Narrow Results

In this paper, the stochastic resonance phenomenon in a tumor growth model under subthreshold periodic therapy and Lévy noise excitation is investigated. The possible reoccurrence of tumor due to stochastic resonance is discussed. The signal-to-noise ratio (SNR) is calculated numerically to measure the stochastic resonance. It is found that smaller stability index is better for avoiding tumor reappearance. Besides, the effect of the skewness parameter on the tumor regrowth is related to the stability index. Furthermore, increasing the intensity of periodic treatment does not always facilitate tumor therapy. These results are beneficial to the optimization of periodic tumor therapy.

Effects of non-Gaussian α-stable Lévy noise on the Gompertz tumor growth model are quantified by considering the mean exit time and escape probability of the cancer cell density from inside a safe or benign domain. The mean exit time and escape probability problems are formulated in a differential-integral equation with a fractional Laplacian operator. Numerical simulations are conducted to evaluate how the mean exit time and escape probability vary or bifurcates when α changes. Some bifurcation phenomena are observed and their impacts are discussed.

In this paper, we examine the global dynamics of the complete Owen–Sherratt model describing the tumor–macrophage interactions. We show for this dynamics that there is a positively invariant polytope. We give upper and lower ultimate bounds for densities of cell populations involved in this model. Besides, we derive sufficient conditions under which each trajectory in tends to the mutant cells-free equilibrium point or to the equilibrium point of macrophages in isolation or to the coordinate plane corresponding to the absence of normal tissue cells depending on initial conditions. The biological sense of our results is discussed as well.

In this paper we study some features of global behavior of a seven-dimensional tumor growth model under immunotherapy described by Joshi et al. [2009]. We find the upper bounds for ultimate dynamics of all types of cell populations involved in this model. A few lower bounds are found as well. Further, we prove the existence of the bounded positively invariant polytope. Finally, we show that if the parameter modeling the flow of antigen presenting cells is very large then the tumor-free equilibrium point attracts all points in the positive orthant.

We report results of a numerical investigation on a two-dimensional cross-section of the parameter-space of a set of three autonomous, eight-parameter, first-order ordinary differential equations, which models tumor growth. The model considers interaction between tumor cells, healthy tissue cells, and activated immune system cells. By using Lyapunov exponents to characterize the dynamics of the model in a particular parameter plane, we show that it presents typical self-organized periodic structures embedded in a chaotic region, that were before detected in other models. We show that these structures organize themselves in two independent ways: (i) as spirals that coil up toward a focal point while undergoing period-adding bifurcations and, (ii) as a sequence with a well-defined law of formation, constituted by two mixed period-adding bifurcation cascades.

A tumor growth system with immune response and chemotherapy is put in a nonlinear dynamical system whose solutions are relative to the initial data. This study presents a phase space analysis of the system. Here, the basin of equilibrium points attraction is determined for a particular class of systems and is subjected to input and state constraints in which all points in phase space would be close to the equilibrium points according to the region of attraction it starts. The addition of a drug term to the system can move the solution trajectory to the desirable basin of attraction. The proposed method gives static output feedback controllers that guarantee the convergence of the generic solutions. Although such a set-point regulation problem is too challenging for general nonlinear systems, the standard surface is found by the proposed approach, which is called separatrix for the controller. This criterion of separating border can perform well even when the mentioned system has limited change parameters. The control is set by separatrix in which the output feedback controller therapy can take all solutions to the healthy state through a constrained chemotherapy protocol. Moreover, this protocol can enable globalization of healthy equilibria.