Bounding the dynamics of a chaotic-cancer mathematical model
type: Item
instance of: scholarly article
title: Bounding the dynamics of a chaotic-cancer mathematical model
zbMATH Open document ID: 1427.92050
DOI: 10.1155/2018/9787015
author: Paul A. Valle
author: Luis N. Coria
author: Diana Gamboa
author: Corina Plata
published in: Mathematical Problems in Engineering
publication date: 2019-02-08T00:00:00Z
review text: Summary: The complexity of cancer has motivated the development of different approaches to understand the dynamics of this large group of diseases. One that may allow us to better comprehend the behavior of cancer cells, in both short- and long-term, is mathematical modelling through ordinary differential equations. Several ODE mathematical models concerning tumor evolution and immune response have been formulated through the years, but only a few may exhibit chaotic attractors and oscillations such as stable limit cycles and periodic orbits; these dynamics are not that common among cancer systems. In this paper, we apply the Localization of Compact Invariant Sets (LCIS) method and Lyapunov stability theory to investigate the global dynamics and the main factors involved in tumor growth and immune response for a chaotic-cancer system presented by \textit{M. Itik} and \textit{S. P. Banks} [Int. J. Bifurcation Chaos Appl. Sci. Eng. 20, No. 1, 71--79 (2010; Zbl 1183.34064)]. The LCIS method allows us to compute what we define as the \textit{localizing domain}, which is formulated by the intersection of all lower and upper bounds of each cells population in the nonnegative octant, \( \mathbb{R}_{+, 0}^3\). Bounds of this domain are given by inequalities in terms of the system parameters. Then, we apply Lyapunov stability theory and LaSalle's invariance principle to establish existence conditions of a global attractor. The latter implies that given any nonnegative initial condition, all trajectories will go to the largest compact invariant set (a stable equilibrium point, limit cycles, periodic orbits, or a chaotic attractor) located either inside or at the boundaries of the localizing domain. In order to complement our analysis, numerical simulations are performed throughout the paper to illustrate all mathematical results and to better explain their biological implications.
Mathematics Subject Classification ID: 92C50
Mathematics Subject Classification ID: 92C42
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C60
zbMATH DE Number: 7019825
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/9787015
OpenAlex ID: W2899935627
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Wikidata QID: Q128973174
description: scientific article; zbMATH DE number 7019825