Stability and bifurcation analysis of a reaction–diffusion equation with spatio-temporal delay

In this paper, a new mathematical model (a system of delay differential equation) is proposed to describe dynamical behaviors of malaria in an infected host with red blood cells (RBCs), infected red blood cells (iRBCs) and immune factors. The basic reproduction number R0 of the malaria infection is derived. If R0≤1, the uninfected equilibrium E0 is globally asymptotically stable. If R0>1, there exists two kinds of infection equilibria. The conditions of these equilibria with respect to the existence, stability and uniform persistence are given. Furthermore, fluctuations occur when the model undergoes Hopf bifurcation, and periodic solution appears near the positive equilibrium. The direction and stability of Hopf bifurcation are also obtained by applying the center manifold method and the normal form theory. Numerical simulations are provided to demonstrate the theoretical results.

In this paper, the diffusion is introduced to an immunosuppressive infection model with delayed antiviral immune response. The direction and stability of Hopf bifurcation are effected by time delay, in the absence of which the positive equilibrium is locally asymptotically stable by means of analyzing eigenvalue spectrum; however, when the time delay increases beyond a threshold, the positive equilibrium loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the norm form and the center manifold theory. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

<p>In this work, we have studied an eco-epidemic model using the Crowley-Martin functional response that includes disease in prey and gestation delay in the predator population. The model possesses three equilibria, namely the disease-free, Predator-free, and the interior equilibrium point. In addition, we examined the stability of the equilibrium points varying the infection rate and time delay parameter. Detailed analysis of Hopf bifurcation of the interior equilibrium point contains two situations: with delay and without delay. Moreover, we have studied the direction of the Hopf bifurcation and the stability of periodic solutions utilizing normal form theory and the center manifold theorem. It is emphasized that Hopf bifurcation occurs when the time delay exceeds the critical value and that the critical value of the delay is strongly impacted by the infection rate in prey. A detailed numerical simulation is provided to verify the analytical results.</p>

This paper is devoted to the analysis of a nutrient-plankton model with delayed nutrient cycling. Firstly, stability and Hopf bifurcation of the positive equilibrium are given, and the direction and stability of Hopf bifurcation are also studied. We show that delay, which is considered in the decomposition of dead phytoplankton, can induce stability switches, such that the positive equilibrium switches from stability to instability, to stability again and so on. One can observe that the influence of delay on the system dynamics is essential. Then, we prove that there exists at least one positive periodic solution as the time delay varies in some regions using the global Hopf bifurcation result of Wu (1998, Trans Am Math Soc 350:4799–4838) for functional differential equations. Furthermore, the impact of input rate of nutrient is discussed along with numerical results, and the role of delay in the nutrient cycling is interpreted ecologically. Finally, several groups of illustrations are performed to justify analytical findings.

Hybrid control of Turing instability and Hopf bifurcation in CDK1-APC feedback systems with diffusion

The article mainly researched the Hopf bifurcation of Lorenz-like system about the coefficient of the quadratic term. When the quadratic term changes, the solution to the Lorenz-like system will become the local periodic solution. Further the stability of this periodic solution and the bifurcation direction of this periodic solution were discussed, and found when the quadratic term comes through a threshold , the direction of hopf bifurcation and stability were given, and the result as follows. If , when , the direction of bifurcation is , when , the direction of bifurcation is ; (b) if , we have the contrary result. That is when , the direction of bifurcation is ; when , the direction of bifurcation is . If , the bifurcation solution is asymptotically stable; if , the bifurcation solution is not asymptotically stable. Finally employing the matlab compute the numerical periodic solution, the results fit the theory well.

Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay

In this paper, the Rosenzweig-MacArthur predator-prey model with the hyperbolic tangent functional response is investigated. We choose capturing efficiency of predator as the bifurcation parameter and mainly discuss the existence, direction and stability of Hopf bifurcation by Poincaré-Andronov-Hopf bifurcation theorem.

This paper presents a detailed analysis on the dynamics of a ring network with short-cut. We first investigate the absolute synchronization on the basis of Lyapunov stability approach, and then discuss the linear stability of the trivial solution by analyzing the distribution of zeros of the characteristic equation. Based on the equivariant branching lemma, we not only obtain the existence of primary steady state bifurcation but also analyze the patterns and stability of the bifurcated nontrivial equilibria. Moreover, by means of the equivariant Hopf bifurcation theorem, we not only investigate the effect of connection strength on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. In particular, we further consider the secondary bifurcation of the nontrivial equilibria. These studies show that short-cut may be used as a simple but efficient switch to control the dynamics of a system.

In this article, we study Hopf bifurcation and Turing instability of a diffusive predator-prey model with hunting cooperation. For the local model, we analyze the stability of the equilibrium and derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, first it is shown that Turing instability occurs, then the direction and stability of the Hopf bifurcation is reached. Our results show that hunting cooperation plays a crucial role in the dynamics of the model, that is, it can be beneficial to the predator population and induce the rise of Turing instability. Finally, numerical simulations are performed to visualize the complex dynamic behavior.

In this paper, we investigate the effect of nonlocal delay on stability and spatiotemporal dynamics of a diffusive predator-prey model with Allee effect in predators. For the system without nonlocal delay, the stability and Turing instability conditions are derived. For the system with nonlocal delay, all Turing bifurcation curves on $ (d_{2}, \tau) $ plane are obtained, and the critical Turing bifurcation curve near the stable region is determined by the discussion of the properties of Turing bifurcation curves. It is found that the presence of nonlocal delay decreases the range of the occurrence for Turing bifurcation. The critical Hopf bifurcation curve is found by the geometric stability switch criterion. The stable region for the positive equilibrium is bounded by the critical Turing and Hopf bifurcation curves. On the boundaries of stable region, Hopf bifurcation, Turing bifurcation, Turing-Turing bifurcation and Turing-Hopf bifurcation may occur. To reveal the complex dynamic near Turing-Hopf bifurcation point, normal form for the system with nonlocal delay is derived, from which different types of bistability are found, including two stable spatially inhomogeneous periodic solutions and two stable spatially inhomogeneous steady states.

Synchronous dynamics of two coupled oscillators with inhibitory-to-inhibitory connection

Stability and Hopf bifurcation for a three-species food chain model with time delay and spatial diffusion

A Cohen-Grossberg neural network with discrete delays is investigated in this paper. Sufficient conditions for the existence of local Hopf bifurcation are obtained by analyzing the distribution of roots of characteristic equation. Moreover, the direction and stability of Hopf bifurcation are obtained by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the obtained results.

This paper investigates the nonlinear dynamics of Transmission Control Protocol and Active Queue Management (TCP/AQM) networks, including the local stability and periodic bifurcation. The parameter of transportation delay affects the stability of the dynamical systems. One of the purposes of our work is to determine the delay stable interval of the transportation network. It is found that there is only one critical value of network delay by the τ decomposition technique. When the delay passes the critical point, the system performs Hopf bifurcation with a pair of symmetry with purely imaginary roots (PIR). In addition, the other purpose is to consider the stability of bifurcating periodic solutions. Combining with τ-decomposition strategy and central manifold theory, the issues of delay stable interval and stability of Hopf bifurcation are all tackled. Finally, numerical examples are illustrated to show the accuracy and effectiveness of the proposed method.