

Hopf bifurcation and control for the delayed predator-prey model with nonlinear prey harvesting

In our study, we focused on investigating a delayed differential-algebraic system. The system incorporates a square root functional response and non-linear prey harvesting. Employing the normal form of differential algebraic systems and the central manifold theory, we conducted a detailed analysis of the system"s stability and bifurcation phenomena, with time delay identified as a critical bifurcation parameter. When the time delay reached a critical value, the system"s equilibrium points underwent the Hopf bifurcation, resulting in system instability. To achieve stability, we introduced a feedback controller, successfully transitioning the system from an unstable to a stable state. Through subsequent numerical simulations, we validated the accuracy and correctness of our research conclusions.     

Global stability and Hopf bifurcation of a plankton model with time delay

A differential delay equation model with a discrete time delay and a distributed time delay is introduced to simulate zooplankton–nutrient interaction. The differential inequalities’ methods and standard Hopf bifurcation analysis are applied. Some sufficient conditions are obtained for persistence and for the global stability of the unique positive steady state, respectively. It was shown that there is a Hopf bifurcation in the model by using the discrete time delay as a bifurcation parameter.     

Hopf bifurcation and stability for a neural network model with mixed delays

A generalized model of the two-neuron network with mixed delays is studied. The main purpose of this paper is to explore the linear stability of the trivial solution and Hopf bifurcation of a two-neuron network with continuous and discrete delays. The general formula of the direction, the estimation formula of period and stability of bifurcated periodic solutions are also studied. Finally, the numerical simulations are given to illustrate the theoretical analysis.     

Hopf bifurcation of a bacteria-immunity system with delayed quorum sensing

On the basis of Zhang’s model (see [P. Fergola, J. Zhang, M. Cerasuolo, Z.E. Ma, On the influence of quorum sensing in the competition between bacteria and immune system of invertebrates, in: Collective Dynamic: Topics on Competition and Cooperation in the Biosciences: A Selection of Papers in the Proceedings of the BIOCOMP2007 International Conference, AIP Conference Proceedings, vol. 1028, 2008, pp. 215-232] for more details), we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells with bacterial quorum sensing mechanism. A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. Subsequently, the length of delay which preserves the stability of the positive equilibrium is estimated and Hopf bifurcation occurs when time delay crosses through a critical value are researched. Further, by using the normal form theory and center manifold theory, the explicit formulaes are calculated which determine the stability, the direction and the period of bifurcating periodical solutions. Finally, numerical simulations are employed to verify the mathematical conclusions.     

Hopf bifurcation and stability for a delayed tri-neuron network model

A neural network model with three neurons and a single time delay is considered. Its linear stability is investigated and Hopf bifurcations are demonstrated by analyzing the corresponding characteristic equation. In particular, the explicit formulae determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form theory and the center manifold theorem. In order to illustrate our theoretical analysis, some numerical simulations are also included in the end.     

Stability and Hopf bifurcation in a predator–prey model with stage structure for the predator

A predator–prey system with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. Further, the existence of a Hopf bifurcation at the positive equilibrium is also studied. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. As a result, the threshold is obtained for the permanence and extinction of the system. Numerical simulations are carried out to illustrate the main results.     

On the Hopf bifurcation for flows

Under fairly general hypotheses, we prove the existence of the families of periodic orbits obtained by Hopf bifurcation, with emphasis on their smoothness. A Banach version of a theorem of Lyapounov is obtained as a corollary. The proofs are complete, simple and original. To cite this article: M. Chaperon, S. López de Medrano, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Hopf bifurcation analysis of a reaction-diffusion Sel'kov system

A reaction-diffusion system known as the Sel'kov model subject to the homogeneous Neumann boundary condition is investigated, where detailed Hopf bifurcation analysis is performed. We not only show the existence of the spatially homogeneous/non-homogeneous periodic solutions of the system, but also derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.     

Hopf bifurcation in an hexagonal governor system with a spring

The complex dynamical behaviors of the hexagonal governor system with a spring are studied in this paper. We go deeper investigating the stability of the equilibrium points in the hexagonal governor system with a spring. These systems have a rich variety of nonlinear behaviors, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. By studying numerical simulations, it is possible to provide reliable theory and effective numerical method for other systems.     

Hopf bifurcation analysis of the Liu system

In this paper, a three dimensional autonomous system which is similar to the Lorenz system is considered. By choosing an appropriate bifurcation parameter, we prove that a Hopf bifurcation occurs in this system when the bifurcation parameter exceeds a critical value. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is presented by applying the normal form theory. Finally, an example is given and numerical simulations are performed to illustrate the obtained results.     

Analysis of stability and Hopf bifurcation for a delayed logistic equation

The dynamics of a logistic equation with discrete delay are investigated, together with the local and global stability of the equilibria. In particular, the conditions under which a sequence of Hopf bifurcations occur at the positive equilibrium are obtained. Explicit algorithm for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.]. Global existence of periodic solutions is also established by using a global Hopf bifurcation result of Wu [Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 350:1998;4799–38.]     

Stability and Hopf bifurcation analysis of a new system

In this paper, a new chaotic system is introduced. The system contains special cases as the modified Lorenz system and conjugate Chen system. Some subtle characteristics of stability and Hopf bifurcation of the new chaotic system are thoroughly investigated by rigorous mathematical analysis and symbolic computations. Meanwhile, some numerical simulations for justifying the theoretical analysis are also presented.