This paper proposes a partial differential equation model based on the model introduced by V. A. Kuznetsov and M. A. Taylor, which explains the dynamics of a tumor–immune interaction system, where the immune reactions are described by a Michaelis–Menten function. In this work, time delay and diffusion process are considered in order to make the studied model closer to reality. Firstly, we analyze the local stability of equilibria and the existence of Hopf bifurcation by using the delay as a bifurcation parameter. Secondly, we use the normal form theory and the center manifold reduction to determine the normal form of Hopf bifurcation for the studied model. Finally, some numerical simulations are provided to illustrate the analytic results. We show how diffusion has a significant effect on the dynamics of the delayed interaction tumor–immune system.
The main purpose of the present paper is to investigate asymptotic properties of a class of neutral differential equations of the formx(s)ds = e(t).(1)Using the Lyapunov direct method, the condition of stability of trivial solution for equation ( 1) with e(t) = 0 is given. With respect to e(t) = 0 , the boundedness of solutions of equation ( 1) is also obtained. Two examples are provided to illustrate the results.
The main purpose of the present paper is to investigate asymptotic properties
of some neutral delay differential equations by means of Lyapunov functions.
The asymptotic stability of the solutions is given in terms of
delay-independent criteria. Two examples are given to illustrate the
results.
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