Bifurcation Analysis of a Discrete-Time Tumor Model with Crowley-Martin Functional Response and its Optimal Control Theory

Functional response is vital in understanding and controlling disease dynamics through mathematical modeling. This study introduces the Crowley-Martin (C-M) functional response in tumor-host cell interaction dynamics, which accounts for the handling time of healthy host cells in destroying tumor cells and the magnitude of interference among these cells. When both interference and handling time of healthy host cells in destroying tumor cells are zero, tumor cells are eradicated effectively. As these parameters increase, the system shifts to a state of coexistence. The complex dynamics of the model are explored by discretize the continuous model using Euler’s method. The local stability and bifurcation of the model for different parameter values are studied, and appropriate conditions are established. Numerical simulations validate the theoretical results of Flip and Neimark-Sacker bifurcation, revealing complex dynamical behaviors, including chaotic patterns, which suggest disease progression. Based on the chaos appearing in the bifurcation diagrams, we propose an optimal control strategy to control tumor cell growth by introducing chemotherapy drugs.