  A considerable number of research works has been devoted to the study of
tumor models. Several biophysical factors, such as cell proliferation,
apoptosis, chemotaxis, angiogenesis and necrosis, have been discovered to have
an impact on the complicated biological system of tumors. An indicator of the
aggressiveness of tumor development is the instability of the shape of the
tumor boundary. Complex patterns of tumor morphology have been explored by Lu,
Min-Jhe et al. [Nonlinear simulation of vascular tumor growth with chemotaxis
and the control of necrosis, Journal of Computational Physics 459 (2022):
111153]. In this paper, we continue to carry out a bifurcation analysis on such
a vascular tumor model with a controlled necrotic core and chemotaxis. This
bifurcation analysis, to the parameter of cell proliferation, is built on the
explicit formulas of radially symmetric steady-state solutions. By perturbing
the tumor free boundary and establishing rigorous estimates of the free
boundary system, %applying the Hanzawa transformation, we prove the existence
of the bifurcation branches with Crandall-Rabinowitz theorem. The parameter of
chemotaxis is found to influence the monotonicity of the bifurcation point as
the mode $l$ increases both theoretically and numerically.
