The Flux Model of the Movement of Tumor Cells and Healthy Cells Using a System of Nonlinear Heat Equations

Abstract

The interaction model is deduced from the concept of the flux motion of heat transfer, and the movement is constrained by space limitation. Furthermore, we analyze the model to obtain the behavior of two-cell populations at time close to initial state and far into the future.

1. Introduction

In mathematics and biomedicine of one-cell population as well as multiple cell populations, there is much research on movement. A consequential early article written by Keller and Segel (1970) modeled a partial differential equation to study the biochemical regulation of bacteria movement; their research has been the basis for models of the movement of diversified cell populations, such as slime mold aggregation (Höfor et al., 1995), tumor angiogenesis (Chaplain and Stuart, 1993), primitive streak formation (Painter et al., 2000), and wound repair (Pettet et al., 1996). Later in the 1980s, the movement of isolated single cells was researched and was modeled by a range of authors (Oster, 1984; Oster and Perelson, 1985; Bottino and Fauci, 1998; Bottino et al., 2002). In recent years, most research on cell movement focused on the interaction of multiple cell populations, precise cell behavior, and the development of mathematics modeling.

We deduce the model from the concept of the flux motion by imitating heat transfer equation. Assuming cell movement is like material flow and the movement is constrained by space limitation, we establish a system of PDEs to describe the interaction of two-cell populations. Simplifying that system to a model of odes, we analyze it and find that two-cell populations will tend to a steady state far into the future.

1.1. Modeling of two-cell populations interaction following the flux motion

To model the motion of biological organisms, there are three major concepts used. We adopt the concept of flux motion imitating heat conduction to establish our model.

This section will show how a PDE of cell movement can be deduced from the concept of flux motion. Then we use the same concept and expand the PDE that has been deduced to establish a system of PDEs describing the interaction of tumor cells and healthy cells.

1.2. Movement of one-cell population under space limitation

We deduce a PDE of cell movement from the concept of flux motion. Before we establish the model of cell movement, we must define and list the functions and variables that will be used. Those functions we need are shown as follows, and we call the cell population the “u-cell.”

u(x, y, z, t) Number of u-cell per unit volume per unit amount of time (density of u-cell per unit volume in unit amount of time)

J(x, y, z, t) Flux of u-cell at (x, y, z) at time t.(number of u-cell crossing unit area at (x, y, z) in the out-word direction per unit time at time t)

Ω The cuboid with the eight points (x, y, z), (x + Δx, y, z), (x, y + Δy, z), (x, y, z + Δz), (x + Δx, y + Δy, z), (x, y + Δy, z + Δz), (x + Δx, y, z + Δz) and (x + Δx, y + Δy, z + Δz)

ΔV = ΔxΔyΔz The volume element of length Δx,Δy, and Δz on x,y,z-axis

Fixing attention on the ambit Ω, we could describe changes in the density by accounting for the flow of u-cell into and out of the ambit Ω. The balance equation could be written in terms of the number of u-cell and the statement would be \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} \left( \begin{matrix} \hbox {Rate of change of u - } \hfill \\ \hbox{cell in the ambit} \ \Omega \hfill \\ \hbox{per unit time} \hfill \\\end{matrix} \right) = \left( \begin{matrix} \hbox{Rate of entry into} \ \Omega \hfill \\ \hbox{per unit time} \hfill \\\end{matrix} \right) - \left( \begin{matrix} \hbox{Rate of departure} \hfill \\ \hbox{from} \ \Omega \ \hbox{per unit} \hfill \\ \hbox{time.} \hfill \\\end{matrix} \right) \end{align*} \end{document}

Described by functions, the above statement leads to the following equation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} \int_ {\Omega } \frac { \partial u } { \partial t } dV &= ( J ( x , y , z , t ) - J ( x + \Delta x , y , z , t ) ) \Delta y \Delta z \\ & + ( J ( x , y , z , t ) - J ( x , y + \Delta y , z , t ) ) \Delta z \Delta x \\ & + ( J ( x , y , z , t ) - J ( x , y , z + \Delta z , t ) \Delta x \Delta y. \tag { 1 } \end{align*} \end{document}

The above statement and equation is illustrated in Figure 1.

FIG. 1.

Rate of change of u-cell in the ambit Ω per unit time.

Moreover, taking the limit as Δx → 0, Δy → 0, and Δz → 0, the above Equation (1) would be translated into \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} \frac { \partial u } { \partial t } = - \left( \frac { \partial J } { \partial x } + \frac { \partial J } { \partial y } + \frac { \partial J } { \partial z } \right) . \tag { 2 } \end{align*} \end{document}

The minus sign on \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \big ( \frac { \partial J } { \partial x } + \frac { \partial J } { \partial y } + \frac { \partial J } { \partial z } \big ) $$\end{document} stems from the fact that the finite difference in equation (2) has a sign opposite to that in the direction of a derivative. Now, we obtain PDE (2), that states the movement of u-cell according to the concept of the flux motion. For convenience, we consider first the one-dimensional case, and the chief equation we adopt is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} \frac { \partial } { \partial t } u ( x , t ) = \frac { \partial } { \partial x } J ( x , t ) . \tag { 3 } \end{align*} \end{document}

Equation (3) is a classic statement of diffusion. According to Fick (1855), the first law of diffusion is that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} material \ flux = D \quad density \ gradient \end{align*} \end{document}

or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} J ( x , t ) = D \frac { \partial d ( x , t ) } { \partial x } , \end{align*} \end{document}

where D is the diffusion coefficient and the density d(x, t) varies from point to point and depends on the position x and time t. The minus sign implies that the particle flow proceeds from high-density region to low-density region. In our work, we consider the situation that the density d(x, t) not only depends on the position x and time t, but also on space limitation.

The mechanism—space limitation—means that a higher density of u-cell at site x will decrease the probability of other u-cell moving to site x. Therefore, we define I(x, t) = 1 \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$I (x , t) = 1 \frac {u {( x , t ) }} { T} $$\end{document} for some T u(x, t) as the probability of u-cell moving to site x. Furthermore, the density d(x, t) could be written as I(u(x, t)) u(x, t), and the flux J(x, t) is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac { \partial } { \partial x } \big ( I ( u ( x , t ) ) \ u ( x , t ) \big ) $$\end{document} . In consequence, the model of the movement of one-cell population is established as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} \frac { \partial u } { \partial t } = \frac { \partial^2u } { \partial x^2 } \quad 1 \quad \frac { u ( x , t ) } { T } \quad u ( x , t ) , \tag { 4 }\end{align*} \end{document}

for some constant T and T u(x, t) initially. The movement of two-cell populations is modeled in the next section.

2. Interaction Of Two-Cell Populations Under Space Limitation

In the preceding section, we established the model that describes the movement of one-cell population. Now we show how to deduce a system of PDEs that characterizes the interaction of two-cell populations.

We define some words and functions first:

u-cell One of the considered cell populations (tumor cells),

v-cell The other of the considered cell populations (health cells),

u(x, t)/v(x, t) The density of u-cell/v-cell,

w(x, t) The density of u-cell and v-cell (w(x, t) = u(x, t) + v(x, t)),

I(x, t) The item which reflects how space limitation influences cell movement.

The function I(x, t) is still vague, and needs to be described and defined clearly.

Given that space limitation influences the movement of cells, the probability of cells moving to position x decreases with how position x is crowded with cells. We choose w(x, t), the total cell density, to express the information of cells on position x (the distribution of cells is sparse or dense on position x at time t.) Hence, we let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$I ( x , t ) = 1 - \frac { w ( x , t ) } { T } $$\end{document} , which shows that the probability of cells moving to position x decreases with the total cell density on position x where T ≥ w(x, t) initially and T is a constant. In our work, the assumption on I(x, t) follows the article written by Painter and Sherratt (2003).

After defining those variables and functions, the model of information of two-cell populations (u-cell and v-cell) can be deduced.

Theorem 2

Following space limitation, the interaction of two-cell populations can be modeled as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \begin{cases} \frac { \partial u } { \partial t } = ( 1 - \frac { 2u } { T } - \frac { \nu } { T } ) \frac { \partial^2 u } { \partial x^2 } - \frac { 2 } { T } \frac { \partial u } { \partial x } \left( \frac { \partial u } { \partial x } + \frac { \partial \nu } { \partial x } \right) - \frac { u } { T } \frac { \partial^2 \nu } { \partial x^2 } , \\ \frac { \partial \nu } { \partial t } = ( 1 - \frac { 2 \nu } { T } - \frac { u } { T } ) \frac { \partial^2 \nu } { \partial x^2 } - \frac { 2 } { T } \frac { \partial \nu } { \partial x } \left( \frac { \partial u } { \partial x } + \frac { \partial \nu } { \partial x } \right) - \frac { \nu } { T } \frac { \partial^2 u } { \partial x^2 } , \end{cases} \tag { 5 } \end{align} \end{document}

where T is a constant and T ≥ u(x, t) + v(x, t) initially.

Proof

According to equation (4), replace \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$1 - \frac { u ( x , t ) } { T } $$\end{document} by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$1 - \frac { w ( x , t ) } { T } $$\end{document} . Then the equation for u-cell movement is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { \partial u } { \partial t } & = \frac { \partial } { \partial x } \left( \frac { \partial } { \partial x } \left( u - \frac { u^2 } { T } - \frac { u \nu } { T } \right) \right) \\ & = \frac { \partial } { \partial x } \left( \frac { \partial u } { \partial x } - \frac { 2 u } { T } \frac { \partial u } { \partial x } - \frac { u } { T } \frac { \partial \nu } { \partial x } - \frac { \nu } { T } \frac { \partial u } { \partial x } \right) \\ & = \frac { \partial^2 u } { \partial x^2 } - \frac { 2 } { T } \left( \frac { \partial u } { \partial x } \right) ^2 - \frac { 2u } { T } \frac { \partial^2 } { \partial x^2 } - \frac { 2 } { T } \frac { \partial u } { \partial x } \frac { \partial \nu } { \partial x } - \frac { u } { T } \frac { \partial^2 \nu } { \partial x^2 } - \frac { \nu } { T } \frac { \partial^2 u } { \partial x^2 } \\ & = \left( 1 - \frac { 2 u } { T } - \frac { \nu } { T } \right) \frac { \partial^2 u } { \partial x^2 } - \frac { 2 } { T } \frac { \partial u } { \partial x } \left( \frac { \partial u } { \partial x } + \frac { \partial \nu } { \partial x } \right) - \frac { u } { T } \frac { \partial^2 \nu } { \partial x^2 } , \end{align} \end{document}

where T is a constant. Similarly, the same processes are applied to v. We obtain the following equation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { \partial \nu } { \partial t } = \left( 1 - \frac { 2 \nu } { T } - \frac { u } { T } \right) \frac { \partial^2 \nu } { \partial x^2 } - \frac { 2 } { T } \frac { \partial \nu } { \partial x } \left( \frac { \partial u } { \partial x } + \frac { \partial \nu } { \partial x } \right) - \frac { \nu } { T } \frac { \partial^2 u } { \partial x^2 } . \end{align} \end{document}

Consequently, system (5) follows. ▪

Furthermore, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$U \equiv U ( x , t ) = \frac { u ( x , t ) } { T } $$\end{document} , with the consequence that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { \partial u } { \partial t } = T \frac { \partial U } { \partial t } , \frac { \partial u } { \partial x } = T \frac { \partial U } { \partial x } , \frac { \partial^2 u } { \partial x } = T \frac { \partial^2 U } { \partial x } . \end{align} \end{document}

Similarly, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$V \equiv V ( x , t ) = \frac { v ( x , t ) } { T } $$\end{document} ; then, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac { \partial v } { \partial t } , \frac { \partial v } { \partial x }$$\end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$T \frac { \partial^2 v } { \partial x^2 } $$\end{document} can be changed into \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$T \frac { \partial V } { \partial t } , \frac { \partial V } { \partial x }$$\end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$T \frac { \partial^2 V } { \partial x^2 } $$\end{document} , respectively. Rewrite system (5) as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \begin{cases} T \frac { \partial U } { \partial t } = ( 1 - 2U - V ) \left( T \frac { \partial^2 U } { \partial x^2 } \right) - \frac { 2 } { T } \left( T \frac { \partial U } { \partial x } \right) \left( T \frac { \partial U } { \partial x } + T \frac { \partial V } { \partial x } \right) - U \left( T \frac { \partial^2 V } { \partial x^2 } \right) , \\ T \frac { \partial V } { \partial t } = ( 1 - 2V - U ) \left( T \frac { \partial^2 V } { \partial x^2 } \right) - \frac { 2 } { T } \left( T \frac { \partial V } { \partial x } \right) \left( T \frac { \partial U } { \partial x } + T \frac { \partial V } { \partial x } \right) - V \left( T \frac { \partial^2 U } { \partial x^2 } \right) . \end{cases} \end{align} \end{document}

Then the constant T in system (5) could be eliminated, and the system of PDEs (5) can be simplified as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \begin{cases} \frac { \partial U } { \partial t } = ( 1 - 2U - V ) \frac { \partial^2 U } { \partial x^2 } - 2 \frac { \partial U } { \partial x } \left( \frac { \partial U } { \partial x } + \frac { \partial V } { \partial x } \right) - U \frac { \partial^2 V } { \partial x^2 } , \\ \frac { \partial V } { \partial t } = ( 1 - 2V - U ) \frac { \partial^2 V } { \partial x^2 } - 2 \frac { \partial V } { \partial x } \left( \frac { \partial U } { \partial x } + \frac { \partial V } { \partial x } \right) - V \frac { \partial^2 U } { \partial x^2 } . \end{cases} \tag { 6 } \end{align} \end{document}

At present, the interaction of u-cell and v-cell has been modeled as a system of PDEs (5), and that model can be simplified (we conceal the constant T) as system (6). Model (6) will be used frequently in the following context, and some properties of two-cell populations can be deduced from analyzing the solution of equation (6). We will show the analyzing procedures and some results in the next section. Moreover, we must remember \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$U ( x , t ) = \frac { u ( x , t ) } { T } $$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$V ( x , t ) = \frac { v ( x , t ) } { T } $$\end{document} , where T is a constant and T ≥ u(x, t) + v(x, t) initially.

3. Analysis Of The Model Of Two-Cell Populations Interaction

So far, we have the system of PDEs (6), which shows the interaction of two-cell populations. In this section, equation (6) will be transformed to a system of odes and then analyzed to obtain some properties of U (x, t) = U(z) as z approaches to zero (time far into the future). In that case, the properties of V(x, t) = V(z) can be deduced from the properties of U(z). We must pay attention to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$U ( x , t ) = \frac { u ( x , t ) } { T } $$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$V ( x , t ) = \frac { v ( x , t ) } { T } $$\end{document} , where u(x, t) and v(x, t) are the density of u-cell and v-cell, respectively. The constant T is not more than the density of total cells initially.

To analyze the solution of the model, we simplify equation (6) into a system of odes Through changing variables, the system of PDEs (6) can be transformed to a system of odes.

Lemma 3.1

The system of PDEs (6) can be viewed as a system of odes as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \begin{cases} \frac { { - z } { 2 }} U^ { \prime } ( z ) = ( 1 - 2U ( z ) - V ( z ) ) U^ { \prime \prime } ( z ) - 2U^ { \prime } ( z ) ( U^ { \prime } ( z ) + V^ { \prime } ( z ) ) - U ( z ) V^ { \prime \prime } ( z ) , \\ \frac { - z } { 2 } V^ { \prime } ( z ) = ( 1 - 2V ( z ) - U ( z ) ) V^ { \prime \prime } ( z ) - 2V^ { \prime } ( z ) ( U^ { \prime } ( z ) + V^ { \prime } ( z ) ) - V ( z ) U^ { \prime \prime } ( z ) , \end{cases} \tag { 7 } \end{align} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$z = \frac { x } { \sqrt { t } } $$\end{document} .

Proof

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$U ( z ) = U ( \frac { x } { \sqrt { t } } ) \equiv U ( x , t ) $$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$V ( z ) = V ( \frac { x } { \sqrt { t } } ) \equiv V ( x , t ) $$\end{document} , then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { \partial U ( x , t ) } { \partial t } \equiv \frac { - 1 } { 2 } xt^ \frac { - 3 } { 2 } U^ { \prime } \left( \frac { x } { \sqrt { t } } \right) = \frac { - 1 } { 2 } zt^ { - 1 } U^ { \prime } ( z ) , \ \frac { \partial V ( x , t ) } { \partial t } \equiv \frac { - 1 } { 2 } xt^ \frac { - 3 } { 2 } V^ { \prime } \left( \frac { x } { \sqrt { t } } \right) = \frac { - 1 } { 2 } zt^ { - 1 } V^ { \prime } ( z )\end{align} \end{document}

and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { \partial^2 U ( x , t ) } { \partial x^2 } \equiv t^ { - 1 } U^ { \prime \prime } \left( \frac { x } { \sqrt { t } } \right) = t^ { - 1 } U^ { \prime \prime } ( z ) , \ \frac { \partial^2 V ( x , t ) } { \partial x^2 } \equiv t^ { - 1 } V^ { \prime \prime } \left( \frac { x } { \sqrt { t } } \right) = t^ { - 1 } V^ { \prime \prime } ( z ) . \end{align} \end{document}

Thus, the above functions of z are substituted for the functions of x and t in model (6), and it can be written as equation (7) immediately. ▪

In that case, the simpler form (model (7)) will be analyzed in the following subsections in order to obtain some properties of U(z) and V(z).

3.1. Properties of total cells as into the future

In this subsection, we investigate the behavior of total cells (u-cell and v-cell).

Lemma 3.2

The behavior of total cells (u-cell and v-cell) can be modeled as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { - z } { 2 } W^ { \prime } ( z ) = W^ { \prime \prime } ( z ) - ( W^2 ( z ) ) ^ { \prime \prime } . \tag { 8 } \end{align} \end{document}

Proof

First, model (7) can be arranged as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \begin{cases} \frac { { - z } { 2 }} U^ { \prime } ( z ) = U^ { \prime \prime } ( z ) - ( U^2 ( z ) + U ( z ) V ( Z ) ) ^ { \prime \prime } , \\ \frac { - z } { 2 } V^ { \prime } ( z ) = V^ { \prime \prime } ( z ) - ( V^2 ( z ) + U ( z ) V ( Z ) ) ^ { \prime \prime } . \end{cases} \tag { 9 } \end{align} \end{document}

We combine two equations of system (9) as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { - z } { 2 } ( U ( z ) + V ( z ) ) ^ { \prime } = ( U ( z ) + V ( z ) ) ^ { \prime \prime } - \left( ( U ( z ) + V ( z ) ) ^2 \right) ^ { \prime \prime } . \end{align} \end{document}

Let W(z) be U(z) + V(z). Thus, W(z) is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac { u ( x , t ) + v ( x , t ) } { T } $$\end{document} , where z is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$${x } \over { \sqrt { t }} $$\end{document} . Then we obtain model (8). ▪

Following equation (8), we spread out the item (W²(z)″; then, equation (8) is shown as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} ( 2W ( z ) - 1 ) W^ { \prime \prime } ( z ) + 2 ( W^ { \prime } ( z ) ) ^2 = \frac { z } { 2 } W^ { \prime } ( z ) . \tag { 10 } \end{align} \end{document}

Assuming W′(2W − 1) is not equal with zero, the above equation can be rewritten as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { W^ { \prime \prime } ( z ) } { W^ { \prime } ( z ) } + \frac { 2W^ { \prime } ( z ) } { 2W ( z ) - 1 } = \frac { z } { 2 } \cdot \frac { 1 } { 2W ( z ) - 1 } , \tag { 11 } \end{align} \end{document}

dividing both sides of equation (10) by W′(2W − 1). For calculating conveniently, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$${ \overline W} ( z ) $$\end{document} be \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) - \frac { 1 } { 2 } $$\end{document} , and replace the item W(z) in equation (11) by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ {\overline{W }} ( z ) + \frac { 1 } { 2 } $$\end{document} . Therefore, the equation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac {{\overline W } ^ { \prime \prime } ( z ) } {{\overline W } ^ {\prime} ( z )} + \frac {{\overline W } ^ { \prime } ( z ) } {{\overline W } ( z ) } = \frac {z} {4W ( z )} \tag {12} \end{align} \end{document}

is obtained trivially.

In order to obtain a homogeneous solution of equation (12), we consider this equation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac {{\overline W } ^ { \prime \prime } ( z ) } { {\overline W } ^ { \prime } ( z ) } + \frac {{\overline W } ^ { \prime } ( z ) } {{ \overline W } ( z ) } = 0. \tag { 13 } \end{align} \end{document}

By integrating the above equation with respect to z twice, a homogeneous solution of equation (12) is obtained as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$${ \overline W} ( z ) = { \sqrt 2}{ \sqrt k_1z + k_2}$$\end{document} , where k₁ and k₂ are constants, with the consequence that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) = \frac { 1 } { 2 } + { \sqrt 2 } { \sqrt k_1z + k_2 } $$\end{document} is a homogeneous solution of equation (11), where k₁ and k₂ are constants. We assume \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) = \frac { 1 } { 2 } + { \sqrt 2 } { \sqrt c_1z + c_2 } $$\end{document} could be a solution of equation (11), where c₁ and c₂ are functions of z. Applying \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) = \frac { 1 } { 2 } + { \sqrt 2 } { \sqrt c_1z + c_2 } $$\end{document} to equation (11), we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} ( { \sqrt c_1z + c_2 } ) ^ \frac { 1 } { 2 } ( { \sqrt c_1z + c_2 } ) ^ { \prime \prime } = \frac { z } { 4 { \sqrt 2 } } ( { \sqrt c_1z + c_2 } ) ^ { \prime } , \tag { 14 } \end{align} \end{document}

where c₁ and c₂ are functions of z.

Through analyzing equation (14) for some very small z, we can know the properties of the item c₁z + c₂ in equation (14) as z approaches zero. Furthermore, the properties of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) ( = \frac { 1 } { 2 } + { \sqrt 2 } \sqrt { c_1z + c_2 } ) $$\end{document} can be gotten.

Now, according to the types of the functions (c₁ and c₂), we have three cases as follows:
(a) Given that c₁ is a constant function and c₂ not, then following equation (14), \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align}( c_1z + c_2 ) ^ \frac { 1 } { 2 } c^ { \prime \prime } _2 = \frac { z } { 4 { \sqrt 2 } } ( c_1 + c_2^ { \prime } ) \sim 0 , \end{align} \end{document}

as z ∼ 0. Moreover, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_2 ( c_2^{ \prime \prime} ) ^2$$\end{document} approaches zero as z approaches zero. Therefore, we know that c₂ or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_2^{ \prime \prime}$$\end{document} is close to zero when z is close to zero. Reasonably, we could assume that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_2 = \sum \nolimits^n_{m = 0 , m \ne 1}a_mz^m$$\end{document} for all n ≥ 0 and n ≠ 0, or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_2 = \sum \nolimits^n_{m = 1}a_mz^m$$\end{document} for all n ≥ 1. Hence, in this case, W(z) is near to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac { 1 } { 2 } + { \sqrt 2a_0 } $$\end{document} or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$\frac {1} {2}$$\end{document} as z is near to zero.

(b) Given that c₂ is a constant function and c₁ not, then following equation (14), \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} ( c_1z + c_2 ) ^ \frac { 1 } { 2 } ( c_1^ { \prime \prime } z + 2c^ { \prime } _1 ) = \frac { z } { 4 { \sqrt 2 } } ( c_1^ { \prime } z + c_1 ) \sim 0 , \end{align} \end{document}

as z ∼ 0. Moreover, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_2 ( c_1^{ \prime} ) ^2$$\end{document} approaches zero as z approaches zero. Reasonably, we could assume that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_1 = \sum \nolimits_{m = 0 , m \ne 1}^na_mz^m$$\end{document} for all n ≥ 0 and n ≠ 0. Hence, in this case, W(z) is near to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac { 1 } { 2 } + { \sqrt 2c_2 } $$\end{document} as z is near to zero.

(c) Given that c₁ and c₂ are constant functions, then following equation (14), \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} ( c_1z + c_2 ) ^ \frac { 1 } { 2 } ( c_1z + c_2 ) ^ { \prime \prime } = \frac { z } { 4 { \sqrt 2 } } ( c_1z + c_2 ) ^ { \prime } \sim 0 , \end{align} \end{document}

as z ∼ 0. Moreover, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_2 ( 2c_1 + c_2^{ \prime \prime} ) ^2$$\end{document} approaches zero as z approaches zero. Reasonably, we could assume that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_1 = \sum \nolimits^n_{m = 1}a_mz^m$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_2 = \sum \nolimits^n_{m = 1}b_mz^m$$\end{document} for all n ≥ 1. Hence, in this case, W(z) is near to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac { 1 } { 2 } $$\end{document} as z is near to zero.

According to the results of case (a), (b), and (c), we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) \le \frac { 1 } { 2 } + d$$\end{document} , where d is a positive constant as z is near to zero. Recall that we set \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$z = \frac { x } { { \sqrt t } } $$\end{document} and let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) = U ( z ) + V ( z ) = \frac { u ( x , t ) } { T } + \frac { v ( x , t ) } { T } $$\end{document} . In consequence, we find that the density of total cells will approach a constant when time is far into the future.

Furthermore, we can calculate a particular solution of equation (14) to verify that W(z) would be close to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac { 1 } { 2 } $$\end{document} as z is close to zero.

Theorem 3

The density of total cells will be near to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac { T } { 2 } $$\end{document} as time is far into the future.

Proof

Following equation (14), let c₁z + c₂ = ξ(z), where c₁ and c₂ are functions of z. Then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { \xi^ { \prime \prime } ( z ) } { \xi^ { \prime } ( z ) } = \frac { z } { 4 { \sqrt 2 } \sqrt { \xi ( z ) } } \tag { 15 } \end{align} \end{document}

is obtained immediately.

Let ξ(z) = k₃z^r and apply ξ(z) to equation (15). We can compute r = 4 and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$k_3 = \frac { 1 } { 288 } $$\end{document} . In consequence, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \xi ( z ) = \frac { z^4 } { 288 } $$\end{document} is a particular solution of equation (14). Namely, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \sqrt { c_1z + c_2 } = \frac { z^2 } { 12 { \sqrt 2 } } $$\end{document} . Consequently, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) = \frac { 1 } { 2 } + \frac { z^2 } { 12 { \sqrt 2 } } $$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) \sim \frac { 1 } { 2 } $$\end{document} as z ∼ 0.

Moreover, we get c₁z + c₂ = c₃z⁴, where c₃ is a function of z. Applying c₃z⁴ to equation (14), we get \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_3{ \sqrt c_3}$$\end{document} approaches zero as z approaches zero. Reasonably, we could assume \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} c_3 = \sum_{m = 1}^n a_mz^m \end{align} \end{document}

for all n ≥ 1 as z approaches zero. Therefore, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_1z + c_2 = z^4 \sum \nolimits^n_{m = 1}a_mz^m$$\end{document} for all n ≥ 1 as z approaches zero, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) = \frac { 1 } { 2 } + { \sqrt 2 } z^2 \big ( \sum \nolimits^n_ { m = 1 } a_mz^m \big) ^ \frac { 1 } { 2 } $$\end{document}

Recall that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) = \frac { u ( x , t ) } { T } + \frac { v ( x , t ) } { T } $$\end{document} . In consequence, we know that the density of total cells will be close to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$\frac {T} {2}$$\end{document} , where T is a constant and T ≥ u(x, t) + v(x, t) initially as time is far into the future. ▪

We can make use of the particular solution \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) = \frac { 1 } { 2 } + { \sqrt 2 } z^2 \big( \sum \nolimits^n_ { m = 1 } a_mz^m \big) ^ \frac { 1 } { 2 } $$\end{document} to analyze the properties of single cell population. Those processes will be shown in the next section.

4. Behavior Of Single Cell Population Into The Future

In the above sections, we discussed the steady points of model (7) and the properties of total cells as time is far into the future. Now, we analyze the properties of single cell population as time far into the future by using results from the above sections.

We have found that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$W ( z ) = U ( z ) + V ( z ) = \frac { 1 } { 2 } + { \sqrt 2 } \sqrt { c_1z + c_2 } $$\end{document} . Following model (9), we apply U(z) + V(z) = W(z) to the first equation of system (9). Then the first equation of system (9) \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { - z } { 2 } U^ { \prime } ( z ) = U^ { \prime \prime } ( z ) - ( U^2 ( z ) + U ( z ) V ( z ) ) ^ { \prime \prime } \end{align} \end{document}

is transformed into \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \frac { - z } { 2 } U^ { \prime } ( z ) = U^ { \prime \prime } ( z ) - W ( z ) U^ { \prime \prime } ( z ) - 2W^ { \prime } ( z ) U^ { \prime } ( z ) - W^ { \prime \prime } ( z ) U ( z ) . \end{align} \end{document}

We arrange the above equation as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} U^ { \prime \prime } ( z ) + \frac { \frac { z } { 2 } - 2W^ { \prime } ( z ) } { 1 - W ( z ) } U^ { \prime } ( z ) + \frac { - W^ { \prime \prime } ( z ) } { 1 - W ( z ) } U ( z ) = 0. \tag { 16 } \end{align} \end{document}

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$${ \overline W} ( z ) = 1 - W ( z ) $$\end{document} . Equation (16) can be simplified as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} U^{\prime \prime} ( z ) + \frac { \frac { z } { 2 } + 2 {\overline W }^{\prime} ( z ) {\overline W } ( z ) } U^ { \prime } ( z ) + \frac {{\overline W } ^ {\prime \prime} ( z )} {{\overline W } ( z ) } U ( z ) = 0. \tag { 17 } \end{align} \end{document}

Through changing variables, the item \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac {\frac {z} {2} + 2 {\overline W } ^ { \prime } ( z ) } {{\overline W } ( z ) } U^ { \prime } ( z )$$\end{document} in equation (17) can be eliminated.

Lemma 4

Equation (17) can be rewritten as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} { \overline U}^{ \prime \prime} ( z ) + a ( z ) { \overline U} ( z ) = 0. \tag{18} \end{align} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$a ( z ) = \frac {{\overline W } ^ { \prime \prime } ( z ) } { W ( z ) } - \frac { 1 } { 2 } \left( \frac { \frac { z } { 2 } + 2 {\overline W } ^ { \prime } ( z ) } { {\overline W } ( z ) } \right) ^ { \prime } - \frac { 1 } { 4 } \left( \frac { \frac { z } { 2 } + 2 {\overline W } ^ { \prime } ( z ) } { {\overline W } ( r ) } \right) ^2$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} U ( z ) = {\overline U } ( z ) \exp \left( \frac { - 1 } { 2 } \int^z \left( \frac { \frac { r } { 2 } + 2 {\overline W } ^ { \prime } ( r ) } {{\overline W } ( r ) } \right) dr \right) . \end{align} \end{document}

Proof

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$p ( z ) = \frac { \frac { z } { 2 } + 2 {\overline W } ^ { \prime } ( z ) } {{\overline W } ( z ) } $$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$q ( z ) = \frac {{\overline W } ^ { \prime \prime } ( z ) } {{\overline W } ( z ) } $$\end{document} . We set \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$U ( z ) = {\overline U } ( z ) e^ { \frac { - 1 } { 2 } \int^z p ( r ) dr } .$$\end{document} Then equation (17) can be translated into \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} {\overline U } ^ { \prime \prime } ( z ) + \left( q ( z ) - \frac { p^ { \prime } ( z ) } { 2 } - \frac { p^2 ( z ) } { 4 } \right) {\overline U } ( z ) = 0. \end{align} \end{document}

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$q ( z ) - \frac { p^ { \prime } ( z ) } { 2 } - \frac { p^2 ( z ) } { 4 } = a ( z ) $$\end{document} . Hence, equation (18) is obtained. ▪

Following equation (18), by Theorem 4.1 in Pettet et al. (1996) we get that U(z) is bounded as z is close to zero.

Theorem 4

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$${ \overline U} ( z ) $$\end{document} is bounded on [0, δ] for some small δ; moreover, U(z) is bounded on [0, δ].

Proof

According to the assumptions in Lemma 4, we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} a ( z ) = q ( z ) - \frac { p^ { \prime } ( z ) } { 2 } - \frac { p^2 ( z ) } { 4 } , p ( z ) = \frac { \frac { z } { 2 } + 2 {\overline W } ^ { \prime } ( z ) } {{\overline W } ( z ) } , q ( z ) = \frac {{\overline W } ^ { \prime \prime } ( z ) } {{\overline W } ( z ) } , \end{align} \end{document}

and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$${\iverline W } ( z ) = 1 - W ( z ) = \frac { 1 } { 2 } - { \sqrt 2 } \sqrt { c_1z + c_2 } .$$\end{document} .

Given that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$c_1z + c_2 = z^4 \sum \nolimits_{m = 1}^n a_m z^m$$\end{document} for all n ≥ 1 as z approaches zero, we get results as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} q (z) & = \frac { - { \sqrt 2 } \left( z^2 \left( \sum_ { m = 1 } ^n a_m z^m \right) ^ \frac {1} {2} \right)^{\prime \prime}} {\frac {1} {2} - {\sqrt 2} z^2 \left( \sum_{ m = 1 }^n a_m z^m \right)^{\frac{1} {2}}} , \\ \frac { - p^ { \prime } ( z ) } { 2 } & = \frac { - 1 } { 2 } \frac { \left( \frac { 1 } { 2 } - 2 { \sqrt 2 } \left( z^2 \left( \sum_ { m = 1 } ^n a_m z^m \right) ^ \frac { 1 } { 2 } \right) ^ { \prime \prime } \right) \left( \frac { 1 } { 2 } - { \sqrt 2 } z^2 \left( \sum_ { m = 1 } ^n z_m z^m \right) ^ \frac { 1 } { 2 } \right) } { \left( \frac { 1 } { 2 } - { \sqrt 2 } z^2 \left( \sum_ { m = 1 } ^n a_m z^m \right) ^ \frac { 1 } { 2 } \right) ^2 } \\ & - \frac { - 1 } { 2 } \frac { \left( \frac { z } { 2 } - 2 { \sqrt 2 } \left( z^2 \left( \sum_ { m = 1 } ^n a_m z^m \right) ^{\frac {1} {2}} \right) ^ { \prime } \right) \left( - { \sqrt 2 } z^2 \left( \sum_ { m = 1 } ^n a_m z^m \right) ^{\frac {1} {2}} \right) ^ { \prime } } { \left( \frac { 1 } { 2 } - { \sqrt 2 } z^2 \left( \sum_ { m = 1 } ^n a_m z^m \right) ^{\frac {1} {2}} \right)^2 } , \\ \frac { - p^2 ( z ) } { 4 } & = \frac { - 1 } { 4 } \frac { \frac { z^2 } { 4 } + 2z ( - { \sqrt 2 } { \sqrt c_1z + c_2 } )^ { \prime } + 4 \left( ( - { \sqrt 2 } { \sqrt c_1z + c_2 } )^ { \prime } \right)^2 } { ( \frac { 1 } { 2 } - { \sqrt 2 } { \sqrt c_1 z + c_2 } ) ^ \frac { 1 } { 2 } } \\ & = \frac { - 1 } { 4 } {\frac { \frac { z^2 } { 4 } + 2z \left( - { \sqrt 2 } z^2 ( \sum_ { m = 1 } ^n a_m z^m ) ^ \frac { 1 } { 2 } \right) ^ { \prime } + 4 \left( \left( - { \sqrt 2 } z^2 ( \sum_ { m = 1 } ^n a_m z^m ) ^ \frac { 1 } { 2 } \right) ^ { \prime } \right) ^2 } { \left( \frac { 1 } { 2 } - { \sqrt 2 } z^2 ( \sum_ { m = 1 } ^n a_m z^m ) ^ \frac { 1 } { 2 } \right) ^2 } .} \end{align} \end{document}

In consequence, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$q ( z ) \sim 0 , \frac { - p^ { \prime } ( z ) } { 2 } \sim \frac { - 1 } { 2 } $$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \frac { - p^2 ( z ) } { 4 } \sim 0$$\end{document} as z is near to zero, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$a ( z ) = q ( z ) - \frac { p^ { \prime } ( z ) } { 2 } - \frac { p^2 ( z ) } { 4 } \sim \frac { - 1 } { 2 } $$\end{document} as z is close to zero.

Equation (18) can be rewritten as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} \overline{U} ^ { \prime \prime } ( z ) + \left( \frac {- 1} {2} + b ( z ) \right) \overline{U} ( z ) = 0 , \tag {19} \end{align} \end{document}

where b(z) approaches zero as z approaches zero. Because the solution of equation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \overline{U} ^ { \prime \prime } ( z ) + \left( \frac { - 1 } { 2 } \right) \overline{U} ( z ) = 0$$\end{document} is bounded as z approaches zero, by Theorem 4.1 in Pettet et al. (1996) the solution of equation (19) is also bounded as z approaches zero. Therefore, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$${ \overline U} ( z ) $$\end{document} , the solution of equation (18) is bounded on [0, δ] for some small δ, saying \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \mid { \overline U} ( z ) \mid \leq M$$\end{document} and M is a constant. Hence, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align} U ( z ) = {\overline U } ( z ) e^ { \frac { - 1 } { 2 } \int^z p ( r ) dr } \leq Me^ { \frac { - 1 } { 2 } \int^z p ( r ) dr } , \end{align} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$p(z) = \frac {\frac{z} {2} - 2{\sqrt 2} (z^2(\sum \nolimits_{m = 1}^n a_{m} z^{m})^{\frac {1} {2}})^{\prime}} {\frac {1} {2} - {\sqrt 2} z^2 (\sum \nolimits_{m = 1}^n a_m z^m)^{\frac {1} {2}}}$$\end{document} . Moreover, since p(z) > 0, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$ \exp \left( \frac { - 1 } { 2 } \int^z p ( r ) dr \right) \leq 1$$\end{document} for all z in [0, δ] for some small δ. Consequently, U(z) is bounded on [0, δ] for some small δ. ▪

It is verified that U(z) is bounded on [0, δ], where z is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$x / { \sqrt t}$$\end{document} and δ is very small. Furthermore, we restore U(z) to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$u ( x / { \sqrt t} ) / T$$\end{document} , where T is a positive constant and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$T \geq \frac { u ( x , t ) } { T } + \frac { v ( x , l ) } { T } $$\end{document} initially. The term z approaches zero expresses that time t is far into the future. Therefore, Theorem 4 indicates that the density of u-cell population approximates finite number as time approaches the infinite. Through writing \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$v ( x / { \sqrt t} ) $$\end{document} as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}$$w ( x / { \sqrt t} ) - u ( x / { \sqrt t} ) $$\end{document} , it can be deduced immediately that the density of v-cell population is finite no matter how much time passes.

Footnotes

Acknowledgments

Thanks are due to Tsai Long-Yi and Tai-Ping Liu for their continuous encouragement and discussions on this work, to Grand Hall and E.-Ton Sola for their financial assistance, and to the referee for interest and helpful comments on this article.

Disclosure Statement

No competing financial interests exist.

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