# Modeling a Tumor Growth with Piecewise Constant Arguments

Authors: F. Bozkurt
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This study is based on an early brain tumor growth that is modeled as a hybrid system such as (A):dx(t)/dt=x(t){r(1-αx(t)-β0x(⟦t⟧)-β1x(⟦t-1⟧))+γ1x(⟦t⟧)+γ2x(⟦t-1⟧)}, where the parameters α,β0,β1, and r denote positive numbers, γ1 and γ2 are negative numbers and ⟦t⟧ is the integer part of t∈[0,∞). Equation (A) explains a brain tumor growth, where γ1 is embedded to show the drug effect on the tumor and γ2 is a rate that causes a negative effect by the immune system on the tumor population. Using (A), we have constructed two models of a tumor growth: one is (A) and the other one is a population model at low density by incorporating an Allee function to (A) at time t. To consider the global behavior of (A), we investigate the discrete solutions of (A). Examination of the characterization of the stability shows that increase of the population growth rate decreases the local stability of the positive equilibrium point of (A). The simulations give a detailed description of the behavior of solutions of (A) with and without Allee effect.


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## Body

## 1. Introduction

Cancer biology has been revolutionized over the past several decades. Genetic alterations that lead to malignant phenotypes have been identified [1, 2], and mechanisms necessary to sustain a solid tumor [3] and that contribute to tumor-cell invasion [4, 5]. Mathematical modeling of both tumor growth and angiogenesis has been active areas of research. Such models can be classified into one of two categories: those that analyze the remodeling of the vasculature while ignoring changes in the tumor mass and those that predict tumor expansion in the presence of a nonevolving vasculature.The works in [6, 7] are very important, since they developed a two-dimensional hybrid cellular automaton model of brain tumor growth. Showing a simple model for a single species, the well-known model is constructed by May [8] and May and Oster [9] such as
(1)dx(t)dt=rx(t){1-x(⟦t⟧)K},
who obtained that the asymptotic behavior of difference solutions can be complex and “chaotic” for certain parameter values of r. In recent years, several studies have been conducted to investigate the difference solutions of specific logistic differential equations with respect to the parameters [10–19]. When γ1=γ2=0 in (A), [13] considered the logistic equation
(2)dx(t)dt=rx(t)(1-αx(t)-β0x(⟦t⟧)-β1x(⟦t-1⟧)),
where t≥0, the parameters α,β0,β1, and r denote positive numbers, and ⟦t⟧ denotes the integer part of t∈[0,∞). Here, the local asymptotic stability of the positive equilibrium point of (2) was proven by using the Linearized Stability Theorem and the global asymptotic stability by using a suitable Lyapunov function.An investigation of (A) can be shown in [14], where it was obtained that under the condition 3β1>α+β0>2α+β1 and γ2>γ1 the positive equilibrium point of (A) is locally asymptotically stable if and only if (γ2+γ1)/(α+β0+β1)<r<(γ2/β1). Furthermore, under specific conditions the global asymptotic stability, the semicycle, and oscillation results of the solutions of (A) were also studied.An important research for population models was conducted in 1931, where Allee [20] demonstrated that “Allee effect” occurs when population growth rate is reduced at low population size. The logistic model assumes that per-capita growth rate declines monotonically when the density increases; it is shown, however, that for population subject to an Allee effect, per-capita growth rate gives a humped curve increasing at low density, up to a maximum intermediate density and then declines again. Many theoretical and laboratory studies have demonstrated the importance of the Allee effect in dynamics of small populations; see, for example, [21–28]. From this reasoning, biological facts lead us to assume the Allee function as follows:(a)
ifN=0, then a(N)=0; that is, there is no reproduction without partners,(b)
a′(N)>0 for N∈(0,∞); that is, Allee effect decreases as density increases,(c)
limN→∞a(N)=1; that is, Allee effect vanishes at high density [29].In this paper, a single species population (here especially about an early brain tumor growth) is modeled such as(3)dx(t)dt=x(t){r(1-αx(t)-β0x(⟦t⟧)-β1x(⟦t-1⟧))x(t)+γ1x(⟦t⟧)+γ2x(⟦t-1⟧)(1-αx(t)-β0x(⟦t⟧)-β1x(⟦t-1⟧))},
where t≥0, the parameters α,β0,β1, and r denote positive numbers, γ1, γ2negative numbers, and ⟦t⟧ denotes the integer part of t∈[0,∞). The parameter r is the population growth rate of this tumor and α,β0, and β1 are rates for the delayed tumor volume and basis of a logistic population model. γ1 is embedded to show the drug effect on the tumor and γ2 is a rate that causes a negative effect by the immune system on the tumor population. In Section 2 we investigate the local and global behaviors of the nonlinear difference solutions of (3) basin under specific conditions. Additionally, a characterization of the stability when the population growth rate increases was also investigated. Section 3 gives results of the local and global asymptotic behaviors of the nonlinear difference solutions of (3) with Allee effect. The discrepancy of the stability behavior with and without Allee effect of (3) will give interesting results, which is discussed in Section 4.


## 2. Local and Global Asymptotic Stability Analysis

An integration of (A) on an interval of the formt∈[n,n+1) leads to
(4)x(t)=x(n)·e∫nt(r+(γ1-β0r)x(n)+(γ2-β1r)x(n-1)-αrx(s))ds.
In (4) if x(n)>0, then x(t)>0. Let t→n+1; it is clear that x(n+1)>0. This implies that we have positive solutions of (A) for positive initial conditions.In addition, on an interval of the formt∈[n,n+1) one can write (A) as
(5)dx(t)dt-{r+(γ1-β0r)x(n)+(γ2-β1r)x(n-1)}x(t)=-αrx2(t).
It is well known that (5) is a Bernoulli differential equation, and so for t→n+1 its solutions are(6)x(n+1)=x(n)·e{r+(γ1-β0r)x(n)+(γ2-β1r)x(n-1)}×(1+αrx(n){e{r+(γ1-β0r)x(n)+(γ2-β1r)x(n-1)}-1r+(γ1-β0r)x(n)+(γ2-β1r)x(n-1)})-1,n=0,1,2,…,where r+(γ1-β0r)x(n)+(γ2-β1r)x(n-1)≠0. The solution of (6) does not give any information about the global behavior of the differential equation. Hence, we can continue to investigate more about (6), since (6) is the difference equation of second order. Let γ1=-δ1 and γ2=-δ2, where δ1 and δ2 are positive numbers. It is important to take into account that the drug therapy γ1 has more destroying effect on the tumor than the immune system. That is why δ1>δ2. Considering (6) again, we obtain, for n=0,1,2,…,
(7)x(n+1)=x(n)(r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1))×((r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1))×exp(-{r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)})+αrx(n)(r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1)))-1,
where hereafter
(8)r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)≠0.
Computations reveal that the equilibrium points of (7) are
(9)x-1=0,x-2=r(α+β0+β1)r+δ1+δ2.
The fundamental study contains the stability analysis of the positive equilibrium point x-2. For this reason we show only the characteristic equation of (7) by linearizing (7) about x-2. Computation gives a quadratic equation such as
(10)μ2-{-(δ1+β0r)+((α+β0)r+δ1)·e-Aαr}×μ-{-(δ2+β1r)(1-e-A)αr}=0,
where A=αr2/((α+β0+β1)r+δ1+δ2).Theorem 1.
Letβ0>β1+α and β1>α. The positive equilibrium point of (7) is locally asymptotically stable if and only if
(11)A<ln((β0-β1+α)r+δ1-δ2(β0-β1-α)r+δ1-δ2).Proof.
By the Linearized Stability Theorem [30] we get that the positive equilibrium point of (7) is locally asymptotically stable if and only if
(12)|-(δ1+β0r)+((α+β0)r+δ1)·e-Aαr|<1+(δ2+β1r)(1-e-A)αr<2
holds. We can write (12) such as(a)
|(-(δ1+β0r)+((α+β0)r+δ1)·e-A)/αr|<1+(δ2+β1r)(1-e-A)/αr,(b)
1+(δ2+β1r)(1-e-A)/αr<2.
Sinceβ1>α, from (b) we have
(13)A<ln(δ2+β1rδ2+(β1-α)r).
From (a) we get(14)A<ln((β0-β1+α)r+δ1-δ2(β0-β1-α)r+δ1-δ2),
where β0>β1+α. Considering both (13) and (14), we will have
(15)A<ln((β0-β1+α)r+δ1-δ2(β0-β1-α)r+δ1-δ2)<ln(δ2+β1rδ2+(β1-α)r),
since (β0+α-β1)r+δ1-δ2>0. This completes our proof.Theorem 2.
Suppose thatr-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1)>0 for n=0,1,2,… and assume that the conditions in Theorem 1 hold.
If(16)r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)<ln(2x-2-x(n)x(n)),x(n)<2r(α+β0+β1)r-γ1-γ2,
then the positive equilibrium point of (7) is globally asymptotically stable.Proof.
We consider a Lyapunov functionV(n) defined by
(17)V(n)={x(n)-x-2}2,n=0,1,2,….
The change along the solutions of (17) is
(18)ΔV(n)=V(n+1)-V(n)={x(n+1)-x(n)}{x(n+1)+x(n)-2x-2}.
Considering (18), we get
(19)x(n+1)-x(n)=U1V,
where
(20)U1=(1-exp(-{r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)}))×x(n)·(r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1)),V=exp(-{r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)})·(r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1))+αrx(n).
Furthermore, from (18) we will have
(21)x(n+1)+x(n)-2x-2=U2V,
where(22)U2=αrx(n)(x(n)-2x-2)×(1-exp(-{r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)}))+(r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1))·(exp(-{r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)})·x(n)+x(n)-2x-2·exp(-{r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)}){r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)}).Since 0<r-(δ1+β0r-αr)x(n)-(δ2+β1r)x(n-1), if
(23)x(n)<x-2,r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)<ln(2x-2-x(n)x(n)),
then
(24)x(n+1)-x(n)>0,x(n+1)+x(n)-2x-2<0.
This implies that ΔV(n)<0, which gives the condition for the global asymptotic stability of the positive equilibrium point of (7).Theorem 3.
Letr1 and r2 be population growth rates of (7) such that r1<r2 and suppose that x-* and x-** are positive equilibrium points of (7) with respect to r1 and r2 that hold the conditions in Theorem 1, respectively. Furthermore, assume that
(25)ln(r2r1)<A2-A1,
where A1=αr12/((α+β0+β1)r1+δ1+δ2) and A2=αr22/((α+β0+β1)r2+δ1+δ2). If
(26)A1>ln((α+β0+β1)r1+δ1+δ2(β0+β1)r1+δ2+δ1),A2<ln((α+β0+β1)r2+δ1+δ2(β0+β1)r2+δ2+δ1),
then the local stability of x-** is weaker than x-*. That is, increase of the population growth rate decreases the local stability of the positive equilibrium point in (7).Proof.
Let us write from (7)(27)x(n+1)=x(n)(r1-(δ1+β0r1)x(n)-(δ2+β1r1)x(n-1))×((r1-(δ1+β0r1+αr1)x(n)-(δ2+β1r1)x(n-1))×exp(-{r1-(δ1+β0r1)x(n)-(δ2+β1r1)x(n-1)})+αr1x(n)(r1-(δ1+β0r1+αr1)x(n)-(δ2+β1r1)x(n-1)))-1,x(n+1)=x(n)(r2-(δ1+β0r2)x(n)-(δ2+β1r2)x(n-1))×((r2-(δ1+β0r2+αr2)x(n)-(δ2+β1r2)x(n-1))×exp(-{r2-(δ1+β0r2)x(n)-(δ2+β1r2)x(n-1)})+αr2x(n)(r2-(δ1+β0r2+αr2)x(n)-(δ2+β1r2)x(n-1)))-1,where r1<r2. In this case, the positive equilibrium points of (27) are
(28)x-*=r1(α+β0+β1)r1+δ1+δ2,x-**=r2(α+β0+β1)r2+δ1+δ2,
respectively. The characteristic equations of (27) are
(29)μ2-{-(δ1+β0r1)+((α+β0)r1+δ1)·e-A1αr1}×μ-{-(δ2+β1r1)(1-e-A1)αr1}=0,
where A1=αr12/((α+β0+β1)r1+δ1+δ2) and
(30)μ2-{-(δ1+β0r2)+((α+β0)r2+δ1)·e-A2αr2}×μ-{-(δ2+β1r2)(1-e-A2)αr2}=0,
where A2=αr22/((α+β0+β1)r2+δ1+δ2), respectively. Since r1<r2, we can write
(31)1αr2<1αr1,(32)δ2+β1r1<δ2+β1r2.
The inequality (32) can be also written as
(33)-(δ2+β1r2)<-(δ2+β1r1).
Since the inequality
(34)(α+β0+β1)r1r2(r2-r1)+(δ1+δ2)(r2-r1)(r2+r1)>0
always holds, we get
(35)αr22(α+β0+β1)r2+δ1+δ2>αr12(α+β0+β1)r1+δ1+δ2.
The inequality (35) can be also written as
(36)-αr22(α+β0+β1)r2+δ1+δ2<-αr12(α+β0+β1)r1+δ1+δ2
or
(37)-A2<-A1.
It is obvious that from (37), we get
(38)δ2e-A2<δ2e-A1.
Furthermore, if
(39)ln(r2r1)<A2-A1,
then the inequality
(40)β1r2e-A2<β1r1e-A1
holds. In view of (39), considering (31), (33), (38), and (40) together, we obtain
(41)-1<-(δ2+β1r2)(1-e-A2)αr2<-(δ2+β1r1)(1-e-A1)αr1.
From the Linearized Stability Theorem we want to obtain the conditions that satisfy the inequality
(42)-1<-(δ1+β0r2)+((α+β0)r2+δ1)·e-A2αr2--(δ2+β1r2)(1-e-A2)αr2<-(δ1+β0r1)+((α+β0)r1+δ1)·e-A1αr1--(δ2+β1r1)(1-e-A1)αr1.
Simplifications of (42) give us the inequality(43)((β1-β0)r2+δ2-δ1)+((α+β0-β1)r2+δ1-δ2)·e-A2αr2<((β1-β0)r1+δ2-δ1)+((α+β0-β1)r1+δ1-δ2)·e-A1αr1.Since β0>α+β1 and δ1>δ2, we get
(44)((α+β0-β1)r2+δ1-δ2)·e-A2<((α+β0-β1)r1+δ1-δ2)·e-A1,(45)(β1-β0)r2+δ2-δ1<(β1-β0)r1+δ2-δ1.
Considering both (44) and (45), we get (42).
Furthermore, from the Linearized Stability Theorem we must also show that the inequality(46)-(δ1+β0r1)+((α+β0)r1+δ1)·e-A1αr1-(δ2+β1r1)(1-e-A1)αr1<-(δ1+β0r2)+((α+β0)r2+δ1)·e-A2αr2-(δ2+β1r2)(1-e-A2)αr2<1
holds. Simplifying (46), we can write
(47)-((β0+β1)r1+δ2+δ1)+((α+β0+β1)r1+δ1+δ2)·e-A1(αr1)-1<-((β0+β1)r2+δ2+δ1)+((α+β0+β1)r2+δ1+δ2)·e-A2(αr2)-1.
If
(48)A1>ln((α+β0+β1)r1+δ1+δ2(β0+β1)r1+δ2+δ1),A2<ln((α+β0+β1)r2+δ1+δ2(β0+β1)r2+δ2+δ1),
then
(49)-((β0+β1)r1+δ2+δ1)+((α+β0+β1)r1+δ1+δ2)·e-A1<0,-((β0+β1)r2+δ2+δ1)+((α+β0+β1)r2+δ1+δ2)·e-A2>0.
The inequalities (49) imply that (46) holds. This result explains that increase of the population growth rate decreases the local stability of the positive equilibrium point in (7), which completes our proof.Theorem 4.
Let{x(n)}n=0∞ be a positive solution of (7). Assume that for n=0,1,… the condition
(50)0<r-(δ1+β0r+αrx(n))x(n)-(δ2+β1r)x(n-1)
holds. Then all positive solutions of (7) are in the interval
(51)x(n)∈(0,1α).Proof.
Let (50) hold. Then we can write
(52)r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)<r.
From (52), we will have
(53)e-(r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1))>e-r.
Considering (52) and (53) together, we get
(54)x(n+1)=x(n)(r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1))×((r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1))×exp(-{r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)})+αrx(n)(r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1)))-1<x(n)r×((r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1))×exp(-r)+αrx(n)(r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1)))-1.
Furthermore, since we have
(55)0<αrx(n)<r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)<r,
we obtain
(56)x(n+1)<x(n)r-αrx(n)exp(-r)+αrx(n)exp(-r)+αrx(n)=1α.
This completes the proof.


## 3. Local and Global Asymptotic Stability Analysis with Allee Effect

In this section we use an Allee function of timet. Let (3) be written as
(57)1xdxdt=r-(δ1+δ2+(α+β0+β1)r)x,
where γ1=-δ1 and γ2=-δ2. Applying to (57) an Allee function
(58)a(x)=xE+x,
where E is an Allee constant, we get
(59)1xdxdt=a(x){r-(δ1+δ2+(α+β0+β1)r)x}.
By defining
(60)g(x)=a(x){r-(δ1+δ2+(α+β0+β1)r)x}
and taking the derivative of g with respect to x, we obtain
(61)g′(x)=-(δ1+δ2+(α+β0+β1)r)x2-2E(δ1+δ2+(α+β0+β1)r)x+Er(E+x)-2.
By showing the sign of (61), we get that g is an increasing function for
(62)x∈(((α+β0+β1)r+δ1+δ2))-1/20,(E2(δ1+δ2+(α+β0+β1)r)2((α+β0+β1)r+δ1+δ2))-1/2+Er((α+β0+β1)r+δ1+δ2))-1/2-E((α+β0+β1)r+δ1+δ2)×((α+β0+β1)r+δ1+δ2)-1((α+β0+β1)r+δ1+δ2))-1/2)
and g is a decreasing function for
(63)x∈((((α+β0+β1)r+δ1+δ2))-1(E2(δ1+δ2+(α+β0+β1)r)2((α+β0+β1)r+δ1+δ2))-1/2+Er((α+β0+β1)r+δ1+δ2))-1/2-E((α+β0+β1)r+δ1+δ2)×((α+β0+β1)r+δ1+δ2)-1,∞).
This also means that if the density is
(64)x<(((α+β0+β1)r+δ1+δ2)E2(δ1+δ2+(α+β0+β1)r)2((α+β0+β1)r+δ1+δ2))-1/2+Er((α+β0+β1)r+δ1+δ2))-1/2-E((α+β0+β1)r+δ1+δ2)×((α+β0+β1)r+δ1+δ2)-1,
then a population model without an Allee function will not give realistic results. But if
(65)x>(((α+β0+β1)r+δ1+δ2)E2(δ1+δ2+(α+β0+β1)r)2((α+β0+β1)r+δ1+δ2))-1/2+Er((α+β0+β1)r+δ1+δ2))-1/2-E((α+β0+β1)r+δ1+δ2)×((α+β0+β1)r+δ1+δ2)-1,
then it is not important to use a model with an Allee function as it is also explained in the introduction. Applying to (3) an Allee function such as
(66)a(x(⟦t⟧))=x(⟦t⟧)E+x(⟦t⟧),
where E>0, we obtain
(67)dx(t)dt=x(t){r(1-αx(t)-β0x(⟦t⟧)-β1x(⟦t-1⟧))-δ1x(⟦t⟧)-δ2x(⟦t-1⟧)(1-αx(t)-β0x(⟦t⟧)-β1x(⟦t-1⟧))}x(⟦t⟧)E+x(⟦t⟧).
It is clear that (67) is a Bernoulli differential equation on the interval t∈[n,n+1). Solving (67) for t∈[n,n+1) and t→n+1, we get for n=0,1,2,…(68)x(n+1)=x(n)(r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1))×((r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1))×exp(-a(x(n)))×{r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)}+αrx(n)(r-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1)))-1.
It can be shown that the equilibrium points of (68) are also the equilibrium points of (7).Linearizing (68) about the positive equilibrium point, we obtain the characteristic equation as follows:
(69)μ2-{-(δ1+β0r)+((α+β0)r+δ1)·e-a(x-)Aαr}×μ-{-(δ2+β1r)(1-e-a(x-)A)αr}=0,
where A=αr2/((α+β0+β1)r+δ1+δ2).Theorem 5.
Letβ0>β1+α and β1>α. The positive equilibrium point of (68) is locally asymptotically stable if
(70)A<1a(x-)ln((β0-β1+α)r+δ1-δ2(β0-β1-α)r+δ1-δ2).Proof.
The proof is similar to that in Theorem1 and will be omitted.Theorem 6.
Suppose thatr-(δ1+β0r+αr)x(n)-(δ2+β1r)x(n-1)>0 for n=0,1,2,… and assume that the conditions in Theorem 5 hold.
If(71)r-(δ1+β0r)x(n)-(δ2+β1r)x(n-1)<ln(2x-2-x(n)x(n)),x(n)<2r(α+β0+β1)r-γ1-γ2,
then the positive equilibrium point of (68) is globally asymptotically stable.Proof.
The proof is similar to that in Theorem2 and will be omitted.Example 7.
The goal of this investigation is to examine the development of monoclonal tumors under the effects of treatment. In view of [6], the carrying capacity of a monoclonal tumor is ca. 38 mm. We select α′=0.00744, β1′=0.007448, and β0′=0.014896. By dividing these values by the carrying capacity, we obtain α=(0.00744/38)=0.000195789, β1=(0.007448/38)=0.000196and β0=(0.014896/38)=0.000392. These values are also suitable for the hypotheses in Theorems 1 and 5. To have a compatible result, a relation between the model and the data is constructed by multiplying these values with 10. Thus, the parameters for (7) are in this case α=0.00195789, β1=0.00196, and β0=0.00392. For a therapy of ca. 75 mg drug and under the assumption that the effect on the tumor is 1.6% we obtain δ1=0.0125. The effect on the immune is ca. 10% compared with the effect of the drug treatment, so we have δ2=0.00125. Figure 1 shows us the behavior of the solution of (7). Differently from this, we can see Figure 2, where we have used the Allee function for E=0.4. Studies demonstrated that Allee effects play an important role in the stability analysis of equilibrium points of a population dynamics model. Generally, an Allee effect has a stabilizing effect on population dynamics. So, in (68) our expectation is that the chaos begins later as it can be also shown in Figure 2.Figure 1
Behavior of the solutions of (7), where α=0.00195789, β1=0.00196, β0=0.00392, δ1=0.0125, and δ2=0.00125.Figure 2
Behavior of the solutions of (68), where α=0.00195789, β1=0.00196, β0=0.00392, δ1=0.0125, δ2=0.00125, and E=0.4.


## 4. Discussion

Section2 was constructed to obtain specific conditions for local and global asymptotic stability of the positive equilibrium point of (7) without Allee effect by applying the Linearized Stability Theorem and the theory of the use of a Lyapunov function, respectively. Furthermore, we showed that increase of the population growth rate decreases the stability of the positive equilibrium point of (7), which is given in Theorem 3. Finally, in Section 2 we proved Theorem 4, which has given information about the bound of the solutions of (7). By using the data of [6] about the radius of tumor at 111 days, we take the radius as 4.96 mm for a population growth rate r=0.31. By multiplying it with 10, we use the growth rate r=3.1. The volume of such a tumor will be then (n)=510.87 mm3. Considering Theorem 4, we can see that the above mentioned value for α is suitable. For α=0.00195789, β1=0.00196, β0=0.00392, δ1=0.0125, and δ2=0.00125 we consider Section 3 in view of E=0.4,
(72)x∈((((α+β0+β1)r+δ1+δ2))-10,(((α+β0+β1)r+δ1+δ2)E2(δ1+δ2+(α+β0+β1)r)2((α+β0+β1)r+δ1+δ2))-1/2+Er((α+β0+β1)r+δ1+δ2))-1/2-E((α+β0+β1)r+δ1+δ2)×((α+β0+β1)r+δ1+δ2))-1/2((α+β0+β1)r+δ1+δ2)-1)=(0,5.322).
In this case, for the volume x(n)=5 mm3 the radius of the tumor must be around 1.0609 mm, which the temporal development of a cross-central section of a tumor growing without angiogenesis show a tumor more than at day 80 and the temporal development of a cross-central section of a tumor growing with angiogenesis around 40 days of the tumor (see [6]). This means that during the 90 days of the tumor we shall use the model given in (68). For days more than 90, both models are suitable ((7) and (68)). Using the above mentioned values, it can be shown that the local stability of both theorems (Theorems 1 and 5) hold. However, Theorem 5 give us that the stable interval for the growth rate is wide as assumed in Theorem 1, which is important for the drug therapy.



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*Source: 841764-2013-05-14.xml*