Analysis of a delayed mathematical model for tumor-immune cell interactions with Holling type II functional response

Abstract

In this study, we analyzed a three-dimensional nonlinear differential system considering Holling type II functional response that describes the dynamics of tumor cells, cytotoxic T lymphocytes, and helper T cells, with a single interaction delay. The linear stability of the internal equilibrium point and the presence of the Hopf bifurcation are examined, with the discrete time delay serving as the bifurcation parameter. To demonstrate the rich dynamic behavior of the model, we present numerical simulations with various values of the time delay τ and the attack rate of cytotoxic T lymphocytes on tumor cells (α1). These simulations exhibit the presence of periodic oscillations and tumor death with survival of the mentioned immune cells, for all α1 greater than a fixed threshold with or without delay.