Research Article | Open Access

# Stability Results for a Class of Differential Equation and Application in Medicine

**Academic Editor:**Yong Zhou

#### Abstract

A *Chemostat* system incorporating hepatocellular carcinomas
is discussed. The model generalizes the classical *Chemostat* model, and it assumes that the *Chemostat* is an increasing function of the concentration. The
asymptotic behavior of solutions is determined. Sufficient conditions for the
local and global asymptotic stability of equilibrium and numerical simulation
are obtained, which is used to select the disease control tactics.

#### 1. Introduction

As we know, the stability of ecological systems and the persistence of species within them are fundamental concerns in ecology. Mathematical models of ecological systems, reflecting these concerns, have been used to investigate the stability of a variety of systems. For example, see [1–7]. The dynamic relationship between predator and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [4], and many good achievements have been reached [1–5, 8–10]. But few works about hepatocellular carcinomas cell model have been done.

According to some medical knowledge, under certain condition, the growth rate of tumor is in proportion to the volume on the time . Let denote this volume, such differential equations take the form where and denote the cell growth and death rate, respectively.

Some research on tumor cell shows that the growth rate is not invariable, but in reciprocal proportion to the factor . Therefore, it becomes

So the growth of the tumor disciplinarian can be denoted as follows, see [2],

Following the accumulation of experimental data, it became evident that the system (1.3) requires modification. Another research shows that the growth speed of the tumor cell is continuously decreased, which can be denoted by cube root function.

In fact, different kinds of tumor cell have different growth mode which behave exponential function, cube root function, or linear equation. It has been well established in experimental literature, so it becomes

The data presented by clinical experience shows that the liver tumor tissue is similar to entity ball. Because the volume of the liver on the time may be regarded as invariable, it is treated as model.

In this paper, we modify the modeling approach developed in some paper [2–7]. It extends the above model by assuming that is constant.

The symbol in paper [7] is used as follows: and denote tumor and normal cell consistency, respectively; and denote the death and growth rate of tumor cell, and a typical choice for and are , respectively; is the growth speed of normal cell, is the consume rate of normal cell. The system (1.3) becomes, see [6, 7] where realistic meaning is the same as the paper in [7, 11].

The study of the system (1.5) shows that it can be global and local asymptotic stability. Specifically, we demonstrate that stability change occurs only when the coefficient varies thus correcting the previously published results.

This paper has the following structure. The basic properties of its solutions and equilibrium are given in Section 2. The stability of the system (1.5) is discussed in Section 3. Applications of the theorem, numerical simulation, and control strategy are presented in Section 4, two examples of oscillatory coexistence are also presented.

#### 2. Equilibrium

In this section, we will study all possible equilibriums of the system (1.5).

For simplicity, the system (1.5) is rescaled with substitutions

, , are still replaced by , , , respectively, then the new system is written as where .

Substituting into the system (2.2), it takes the form

By straightforward computing:

two equilibriums of the system (2.3) are obtained, and , where

Obviously, according to the first equation of the system (2.3), if , then . Consequently, if , then . That is, the system (2.3) limit set of all positive solutions are in the point .

We assume that if the liver volume does not obviously change, we call it health equilibrium, otherwise we named it disease equilibrium. Therefore, is the former and is the latter.

A necessary condition for existence of positive equilibrium in the system (2.3) is that

Lemma 2.1. *The system (2.3) has positive equilibrium if and only if is hold.*

#### 3. Stability

People always expect that any disease can be cured no matter what stage it is, that is to say that this differential equation is asymptotically stable.

Following Section 2, we apply Lyapunov's stability theorem to analyse the two equilibriums of the system (1.5).

Theorem 3.1. *If , then the system (2.3) has only one equilibrium, and it is locally asymptotically stable.*

*Proof. *If , then does not exist, the system has only . Therefore, we obtain the *Jacobic* matrix

Because , is a stable crunode, and it is locally asymptotically stable. This completes the proof.

Theorem 3.2. *If and hold, the positive equilibrium of the system (2.3) is globally asymptotically stable.*

*Proof. *Substituting into the system (2.3), we get

Construct *Lyapunov* function:
therefore, the derived function of about the system (2.3) take the form

According to the formal conclusion: , , , we obtain

Therefore, if and hold, is globally asymptotically stable. This completes the proof.

Theorem 3.3. *If and hold, the positive equilibrium of the system (2.2) is globally asymptotically stable.*

#### 4. Numerical Simulation and Control Policy

In this section, we will select proper parameter and present numerical computer simulation to obtain control policy.

##### 4.1. When

Then . To illustrate the result of this subsection numerically, we fix the initialization points , and , , respectively and present computer simulations of the system (1.5), which based on the experiment and clinic date. The critical parameters are as follows: , where ; , where , respectively. Figures 1 and 2 show that it is asymptotically stable.

Therefore, the system has only one equilibrium in which the volume of liver does not obviously change. This probably is in the delitescence or no disease in which the liver has changed at functionality at most, but not organic. If proper measure is taken which will control the growth of the tumor, the development of tumor will be stagnancy.

We suppose that if the volume of liver does not obviously change, disease will not exacerbate, not to mention proper measure is adopted; therefore, hepatocellular carcinomas early diagnosis is very important which can control the development of the disease. Therefore we advocate early detection, early diagnosis, and early treatment.

##### 4.2. When

The system has one equilibrium on which the volume of liver has obviously changed, that is to say it is in serious condition.

Therefore, . Based on the experiment and clinic date, we fix the initialization points of the system (1.5): , , the critical parameters are as follows: , where . It is shown in Figures 3 and 4.

Otherwise, we fix the initialization points of the system (2.3): , the critical parameters are as follows: , where . Figure 5 shows that such system may exhibit a stable periodic solution.

We suppose that when the volume of liver has obviously changed, if proper measure such as chemotherapy or radiology is adopted, disease will not exacerbate at least, then the development of the tumor is in the phase of stagnation; therefore, hepatocellular carcinomas diagnosis is very important as it can control the development of disease. So we prefer medication treatment.

#### 5. Concluding Remarks

In this paper we have considered a *Chemostat* system incorporating hepatocellular carcinomas. We obtained a more realistic model by incorporating hepatocellular carcinomas in system (1.5). We have proved that local asymptotic stability of the positive equilibrium implies that it is global asymptotic stability. All the results indicated that medication treatment had a stabilizing effect on hepatocellular carcinomas development. We have given a numerical simulation to verify some of the key results we have obtained.

#### Acknowledgment

This work is supported by the Natural Science Foundation of Fujian Province (2008J0185, 2008J0202).

#### References

- L. S. Chen, X. Song, and Z. Lu,
*Mathematical Models and Methods in Ecology*, Sichuan Science and Techlogy, Chendu, China, 2003. - Z. Yixing, “Differential equation applied to research of quantitative analysis in biomedicine,”
*Journal of Southwest University for Nationalities*, vol. 37, no. 2, pp. 231–233, 2001. View at: Google Scholar - Z. Jiakun, Z. Lingli, and Z. Wei, “Epidemic model of hepatitis B without vaccination,”
*Journal of Xuzhou Normal University*, vol. 20, no. 4, pp. 7–11, 2002. View at: Google Scholar | Zentralblatt MATH - Q. Zhan, X. Xie, C. Wu, and S. Qiu, “Hopf bifurcation and uniqueness of limit cycle for a class of quartic system,”
*Applied Mathematics: A Journal of Chinese Universities*, vol. 22, no. 4, pp. 388–392, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Yang,
*Qualitative Theory of Dynamical Systems*, Cheung Shing, Hong Kong, 2004. - S.-B. Hsu and T.-W. Huang, “Global stability for a class of predator-prey systems,”
*SIAM Journal on Applied Mathematics*, vol. 55, no. 3, pp. 763–783, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Pang and L. Chen, “Global stability of chemostat models with ratio-dependent increase rate,”
*Journal of Guangxi Normal Universiy*, vol. 24, no. 1, pp. 37–40, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - X. Xie and F. Chen, “Bifurcation of limit cycles for a class of cubic systems with two imaginary invariant lines,”
*Acta Mathematica Scientia. Series A*, vol. 25, no. 4, pp. 538–545, 2005. View at: Google Scholar | MathSciNet - F. Chen, “The permanence and global attractivity of Lotka-Volterra competition system with feedback controls,”
*Nonlinear Analysis: Real World Applications*, vol. 7, no. 1, pp. 133–143, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. S. Pilyugin and P. Waltman, “Multiple limit cycles in the chemostat with variable yield,”
*Mathematical Biosciences*, vol. 182, no. 2, pp. 151–166, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Zhang, T. Ding, W. Huang, and Z. Dong,
*Qualitative Theory of Differential Equations*, Scientific, Beijing, China, 1985.

#### Copyright

Copyright © 2009 Qingyi Zhan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.