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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 [

According to some medical knowledge, under certain condition, the growth rate of tumor is in proportion to the volume on the time

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

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

Following the accumulation of experimental data, it became evident that the system (

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

In this paper, we modify the modeling approach developed in some paper [

The symbol in paper [

The study of the system (

This paper has the following structure. The basic properties of its solutions and equilibrium are given in Section

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

For simplicity, the system (

Substituting

By straightforward computing:

two equilibriums of the system (

Obviously, according to the first equation of the system (

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

A necessary condition for existence of positive equilibrium in the system (

The system (

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

If

If

Because

If

Substituting

Construct

According to the formal conclusion:

Therefore, if

If

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

Then

Solution curves for

Solution curves for

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.

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

Therefore,

Solution curves for

Solution curves for

Otherwise, we fix the initialization points of the system (

Solution curves for

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.

In this paper we have considered a

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