Bifurcation, multistability in the dynamics of tumor growth and electronic simulations by the use of Pspice

This contribution throws more light on the nonlinear dynamics of a cancer HET model proposed by Pillis and Radunskaya [23] and adjusted by Itik and Banks [13]. Research shows that the number of equilibrium points in the model fluctuates between 3 and 8 depending on whether the biological parameters of the system vary. For some well precise parameters, the system has 5 equilibrium points, 3 of which are always unstable, 1 always quasi-stable and the last, on the other hand, the only point where all cell populations change stability according to the values of the rate of growth parameter of the host cells. We show that the Hopf bifurcation occurs in this system, when the growth rate of the host cells varies and reaches a critical value. Applying the theory of the normal bifurcation form, we describe the formulae for determining the direction and stability of the periodic solutions of the Hopf bifurcation. Numerical simulations are performed to illustrate its theoretical analysis. Numerical simulations, performed in terms of bifurcation diagrams, Lyapunov exponent graph and phase portraits, permits to highlight the rich and complex phenomena presented by the model. The exploitation of these numerical results reveals that the system degenerates a transition to chaos intermittently through the saddle-node bifurcation. We find that the system presents the diversity of bifurcations such as the chaos, periodic window, saddle-node bifurcations, internal crises, when mentioned parameters are varied in small steps. In addition, we find that the model also has multiple attractors for a precise set of parameters. The basins of attraction of various coexisting attractors present extremely complex structures, thus justifying jumps between coexisting attractors. Finally, in addition to mathematical analysis developed in the work and digital processing, we propose an electronic implementation of the three-dimensional model of cancer capable of imitating the model of mathematical evolution. We determine a circuit equivalent to the ODE model, we determine the electrical components equivalent to the ODE parameters of the mathematical model and the equivalent electronic circuit is simulated numerically with the Orcad-Pspice software. The numerical results obtained from the proposed Orcad-Pspice electronic circuit exhibit the same behavior as that obtained by the numerical simulations and the comparison results provide proof of the reliability and precision of the proposed circuit.