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In this paper, we propose and analyze an age-structured tumor immune model with time delay. We divide immune cells into two kinds. One is those whose growth is independent of tumor and the other is those whose growth depends on the simulation of the tumor. For these cells, their physiological ages are considered. A mature time delay τ is introduced to the tumor-simulation-dependent immune cells to restrict those cells who participate in the immune response to grow to a minimum physiological age. The existence and stability threshold ℜ is established for the tumor-free equilibrium state. If ℜ < 0 , the tumor-free equilibrium state is both locally and globally asymptotically stable. Whereas, when ℜ > 0 , the tumor equilibrium state is locally asymptotically stable if τ ∈ [ 0 , τ k = 0) and a Hopf bifurcation occurs when τ passes through the threshold τ k = 0 . This may partly explain the periodic recurrence of some tumors. Finally, theoretical results are verified by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a multiple delayed HIV-1 infection model which describes the interactions of humoral, cytotoxic T lymphocyte (CTL) immune responses, and two modes of transmission that are the classical virus-to-cell infection and the direct cell-to-cell transmission are investigated. The model incorporates three delays, including the delays of cell infection, virus production and activation of immune response. We first prove the well-posedness of the model, and calculate the biological existence of equilibria and the reproduction numbers, which contain virus infection, humoral immune response, CTL immune response, CTL immune competition, and humoral immune competition. Further, the threshold conditions for the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response, and interior are established by utilizing linearization method and the Lyapunov functionals. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics. [ABSTRACT FROM AUTHOR]

Delayed dynamical system plays a vital role in describing the dynamical phenomenon of neural networks. In this article, we proposed a class of new BAM neural networks involving time delay. The traits of solution and bifurcation behavior of the established BAMneural networks involving time delay were probed into. First, the existence and uniqueness is discussed using a fixed point theorem. Second, the boundedness of solution of the formulated BAM neural networks involving time delay was analyzed by applying an appropriate function and inequality techniques. Third, the stability peculiarity and bifurcation behavior of the addressed delayed BAM neural networks were investigated. Fourth, Hopf bifurcation control theme of the formulated delayed BAM neural networks was explored by virtue of a hybrid controller. By adjusting the parameters of the controller, we could control the stability domain and Hopf bifurcation onset, which was in favor of balancing the states of different neurons in engineering. To verify the correctness of gained major outcomes, computer simulations were performed. The acquired outcomes of this article were new and own enormous theoretical meaning in designing and dominating neural networks. [ABSTRACT FROM AUTHOR]

In this study, a class of novel fractional‐order bi‐directional associative memory (BAM) neural networks involving double time delays are put up and investigated. First of all, we prove that the solution of the involved neural networks exists and is unique and bounded. Second of all, we investigate the stability behavior and the onset of Hopf bifurcation of the involved network models by applying the stability theory and the related Hopf bifurcation knowledge on fractional‐order differential equations. A sufficient condition to ensure the stability and onset of Hopf bifurcation of the considered network models is established. The importance of time delay in dynamical behavior of the fractional‐order delayed BAM neural networks has been displayed. Lastly, computer simulations with Matlab software are presented to test the effectiveness of the established theoretical results. The obtained analysis fruits of this manuscript play a vital role in designing and controlling networks. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of bifurcation for a ring fractional Hopfield neural network with leakage time delay and communication time delay. The stability and the Hopf bifurcations of such a network without and with time delays are investigated by analyzing the associated characteristic equations. Specifically, some criteria for the occurrence of Hopf bifurcations at the trivial steady state are established. It is shown that the dynamical property of the network is not only crucially dependent on the communication time delay, but also significantly influenced by the leakage time delay. Furthermore, the effects of the order on the Hopf bifurcation are numerically demonstrated. Finally, four numerical examples are provided to illustrate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]