Dynamical Analysis of a Melanoma Model with Immune Response

Abstract

In this paper, we study the stability and bifurcation behavior of a three-dimensional melanoma model with immune response. The system has at least one and at most three positive equilibria. It is proved that the system undergoes Hopf bifurcation and saddle-node bifurcation at the positive equilibrium. We investigate the direction of Hopf bifurcation and stability of the bifurcating periodic solution by center manifold theorem and normal form theory. Moreover, codimension two bifurcations of the system are analyzed. We demonstrate the existence of Bautin bifurcation and Bogdanov–Takens bifurcation of the system. The normal form of Bautin bifurcation and Bogdanov–Takens bifurcation are given. Finally, some numerical simulations are demonstrated to support our theoretical results, and the importance of some parameters of the system is discussed, in particular the activation rate of CD8[Formula: see text]T cells.