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Abstract

In this research paper, we examined a recently proposed three-dimensional dynamical model of cancer cell growth that explicitly couples the populations of tumour cells, healthy host cells, and immune cells (Pardeep et al.). We aim to elucidate the model’s novel biological features relative to classical tumour–immune interaction models and to characterize its dynamical regimes. This model is governed by nonlinear differential equations featuring a quadratic tumour proliferation term and bilinear coupling terms for tumour–immune and tumour–host interactions. Then, we performed the rigorous analysis by solving the fixed-point equations and from the Jacobian matrices at the resulting equilibria to identify the system stability. To probe the global behaviour of this model, we have computed the Lyapunov exponents and the Lyapunov dimension (Kaplan–Yorke) across the parameter space. A positive maximal Lyapunov exponent confirms the presence of deterministic chaos, and the fractal dimension quantifies the complexity of the attractor. Further, through computational analysis, we found that a robust chaotic attractor emerges near a saddle-type equilibrium and observed the existence of Shilnikov-homoclinic orbits. These findings indicate that the tumour-host-immune system can exhibit rich nonlinear dynamics, including strange attractors, and open the path for experimental studies aimed at detecting and understanding chaotic behavior in cancer progression.

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