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We establish the interaction model of two cell populations following the concept of the random-walk, and assume the cell movement is constrained by space limitation primarily. Furthermore, we analyze the model to obtain the behavior of two cell populations as time is closed to initial state and far into the future.

In the 1980s, the movement of isolated single cells was researched and was modelled by a range of authors (Oster [

A consequential early paper written by Keller and Segel [

In the recent years, most of the researches on cell movement focused on the interaction of multiple cell populations, precise cell behavior, and the development of the mathematics modelling. In this study we follow the contour of two-cell interaction developed by Painter and Sherratt [

To model the motion of biological organisms, there are three major concepts which would be used:

the space-jump process in which the individual jumps between sites on a lattice,

the velocity-jump process in which discontinuous changes in the speed or direction of an individual are generated by a Poisson process,

the flux motion in which the movement of cells are treated as the flux motion.

In this work we adopt space-jump concept to establish our model and from it we show how a PDE of cell movement could be deduced. Then we use the same concept and expand the PDE which has been deduced to reason a system of PDEs describing the interaction of two cell population.

We will deduce an equation of cell movement on a lattice from the space-jump concept; moreover, we translate that equation into a PDE of cell movement through changing variables. First, we list the functions and variables that will be used in this content and call the considering cell population by

Moreover, the meaning of

For example, we choose that the cell density on position

Supposing that cells move continuously in time on a lattice (discrete space), a PDE of

In the lattice space, the

We explain our idea as shown in Figure

The movement of cells.

Figure

The model of

For a continuum flow we consider that the jumping coefficient

In consequence,

Consequently,

The

Now we show how to deduce a system of PDEs which describes the interaction of two cell populations. Here the two considered cell populations are called by

Given that space limitation influences the movement of cells, the probability of cells moving to position

After defining those variables, the model of interaction of two cell populations (

Following space limitation, the interaction of two cell populations can be modelled as

Furthermore, through changing variables,

Now, the interaction of

We have got the system of PDEs (

Our purpose is to obtain a simpler form of (

Given two cell populations with the same diffusion coefficient, the system of PDEs (

According to the system of PDEs (

Let

The system of PDEs (

In that case, the simpler form (model (

Before deducing that

The movement of total cells (

Adding the two equations in the system (

Imposing

In consequence,

According to above assumptions,

After describing the behavior of total cells, following (

Hence,

Given that

Equation (

Assuming that

Therefore,

Hence,

In order to simplify the representation of the following equations, we let

Before we make the following theorem complete, the substantiation of the next lemma must be finished.

The solution of

Assume

We say that

Let

The equation

Let

By Granwall's inequality and

Hence, the solution of

Supposing that

According to the above assumptions, we have

For

Thus the equation

Because the solution of

Hence,

It is verified that

Near

The solution of

Supposing

Consequently,

Now let

After substantiating that

Given

In consequence,

Restoring

If it is possible, we hope the solutions of (

The authors declare that there is no conflict of interests regarding the publication of this paper.

Thanks are due to Professor Long-Yi Tsai, Professor Tai-Ping Liu, Professor Ton Yang, and Professor Shih-Shien Yu for their continuous encouragement and discussions over this work, to Metta Education, Grand Hall, and Auria Solarfor for their financial assistance, and to the referee for his interest and helpful comments on this paper.