Neimark–Sacker bifurcation of a chemotherapy treatment of Glioblastoma multiform (GBM)

Bozkurt Yousef F. , Yousef A.

Advances in Difference Equations, cilt.397, no.2019, ss.1-25, 2019 (SCI İndekslerine Giren Dergi) identifier

  • Cilt numarası: 397 Konu: 2019
  • Basım Tarihi: 2019
  • Doi Numarası: 10.1186/s13662-019-2324-9
  • Dergi Adı: Advances in Difference Equations
  • Sayfa Sayıları: ss.1-25


In this paper, we propose a system of differential equations with piecewise constant

arguments to describe the growth of GBM under chemotherapeutic treatment and

the interaction among the glial cells, the cancer cells, and the chemotherapeutic

agents. In this work, the cancer cells are considered as two populations: the sensitive

cancer cells and the resistant cancer cells. The sensitive tumor cells produce a

population that is known as the resistant cell population, where this population has

more resistance to the drug treatment than the sensitive tumor cell population. We

analyze at first the local and global stability of the positive equilibrium point by

considering the Schur–Cohn criteria and constructing a suitable Lyapunov function,

respectively. Moreover, we use the center manifold theorem and bifurcation theory to

show that the model undergoes Neimark–Sacker bifurcation. To investigate the case

for the extinction of the tumor population, we consider the Allee threshold at time t.

Simulation results support the theoretical study.