Search

In this work, we develop a mathematical model to study the effect of drug on the development of cancer including the quiescent compartment. The model is governed by a system of delay differential equations where the delay represents the time that the cancer cell take to proliferate. Our analytical study of the stability shows that by considering the time delay as a parameter of bifurcation, it is possible to have stability switch and oscillations through a Hopf bifurcation. Moreover, by introducing the drug intervention term, the critical delay value increases. This indicates that the system can tolerate a longer delay before oscillations start. In the end, we present some numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In the machining process, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece, hence the relevant measures must be taken to predict and avoid this phenomenon of instability. In this paper, we propose a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms, to represent self-excited vibrations in machining. It is proved that Hopf bifurcation exists when bifurcation parameter equals a critical value, a formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are given by using the normal form method and center manifold theorem. Numerical simulations show excellent agreement with the theoretical results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a commensalism model with a discrete delay and a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the interior equilibrium is investigated. It is found that the interior equilibrium is asymptotically stable when the discrete delay is less than a certain critical value and unstable when the discrete delay is greater than this critical value. By regarding the delay as the bifurcation parameter, the existence of Hopf bifurcation is also considered. Furthermore, the properties of Hopf bifurcation are determined. In particular, the global existence of bifurcated periodic solutions is established by applying the topological global Hopf bifurcation theorem, which shows that the local Hopf bifurcations imply the global ones after the second critical value of parameter. Finally, numerical simulations supporting the theoretical analysis are also included. [Copyright &y& Elsevier]

Abstract: In this paper, a delayed differential equation model that describes infection of thymus with HIV-1 is considered. We first investigate the existence and stability of the equilibria and then we study the effect of the time delay on the stability of the infected equilibrium. Criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. Finally, by using a delay as a bifurcation parameter, the existence of Hopf bifurcation is investigated. Numerical simulations are presented to illustrate the analytical results. [Copyright &y& Elsevier]

Abstract: In this paper, we construct a two-layer feedback neural network to theoretically investigate the influence of symmetry and time delays on patterned rhythmic behaviors. Firstly, linear stability of the model is investigated by analyzing the associated transcendental characteristic equation. Next, by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, we not only investigate the effect of synaptic delays of signal transmission on the pattern formation, but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Thirdly, based on the normal form approach and the center manifold theory, we derive the formula to determine the bifurcation direction and stability of Hopf bifurcating periodic solutions. Finally, some numerical examples and the corresponding numerical simulations are used to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract: A delayed Lotka–Volterra two-species predator–prey system with discrete hunting delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that under suitable conditions on the parameters the positive equilibrium is asymptotically stable when the hunting delay is less than a certain critical value and unstable when the hunting delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs), we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the hunting delay crosses through a sequence of critical values. In particular, by applying the normal form theory and the center manifold reduction for FDEs, an explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations is given. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, Hopf bifurcation for a delayed SIS epidemic model with stage structure and nonlinear incidence rate is investigated. Through theoretical analysis, we show the positive equilibrium stability and the conditions that Hopf bifurcation occurs. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. In addition, we also study the effect of the inhibition effect on the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included. [Copyright &y& Elsevier]