_{1}

^{*}

The main theorem of the present paper is the bistability theorem for a four dimensional cancer model, in the variables
representing primary cancer C, metastatic cancer
, growth factor GF and growth inhibitor GI, respectively. It says that for some values of the para- meters this system is bistable, in the sense that there are exactly two positive singular points of this vector field. And one is stable and the other unstable. We also find an expression for
for the discrete model T of the introduction, with variables
, where C is cancer, are growth factors and growth inhibitors respectively. We find an affine vector field Y whose time one map is T
^{2} and then compute
, where
is an integral curve of Y through
. We also find a formula for the first escape time for the vector field associated to T, see section four.

We continue the study of the cancer model from Larsen (2016) [

where

and

when

The eigenvalues are

In section two we prove the Bistability Theorem for a mass action kinetic system of metastatic cancer

For

The phase space of our model T is

uPAR (urokinase plasminogen activator receptor) is a cell surface protein, that is associated with invasion and metastasis of cancer cells. In Liu et al. (2014) [

According to Luo and Fu (2014), [

There are several important monographs relevant to the present paper, see Adam & Bellomo (1997), [

The main result of this section is Theorem 1 below that proves the bistability of the mass action kinetic system (1) to (8). Consider then the mass action kinetic system from Larsen (2016), [

The complexes are

all

But first we state a theorem, we shall next prove. A positive (nonnegative) singular point

of f is a singular point of f, such that

Theorem 1 Assume

where

is stable when

Consider a singular point

Setting the last coordinate of f equal to zero gives

when

and

When

and from (10) we get

This means that B simplifies to

Let

Also

The characteristic polynomial of

Finally

In Larsen (2016) [

whenever

Lemma 1 Assume

where

Proof. The coefficient to

Everything cancels out and leaves a zero. The coefficient to

Square

Everything cancels out except

The coefficient to

Multiply

Everything cancels out except

Finally the constant term is

The lemma follows.

Theorem 2 Assume

where

Proof. We have

where

and

due to symmetry of

in P and

in

are singular points of f and that

are not singular points of f. Here

and

Also

We have

and logically equivalent

where

So

and from this the formula follows. And (16) is a similar computation.

We shall insert (15), (16) in the first coordinate of f, multiplied with

Now abbreviate

Multiply with

But this amounts to

and this vanishes due to the formula for roots of quadratic polynomials. That the second coordinate vanishes is logically equivalent. So (11) are singular points of f.

We shall now argue, that

is not a singular point of f. To this end define

Insert the formulas (15), (16) for

Multiply with

But (17) is zero by the above and (18) is nonzero. So

In the remainder of the proof of Theorem 1, we assume, that

We shall now verify that

But we have

Simply insert (15) and (16) in the numerator

Now we use that

so

is equivalent to

The right hand side here is negative and the left hand side is positive. Thus

We shall now show that

But this amounts to

which is equivalent to

and this again is equivalent to

and from this it follows that

And a formula for

Define

so that

Now introduce these two formulas in the formulas for

Notice that

is negative for small

This equation holds for small

where

And from this it follows that

Consider the mass action kinetic system in the species

The complexes are

see Horn and Jackson (1972), [

In this section

on the hyperplane

and compute, when

If

By Larsen (2016), [

Then

Define the vector field

where

where

If

then

Assume that

But this means that

because we have

i.e.

So assuming

We want to have

and

such that

Here

Then

Thus

Now

Define

Let

where

Now suppose

Then

when

Then

Define the vector field

and the time one map is

and we want this to be

Then define the vector field

This vector field has time one map

Then arguing as before

and

We can now find

Next consider the chemo therapy model

and initially, that

We want this vector field to have time one map

Then we find

Now compute arguing as above

Finally we can find

and this becomes

Now consider the chemo therapy model, when

The second coordinate here should be equal to

while the third coordinate should be equal to

in order that the time one map of

and this is simplified to

Remark 1 When

by the above you can find an affine vector field whose time one map is

In this section

and

U as in section 3. This vector field, B, has time one map T, see Larsen (2016), [

The purpose of this section is to prove, that there exists a first escape time

When

or there exists a smallest

Proposition 3 Suppose

Proof. We have the following formula for the flow of B

Here

and

Define

Since

It follows that we have the following formula

Since

Remark 2 By the proof we have

implies

When

These formulas are explained in the proof of Proposition 4.

Let

D as in section 3. B has time one map T, see Larsen (2016), [

Proposition 4 Suppose

If

for all

(ii) If

If

for all

Proof. First of all the flow of F is

We have the following formula

where

From this formula, (ii) follows.

Remark 3 In case (i) of the proposition, if

implies

In case (ii) of the proposition, if

implies

We shall now derive a formula for the first escape time

and

where

i.e.

Compute

where

If

Then we have the following formulas

Assume that

for

we claim that there are atmost finitely many such solutions and hence that there exists a smallest

Assume for contradiction, that there are infinitely many solutions to

By (31) there are exactly

Since there are infinitely many solutions to

in

By the mean value theorem, there exists

If

Since

so

By

let

By

Suppose

If

We shall now find the first escape time when

and

where

i.e.

Assume in the notation of Proposition 4, that

If

and

There are atmost two solutions to

In this paper we proved that the model of primary and metastatic cancer in Section 2 is bistable, in the sense, that there are exactly two positive singular points. One of them is unstable, and when one of the rate constants is small the other is stable. Then we found formulas for the rate of change of cancer growth for the model T of the introduction, when for

It would be interesting to figure out what happens if the polynomials

How do cancer cells coordinate glycolysis and biosynthesis. They do that with the aid of an enzyme called Phosphoglycerate Mutase 1. In the reference [

Jens Christian Larsen, (2016) The Bistability Theorem in a Model of Metastatic Cancer. Applied Mathematics,07,1183-1206. doi: 10.4236/am.2016.710105