Stability and Bifurcation Analysis of a Discrete Tumor-Immune System with Allee Effects

Differential equations are usually employed to accurately represent the ongoing relationships between tumor cells and immune effector populations, enabling scientists to discover how variation in growth and response rates affects tumor development or elimination. The essential objective of this work is to analyze the dynamical development of a discrete tumor-immune interaction model, with a particular focus on finding out how the combined effects of tumor growth and immune response influence tumor progression. The forward Euler approach is effectively used to discretize the governed system. The bifurcation theory is used to establish the fixed points of the considered system, the stability about the fixed points, and Neimark–Sacker and period-doubling bifurcations. We identify parameter domains that result in tumor existence, restricted oscillations, or full-tumor elimination utilizing stability evaluation, bifurcation examination, and computational simulations. In addition, the 0–1 test is presented. Chaos control is also developed. This article successfully discusses some numerical simulations to verify the results obtained. In general, the research gives an overall insight into this interaction and highlights the circumstances under which the immune system is capable of suppressing or removing tumor cells.