Multiple bifurcations in a mathematical model of glioma-immune interaction

In this paper, a mathematical model describing the interaction of malignant glioma cells, macrophages and glioma specific CD8+T cells is discussed. The biologically feasible equilibria and corresponding local stability are deduced. The bifurcations and related dynamical behaviors of this model are further studied thoroughly. The existence of transcritical bifurcation and saddle–node bifurcation is derived based on Sotomayor’s theorem and Hopf bifurcation is well discussed. The codimension 2 bifurcation such as Bogdanov–Takens bifurcation is investigated using the normal form theory and center manifold theorem in more detail. Finally, numerical simulations are obtained to validate our analytical findings by varying the parameters. •The dynamics of a mathematical model describing the interaction of malignantglioma cells, macrophages and glioma specific CD8+T cells are discussed.•The co-existence of equilibria of the model are obtained by numerical simulation.•The existence of codimension 1 and codimension 2 bifurcations is investigated.•Extensive numerical simulations are obtained to validate our analytical findings.