%%% use two columns and PHYSCON 2011 format \documentclass[twocolumn]{physcon13} \usepackage{graphicx,array} \usepackage{amssymb,amsmath} % If you want to use BibTex, then uncomment the following command: %\usepackage[dcucite]{harvard} \title{Analysis of global dynamics based on localization of compact invariant sets of \ some cancer tumor growth models } %% Use the \author command to put authors horizontally on one row. %% Separate the authors with the \and command. Put as may authors as possible on one row. %% If necessary, then use a second \author command as shown below, which will put additional authors on another row. %%% first two or three authors \author{ \twlsfb Konstantin E. Starkov \affiliation{ Departamento de Control\\ CITEDI-IPN\\ Tijuana, B.C\\ konst@citedi.mx, konstarkov@hotmail.com } % \and \twlsfb FirstName2 FamilyName2 % \affiliation{ % Department2\\ % University2\\ % Country2\\ % name2@university2.country2 % } } %%% another two or three authors, uncomment if necessary %\author{ \twlsfb C. Third % \affiliation{ % Institute\\ % University\\ % Country\\ % a.name@university.country % } % \and \twlsfb D. Fourth % \affiliation{ % Institute \\ % University\\ % Country\\ % anothername@university.country % } % \and \twlsfb E. Fifth % \affiliation{ % Institute \\ % University\\ % Country\\ % fifth@university.country % } %} \begin{document} \maketitle \begin{abstract} In this work we describe recent results of dynamical analysis of two cancer growth models with help of the localization method of compact invariant sets. These models are the Owen-Sherratt system and the Kirschner-Panetta model. \end{abstract} \begin{keywords} Polynomial system, cancer tumor growth model, localization, compact invariant set, global stability \end{keywords} \section{ Introduction} Nowadays it is well-recognized that the cancer is one of the most dangerous killers of the humankind. Many dynamic models are elaborated in order to study the progress of the cancer tumor and comprehensively apply efficient treatment strategies. Dynamical analysis of models describing interactions between the cancer tumor and the immune system of a patient is one of topics in mathematical biology which attracts the attention of many researchers, see the review paper \cite{Eftime} and \cite{dOnofrio}. For example, such mathematical models as the Kirschner-Panetta model, \cite{Kirsh}; Kuznetsov et. al. model \cite{kuz} and the dePillis- Radunskaya models \cite{Pillis,Pilliss} are examined in many publications which can be easily seen from web pages in Scholar Google. \ The goal of this work is to review main results on dynamical analysis of two cancer tumor models which were very recently published\ by the author and his colleagues, \cite{starpo,starco}. \section{ The 1st model: the Owen- Sherratt system} In 1997 Owen and Sherratt in \cite{owen} obtained the spatially independent model describing the macrophage- tumor interactions: \begin{equation} \begin{array}{l} \dot{x}_{1}=\frac{a_{1}x_{1}x_{4}}{a_{2}+x_{1}+x_{2}+x_{3}}% +a_{3}+a_{4}x_{4}-a_{5}x_{1}x_{2}x_{4} \\ +a_{6}x_{5}-a_{7}x_{1}, \\ \dot{x}_{2}=\frac{a_{8}x_{2}}{a_{2}+x_{1}+x_{2}+x_{3}}% -a_{9}x_{2}-a_{5}x_{1}x_{2}x_{4}, \\ \dot{x}_{3}=\frac{a_{10}x_{3}}{a_{2}+x_{1}+x_{2}+x_{3}}-a_{9}x_{3}, \\ \dot{x}_{4}=a_{11}x_{2}-a_{12}x_{4}, \\ \dot{x}_{5}=a_{5}x_{1}x_{2}x_{4}-a_{13}x_{5}.% \end{array} \label{CancerModel} \end{equation} In these equations by $x_{1}(t),...,x_{5}(t)$ we denote vectors of densities of the macrophages, the mutant cells, the normal-tissue cells, the chemical concentration and the macrophage- mutant complexes. In (\ref{CancerModel}) we exploit parameters $a_{j},j=1,...,13,$ which depend on biological parameters given in \cite{owen} by the following formulae% \begin{eqnarray} a_{1} &=&\alpha (N+N_{e});a_{2}=N;a_{3}=I;a_{4}=I\sigma; \nonumber \\ a_{5} &=& k_{1};a_{6}=k_{2};a_{7}=\delta _{l}; \label{parameters} \\ a_{8} &=& \xi \delta (N+N_{e});a_{9}=\delta ;a_{10}=\delta(N+N_{e}); \nonumber \\ a_{11} &=& \beta ;a_{12}=\delta _{f};a_{13}=k_{2}+\delta _{c}, \nonumber \end{eqnarray} Here: 1) $k_{1},k_{2}$ are two positive parameters characterizing the lysis process in the two upper equations of the system (\ref{CancerModel}); 2) $N$ is a measure of the crowding response; 3) $N_{e}$ is the total equilibrium density of all cell types in the normal tissue; 4) $\xi $ is the scaling parameter of the mutant cell growth rate, $\xi >1$; 5) $\delta $ is the growth rate in the normal tissue; 6) $\beta $ is the constant secretion rate per unit of mutant-cell density; 7) $\delta _{f}$ is the rate of a linear natural decay of the mutant density; 8) $\delta _{c}$ is the rate of a linear natural decay of the complex density; 9) $\delta _{l}$ is the rate of a linear natural decay of the macrophage density; 10) $\alpha $ is parameter characterizing the proliferation term in the first equation in (\ref% {CancerModel}); 11) $I$ and $\sigma $ are two positive parameters characterizing the constant influx of tissue macrophages. Despite of the existence of more than one hundred papers with references on these three articles \ up to now dynamic properties of the system (\ref% {CancerModel}) are not well investigated. The proof of the existence of a bounded positively invariant domain (BPID)\ inside the positive orthant of biologically feasible states for cancer growth models is a typical problem which is investigated in a number of publications; some of them will be mentioned below. \ This is due to the fact that ultimate upper and lower bounds for trajectories have a biological sense characterizing the "ultimate" health condition of a patient and efficiency of applied drug treatment and immunotherapy. Therefore finding of bounds for a BPID is of essential interest as well. These bounds are expressed as functions of the model parameters. If some of these parameters can be manipulated by biologists the latter have chance to regulate global dynamics via the process of a variation of parameters admissible for manipulations. In this case this type of model parameters may be considered as control parameters. In this relation one may mention the rate of tumor cell lysis and the density of engineered macrophages in the blood vessel. Further, if we may decrease sufficiently the upper bound for the density of tumor cells by manipulating of some control parameters the health condition of a patient may be characterized as a cancer remission. Not only upper bounds but some lower bounds for the ultimate dynamics may be essential for a characterization of health conditions because they provide conservative estimates of densities of cells populations. For example, it concerns the density of the macrophages population because macrophages are important for general homeostasis, see comment e.g. in \cite{owen}. However, up to the author knowledge, computing explicit upper and lower bounds for the BPID is a problem which is not well-studied for the most of cancer tumor growth models. For example, we mention that 1) for the de Pillis- Radunskaya model, see \cite{Pilliss}, the upper bound respecting the ultimate value for effector immune cells has not \ been derived in \cite{Itik}; no lower bounds have not been found there as well; 2) for the Kirschner- Panetta model only \ the upper bound respecting the ultimate value for the density of tumor cells and the lower bound respecting the ultimate value for the concentration of effector molecules were given in \cite{Kirshh}; 3) Nani and Freedman established in \cite{nani} the existence of a bounded positively invariant domain is established for one four-dimensional cancer growth model; however, values for its bounds were not computed explicitly. In some publications concerning cancer growth models, see e.g. \cite{Pilliss,Kirshh,lej,nani} one can meet results related to the study of $\omega -$ limit sets of trajectories in the BPID. If the structure of these $\omega -$ limit sets is not complex as it occurs in low- dimensional models, for example, if they contain only equilibrium points then the ultimate health condition of a patient may be characterized by the complete cancer clearance equilibrium point, or the "small" tumor size equilibrium point, or the "death" equilibrium point. Below by the positive orthant $\mathbf{R}_{+}^{n}=\{x=(x_{1},...,x_{n})\in \mathbf{R}^{n},x_{i}>0,i=1,...,n\}.$ Let $\mathbf{R}_{+,0}^{n}$ be the closure of $\mathbf{R}_{+}^{n}.$ \section{Some preliminaries and notations} The principal idea of the localization method of compact invariant sets is to study extrema of some differentiable functions specially constructed and to use the fact that they are bounded on trajectories taken from compact invariant sets. This idea is realized with help of exploiting the first order extremum conditions and in some cases of the high order extremum conditions, see e.g. in \cite{Krishko,starkov}. Let us introduce some objects and recall useful assertions, see in \cite{Krishko}. We consider a $C^{\infty }-$ differentiable system \begin{equation} \dot{x}=F(x), \label{main-system-pol} \end{equation}% with $x\in \mathbf{R}^{n}$, $F(x)=(F_{1}(x),\dots ,F_{n}(x))^{T}$ and $% F_{i}(x)\in C^{\infty }(\mathbf{R}^{n})$, $i=1,\dots ,n.$ Let $h(x)\in C^{\infty }(\mathbf{R}^{n})$ be a function such that $h$ is not the first integral of the system (\ref{main-system-pol}). The function $h$ is used in the solution of the localization problem of compact invariant sets and is called a localizing function. Suppose that we are interested in the localization of all compact invariant sets located in some set $N\subset \mathbf{R}^{n}$ where $N$ \ is an invariant set for the system (\ref% {main-system-pol}) or a domain. By $S(h)$ we denote the set $\{x\in \mathbf{R% }^{n}:L_{F}h(x)=0\},$ where $L_{F}h(x)$ is a Lie derivative with respect to $% F$. Further, we define \begin{eqnarray*} h_{\inf }(U) &:&=\inf \{h(x)\mid x\in U\cap S(h)\}; \\ h_{\sup }(U) &:&=\sup \{h(x)\mid x\in U\cap S(h)\}. \end{eqnarray*}% \textbf{Assertion 1. }\textit{1. For any }$h(x)\in C^{\infty }(R^{n})$% \textit{\ all compact invariant sets of the system (\ref{main-system-pol}) located in }$N$\textit{\ are contained in the set defined by the formula }$% K(N)=\{x\in N:h_{\inf }(N)\leq h(x)\leq h_{\sup }(N)\}$\textit{\ as well. 2. If }$N\cap S(h)=\emptyset $\textit{\ then the system (\ref{main-system-pol}) has no compact invariant sets located in }$N$\textit{.} Any of sets $K(N)$\ is called a localization set. Now we remind another result called the iteration theorem: \textbf{Assertion 2.}\textit{\ Let }$h_{m}(x),m=1,2,\dots $\textit{\ be a sequence of functions from }$C^{\infty }(R^{n})$\textit{. Sets }% \[ K_{1}=K_{h_{1}},\quad K_{m}=K_{m-1}\cap K_{m-1,m},\quad m>1, \]% \textit{with} \[ \begin{array}{l} K_{m-1,m}=\{x:h_{m,\inf }\leq h_{m}(x)\leq h_{m,\sup }\}, \\ h_{m,.\sup }=\sup\limits_{S_{h_{m}}\cap K_{m-1}}h_{m}(x), \\ h_{m,\inf }=\inf\limits_{S_{h_{m}}\cap K_{m-1}}h_{m}(x),% \end{array}% \]% \textit{contain all compact invariant sets of of the system (\ref% {main-system-pol}) and }$K_{1}\supseteq K_{2}\supseteq \dots \supseteq K_{m}\supseteq \dots \;.$ Below by $f$ we denote the vector field which corresponds the system under investigation. \section{Compact localization set for the system (\protect\ref{CancerModel})} Below in this section we derive the compact localization set containing all compact invariant sets in $\mathbf{R}_{+,0}^{5}$ of the system (\ref% {CancerModel}). It is easy to see that the domain $\mathbf{R}_{+,0}^{5}$ is positively invariant which means that each trajectory in $\mathbf{R}% _{+,0}^{5}$ cannot escape from $\mathbf{R}_{+,0}^{5}.$ Firstly, we present \textbf{Theorem 1.}\textit{\ All compact invariant sets of the system (\ref% {CancerModel}) are placed in the set }$\cap _{i=1}^{4}K(h_{i}),$\textit{\ with} \begin{eqnarray*} K(h_{1}) &=&\{0\leq x_{3}\leq x_{3\max }:=\frac{a_{10}-a_{2}a_{9}}{a_{9}}\}; \\ K(h_{2}) &=&\{0\leq x_{2}\leq x_{2\max }:=\frac{a_{8}-a_{2}a_{9}}{a_{9}}\}; \\ K(h_{3}) &=&\{0\leq x_{4}\leq x_{4\max }:=\frac{(a_{8}-a_{2}a_{9})a_{11}}{% a_{9}a_{12}}\}; \\ K(h_{4}) &=&\{0\leq x_{1}+x_{5}\leq h_{4\max }: \\ &=&(a_{3}+ \\ &&\frac{(a_{1}+a_{4})(a_{8}-a_{2}a_{9})a_{11}}{a_{9}a_{12}})(\frac{1% }{a_{7}}+\frac{1}{a_{13}-a_{6}})\}. \end{eqnarray*} Examining formulae for $L_{f}h_{i},i=1,...,4,$ one can see that all trajectories in $\mathbf{R}_{+}^{5}$ enter into some compact domain and remain there. More precisely, we establish \textbf{Theorem 2. }\textit{The domain \ }$\cap _{i=1}^{4}K(h_{i})$\textit{\ is a positively invariant polytope for the system (\ref{CancerModel}).} \section{Lower bounds} Here we prove \textbf{Proposition 1.}\textit{\ \ All compact invariant sets of (\ref% {CancerModel}) are located in the set}% \[ \{x_{1}+x_{5}>\frac{a_{3}}{M_{1}}\}, \] \textit{with }$M_{1}:=\max \{a_{7};a_{13}-a_{6}\}.$ \textbf{Proposition 2.} \textit{All compact invariant sets of (\ref% {CancerModel}) are located in the set}% \[ \{x_{1}+x_{2}+2x_{5}>\frac{a_{3}}{M_{2}}\}, \] \textit{with }$M_{2}:=\max \{a_{7};a_{9};a_{13}-\frac{1}{2}a_{6}\}.$ \textbf{Proposition 3.} \textit{All compact invariant sets of (\ref% {CancerModel}) are located in the set} \[ K(h_{6})=\{x_{1}\geq x_{1\min }:=\frac{a_{3}}{a_{5}x_{2\max }x_{4\max }+a_{7}% }\}. \] \textbf{Proposition 4 }\textit{All compact invariant sets of (\ref% {CancerModel}) are located in the set}% \[ K(h_{7})=\{x_{1}-x_{2}\geq (a_{9}-a_{7})a_{9}^{-1}x_{1\min }+(a_{3}-a_{8})a_{9}^{-1}\} \] \textit{provided }$a_{9}\geq a_{7}$\textit{\ \ or in }% \[ K(h_{7})=\{x_{1}-x_{2}\geq (a_{9}-a_{7})a_{9}^{-1}x_{1\max }+(a_{3}-a_{8})a_{9}^{-1}\} \] \textit{provided }$a_{9}\eta _{1}>a_{10} $\textit{\ holds. Then all compact invariant sets located in }$% \mathbf{R}_{+,0}^{5} $\textit{\ are contained in the set }% \[ \{x_{2}=x_{5}=0;x_{1},x_{3},x_{4}\geq 0\}\cup \{x_{3}=0;x_{1},x_{2},x_{4},x_{5}\geq 0\}. \]% \textit{2. If }$\eta _{1}>a_{8}$\textit{\ holds then all compact invariant sets located in }$\mathbf{R}_{+,0}^{5}$\textit{\ are contained in the set}% \[ \{x_{2}=x_{5}=0;x_{1},x_{3},x_{4}\geq 0\}. \] \textbf{Corollary 1}\textit{. In the first case of Theorem 3 the }$\omega -$% \textit{\ limit set }$\omega (x)$\textit{\ of any trajectory }$\varphi (x,t)$% \textit{\ in }$\mathbf{R}_{+,0}^{5}$\textit{\ is contained in the set} \begin{eqnarray*} \{\{x_{2} &=&x_{5}=0;x_{1},x_{3},x_{4}\geq 0\}\cup \{x_{3}=0;x_{1},x_{2},x_{4},x_{5}\geq 0\}\} \\ &&\cap K(h_{2})\cap K(h_{3})\cap K(h_{6}) \end{eqnarray*}% \textit{while in the second case of Theorem 3 the }$\omega -$\textit{\ limit set }$\omega (x)$\textit{\ of any trajectory }$\varphi (x,t)$\textit{\ in} $% \mathbf{R}_{+,0}^{5}$ \textit{is located in the set} \begin{equation} \{x_{2}=x_{5}=0;x_{1},x_{3},x_{4}\geq 0\}\cap K(h_{2})\cap K(h_{3})\cap K(h_{6}) \label{40} \end{equation} \section{ The Kirschner- Panetta model} In what follows, we describe recent results concerning dynamical analysis of the Kirschner- Panetta model, see [6]. The paper [6] has more than 180 citations and is of essential interest for biologists and, in particular, for specialists in mathematical biology. Though dynamics \ of the Kirschner- Panetta model is rich and may include e.g. periodic orbits; homoclinic orbits, see [6], its rigorous analysis is far from being completed, [7]. This model is written in the form \begin{eqnarray} \dot{x} &=&cy-\mu _{2}x+\frac{p_{1}xz}{g_{1}+z}+s_{1}; \nonumber \\ \dot{y} &=&r_{2}y\left( 1-by\right) -\frac{axy}{g_{2}+y}; \label{Cancer Eq} \\ \dot{z} &=&\frac{p_{2}xy}{g_{3}+y}+s_{2}-\mu _{3}z \nonumber \end{eqnarray} where by $x(t);y(t);z(t)$ we denote vectors of densities of immune (effector) cells; tumor cells; the concentration of effector molecules correspondingly at the moment $t$ in the single tumor-site compartment. Dynamics of (\ref{Cancer Eq}) is biologically feasible on the positive orthant $\mathbf{R}_{+}^{3}.$ All parameters are supposed to be positive excepting $s_{1}$ and $s_{2}.$ In more details, the parameter $c$ is known as the \textquotedblright antigenicity\textquotedblright\ which characterizes the strength of the tumor to generate effector immune cells; $% \mu _{2}$ is the death rate of immune cells; $p_{1}$ is the proliferation rate of immune cells; $g_{1}$ is the half saturation for proliferation term; $r_{2}$ is the cancer growth rate; $b$ is the logistic growth of cancer capacity; $a$ is the cancer clearance term; $g_{2}$ is the half saturation for cancer clearance; $p_{2}$ is the production rate of immune molecule; \ $% g_{3}$ is the half saturation of production; $\mu _{3}$ is the half-life of effector molecule. Parameters $s_{1}$ and $s_{2}$ are considered as %TCIMACRO{\U{b4}}% %BeginExpansion \'{}% %EndExpansion %TCIMACRO{\U{b4}}% %BeginExpansion \'{}% %EndExpansion controls% %TCIMACRO{\U{b4}}% %BeginExpansion \'{}% %EndExpansion %TCIMACRO{\U{b4}}% %BeginExpansion \'{}% %EndExpansion . They correspond to applying two kinds of medical treatment and it is supposed that they may be zeroes but not simultaneously. More precisely, $% s_{1}$ represent a treatment term where by a physician administers effector cells that have been taken from a patient, stimulated to a large degree, and then subsequently infused back into the patient. The term $s_{2\text{ }}$ is a treatment term that represents administration of the cytokine interleukin-2 (manufactured) by a physician to a patient, to stimulate again \ effector cell growth and proliferation, see [6,7]. \section{Upper and lower bounds for the localization polytope} In this section we give results on computations of upper and lower bounds for a polytope containing all compact invariant sets of the system (\ref% {Cancer Eq}) by using the localization method of compact invariant sets. \textbf{Lemma 1}. 1. \textit{All compact invariant sets in }$\mathbf{R}% _{+}^{3}$\textit{\ are located in the set }$K\left( h_{1}\right) =\left\{ 0\leq y\leq b^{-1}\right\} \cap \mathbf{R}_{+}^{3}.$ \textit{2. If }$s_{1}>0$ \textit{then} \textit{all compact invariant sets in }$\mathbf{R}_{+}^{3}$% \textit{\ are located in the set }$K_{1}\left( h_{2}\right) =\left\{ x\geq x_{\min }:=s_{1}\mu _{2}^{-1}\right\} \cap \mathbf{R}_{+}^{3}.$ 3. \textit{% If }$s_{2}>0$\textit{\ then all compact invariant sets in }$\mathbf{R}% _{+}^{3}$\textit{\ are located in }$K\left( h_{3}\right) =\left\{ z\geq \frac{s_{2}}{\mu _{3}}\right\} \cap \mathbf{R}_{+}^{3}.$ Further, we have \textbf{Proposition 6. }\textit{All compact invariant sets are located in the domain}% \[ K_{\ast }\left( h_{2}\right) =\{x\geq x_{\min \ast }:=\frac{% s_{1}(s_{2}+g_{1}\mu _{3})}{\mu _{2}g_{1}\mu _{3}+s_{2}(\mu _{2}-p_{1})}\} \]% \textit{where there are no additional conditions on }$s_{1};s_{2}$\textit{\ if }$p_{1}\leq \mu _{2},$\textit{\ and satisfy}% \[ s_{2}<\frac{g_{1}\mu _{2}\mu _{3}}{p_{1}-\mu _{2}},\text{ if }p_{1}>\mu _{2}. \] Next, we refine the localization set $K\left( h_{1}\right) $ as follows: \textbf{Proposition 7. }\textit{\ Let }$bg_{2}>1.$ \textit{If }% \begin{equation} 00;s_{2}\geq 0.$ \textbf{Theorem 4.} \ 1.\textit{\ Let\ }$g_{3}\geq g_{2}.$\textit{\ Then we get the localization set defined by} \begin{equation} K(h_{5})=\{z\leq z_{\max }:=\frac{s_{2}}{\mu _{3}}+\frac{(\mu _{3}+r_{2})^{2}p_{2}}{4abr_{2}\mu _{3}}\}. \label{zmax} \end{equation} 2.\textit{\ Let \ }$g_{3}\geq g_{2};$\textit{\ }$\mu _{2}>p_{1}$\textit{\ and (\ref{30}) holds. Then we get the localization set defined by} \[ K_{up}(h_{2})=\{x\leq x_{\max }:=\frac{1}{\mu _{2}-p_{1}}(cy_{\max }+s_{1}-% \frac{p_{1}g_{1}x_{\min }}{g_{1}+z_{\max }})\}. \] \textbf{Corollary 2. }\textit{Suppose that }$g_{3}\geq g_{2};bg_{2}\geq 1$% \textit{\ and 1) }$p_{1}=\mu _{2}$\textit{\ or 2) }$p_{1}>\mu _{2}$\textit{\ and }% \[ z_{\max }<\frac{\mu _{2}g_{1}}{p_{1}-\mu _{2}}. \] \textit{In addition, we suppose that parameters }$s_{1}>0$ and $s_{2}$% \textit{\ satisfy conditions of Proposition 6. Then all compact invariant sets are located in the domain} \begin{eqnarray*} K_{\ast up}(h_{2}) &=& \{x\leq x_{\max \ast }\}, \\ x_{\max \ast }:&=& \frac{(g_{1}+z_{\max})(cy_{\max }+s_{1})}{p_{1}g_{1}},\\ \text{ \textit{in case 1);}} \\ K_{\ast \ast up}(h_{2}) &=& \{x\leq x_{\max \ast \ast }\} , \\ x_{\max \ast \ast } :&=& \frac{(g_{1}+z_{\max})(cy_{\max }+s_{1})}{\mu _{2}(g_{1}+z_{\max })-p_{1}z_{\max }}\},\\ \text{\textit{in case 2).}} \end{eqnarray*} \section{The existence and bounds for the BPID} In this section we describe sufficient conditions under which the system has a positively invariant polytope which is contained in $\mathbf{R}_{+,0}^{3}$ : \textbf{Theorem 5. }\textit{\ Suppose that parameters of (\ref{Cancer Eq}) satisfy }% \begin{equation} r_{2}g_{2}>a. \label{23} \end{equation}% \textit{and }$s_{i}\geq 0,i=1,2$\textit{.} \textit{Then the polytope }% \[ K_{BPID}:=K_{\ast }(h_{1})\cap K_{1}(h_{2})\cap K_{0,up}(h_{2})\cap K(h_{3})\cap K(h_{5}), \]% \textit{\ with }$K_{0,up}(h_{2}):=K_{up}(h_{2})$\textit{\ or }$K_{\ast up}(h_{2})$\textit{\ or }$K_{\ast \ast up}(h_{2})$\textit{\ in corresponding cases, is a BPID. \ } \section{Conditions of the tumor free limit behavior} In this section we describe one assertion in which sufficient conditions of the tumor clearance limit dynamics are given. Its statement is expressed in terms of inequalities imposed on parameters of the system and on treatment terms under which $y(t)\longrightarrow 0,$ with $t\longrightarrow \infty .$ Let $s_{1};s_{2}>0.$ The tumor-free equilibrium point containing in $\mathbf{% R}_{+,0}^{3}\cap \{y=0\}$ is given by\textit{\ }% \[ O_{1}=(O_{11},O_{12},O_{13})=(\frac{s_{1}(g_{1}\mu _{3}+s_{2})}{\mu _{2}(g_{1}\mu _{3}+s_{2})-p_{1}s_{2}},0,\frac{s_{2}}{\mu _{3}}). \] Here% \begin{eqnarray} 0 &<&s_{2}<\frac{\mu _{2}\mu _{3}g_{1}}{p_{1}-\mu _{2}},\text{ with }% p_{1}>\mu _{2}, \label{38} \\ \text{or }s_{2} &>&0,\text{ with }p_{1}\leq \mu _{2}. \nonumber \end{eqnarray} The point $O_{1}$ taken from $\mathbf{R}_{+,0}^{3}\cap \{y=0\}$ is locally asymptotically stable if, in addition, we claim that% \begin{equation} s_{1}>\frac{r_{2}g_{2}[\mu _{2}(g_{1}\mu _{3}+s_{2})-p_{1}s_{2}]}{a(\mu _{3}g_{1}+s_{2})}. \label{39} \end{equation} Now we state \textbf{Theorem 6.} \textit{Suppose that } \textit{1) conditions (\ref{38})-(\ref{39}) are fulfilled and} \textit{2) inequalities } \textit{\ }% \begin{equation} \frac{\mu _{2}(bg_{2}+1)^{2}r_{2}}{ab}\geq s_{1}>\max \{\frac{\mu _{2}g_{2}^{2}r_{2}b}{a};\frac{\mu _{2}(bg_{2}+1)^{2}r_{2}}{4ab}\}, \label{49b} \end{equation} \textit{hold, or instead of (\ref{49b}),} \textit{3) inequality }% \begin{equation} \frac{\mu _{2}(bg_{2}+1)^{2}r_{2}}{ab}0;z>0\}$\textit{\ tend to the tumor-free equilibrium point }$O_{1}.$ \textbf{Corollary 3.} \textit{Assume that conditions of Theorems 5 and 6 hold. Then all trajectories in }$\mathbf{R}_{+,0}^{3}\cap \{x>0;z>0\}$% \textit{\ tend to }$O_{1}.$ \section{Concluding remarks} In this work we demonstrate for that the localization method of compact invariant sets [9] is helpful in studies of global aspects of population dynamics, especially for finding upper and lower bounds of ultimate densities of specific types of interacting populations. The biological significance of compact invariant sets is connected with the fact that they carry information about the long-time behavior of our model in future. With help of finding bounds for compact invariant sets we compute bounds for the bounded positively invariant domain. The goal of this computation is two-fold. From the first side, the bounded positively invariant domain consists of states whose evolution in future always satisfies bounds for the bounded positively invariant domain; this makes tumor growth dynamics predictable in the sense that the density of mutant cells has a more accurate and predictable estimate of changes in a tumor size. From another side, these bounds are used in statements and proofs of assertions relating to a location of $\omega $- limit sets and conditions of the attractivity to the tumor-free equilibrium point. Main results concerning mutant-cells-free limit behavior are contained in Theorem 3 and Corollary 1 for the Owen-Sherratt system and concerning tumor-free limit behavior are contained in Theorem 6 and Corollary 3 for the Kirchner-Panetta system. Finally, we indicate that mathematical models describing the cancer tumor growth contain a lot of parameters which cannot be measured with an acceptable accuracy. It takes place because these models are usually derived in the situation of the deficiency of clinical results/ experimental observations. Therefore our results on global dynamics of the cancer tumor growth models expressed in terms of simple algebraic conditions may be helpful for biological applications. In particular, these conditions give thresholds of the uncertainty of model parameters under which the given type of global dynamics preserves. \begin{thebibliography}{} %\bibitem[d`Onofrio, 2005]{dOnofrio} A. d`Onofrio (2005). A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences. {\em Physica D} , vol.208, pp.~ 202--235. % %\bibitem[dePillis and Radunskaya, 2001] {Pillis} L. dePillis and A. Radunskaya (2001). 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Kirschner and A.V. Tsygvintsev (2009). On the global dynamics of a model for tumor immunotherapy. {\em Math Biosci Eng},vol.6, pp.~579--583. % %\bibitem[Krishchenko and Starkov, 2006] {Krishko} A.P. Krishchenko and K.E. Starkov (2006). Localization of compact invariant sets of the Lorenz system. {\em Phys Lett A},vol.353, pp.~383--388. % %\bibitem[Kuznetsov,Makalkin,Taylor and Perelson, 1994] {kuz} V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor and A.S. Perelson (1994). Nonlinear dynamics of immunogenic tumors: Parameters estimation and global bifurcation analysis. {\em Bull Math Biol},vol.56, pp.~295--321. % %\bibitem[Lejeune,Chaplain and Akili, 2008] {lej} O. Lejeune, M.A.J. Chaplain and I. El. Akili (2008). Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumors. {\em Math Comput Model},vol.47, pp.~649--662. % %\bibitem[Nani and Freedman, 2000] {nani} F. Nani and H.I. Freedman (2000). A mathematical model of cancer treatment by immunotherapy. {\em Math Biosci},vol.163, pp.~159--199. % %\bibitem[Owen and Sherratt, 1998] {owen} M.R. Owen and J.A. Sherratt (1998). Modelling the macrophage invasion of tumors: Effects on growth and composition. {\em IMA J Math Appl Med Biol},vol.15, pp.~165--185. % %\bibitem[Starkov, 2011] {starkov} K.E. Starkov (2011). Compact invariant sets of the Bianchi VIII and Bianchi IX Hamiltonian systems. {\em Phys Lett A},vol.375, pp.~3184--3187. % %\bibitem[Starkov and Pogromsky, 2013] {starpo} K.E. Starkov and A. Pogromsky (2013). On the global dynamics of the Owen-Sherratt model describing the tumor- macrophage interactions. {\em Int J Bifurcat Chaos},vol.23, pp.~9. % %\bibitem[Starkov and Coria, 2013] {starco} K.E. Starkov and L. Coria (2013). Global dynamics of the Kirschner-Panetta model for the tumor immunotherapy. {\em Nonlinear Analysis: Real World Applications},vol.14, pp.~1425--1433. \bibitem[d`Onofrio, 2005]{dOnofrio} A. d`Onofrio (2005). A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences. {\em Physica D} , vol.208, pp.~ 202--235. \bibitem[Krishchenko and Starkov, 2006] {Krishko} A.P. Krishchenko and K.E. Starkov (2006). Localization of compact invariant sets of the Lorenz system. {\em Phys Lett A},vol.353, pp.~383--388. \bibitem[Kirschner and Panetta, 1998] {Kirsh} D. Kirschner and J. Panetta (1998). Modelling immunotherapy of the tumor- immune interaction,. {\em J Math Biol},vol.37, pp.~235--252. \bibitem[Kirschner and Tsygvintsev, 2009] {Kirshh} D. Kirschner and A.V. Tsygvintsev (2009). On the global dynamics of a model for tumor immunotherapy. {\em Math Biosci Eng},vol.6, pp.~579--583. \bibitem[Nani and Freedman, 2000] {nani} F. Nani and H.I. Freedman (2000). A mathematical model of cancer treatment by immunotherapy. {\em Math Biosci},vol.163, pp.~159--199. \bibitem[Starkov, 2011] {starkov} K.E. Starkov (2011). Compact invariant sets of the Bianchi VIII and Bianchi IX Hamiltonian systems. {\em Phys Lett A},vol.375, pp.~3184--3187. \bibitem[Starkov and Pogromsky, 2013] {starpo} K.E. Starkov and A. Pogromsky (2013). On the global dynamics of the Owen-Sherratt model describing the tumor- macrophage interactions. {\em Int J Bifurcat Chaos},vol.23, pp.~9. \bibitem[Starkov and Coria, 2013] {starco} K.E. Starkov and L. Coria (2013). Global dynamics of the Kirschner-Panetta model for the tumor immunotherapy. {\em Nonlinear Analysis: Real World Applications},vol.14, pp.~1425--1433. \bibitem[dePillis and Radunskaya, 2001] {Pillis} L. dePillis and A. Radunskaya (2001). A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. {\em Comput Math Methods Med 3}, pp.~78--100. \bibitem[dePillis and Radunskaya, 2003] {Pilliss} L. dePillis and A. Radunskaya (2003). The dynamics of the optimally controlled tumor model: a case study. {\em Math Comput Model},vol.37, pp.~78--100. \bibitem[Itik and Banks, 2010] {Itik} M. Itik and S. Banks (2010). Chaos in a three-dimensional cancer model. {\em Int J Bifurcat Chaos},vol.20, pp.~71--79. \bibitem[Owen and Sherratt, 1998] {owen} M.R. Owen and J.A. Sherratt (1998). Modelling the macrophage invasion of tumors: Effects on growth and composition. {\em IMA J Math Appl Med Biol},vol.15, pp.~165--185. \bibitem[Lejeune,Chaplain and Akili, 2008] {lej} O. Lejeune, M.A.J. Chaplain and I. El. Akili (2008). Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumors. {\em Math Comput Model},vol.47, pp.~649--662. \bibitem[Eftimie,Bramson and Earn, 2011] {Eftime} R. Eftimie, J.L. Bramson and D.J.D Earn (2011). Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. {\em Bull Math Biol},vol.73, pp.~2--32. \bibitem[Kuznetsov,Makalkin,Taylor and Perelson, 1994] {kuz} V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor and A.S. Perelson (1994). Nonlinear dynamics of immunogenic tumors: Parameters estimation and global bifurcation analysis. {\em Bull Math Biol},vol.56, pp.~295--321. \end{thebibliography} \end{document}