# br Then one has the following assertions

Then one has the following assertions:

(i) Androgen-independent KIN59 X2 are persistent in mean.

then androgen-dependent cells X1 are persistent in mean.

then androgen-dependent cells X1 go to extinction and androgen-independent cells are persistent in probability, that is,
P(limt inf X2 (t ) > 0) = 1.
(4.19)
→∞

for all

The solution of the first equation of (4.1) is represented by

K

t

R

t

K

s

Then

K

t

K

for all

s

almost surely. Without loss of generality, we assume that (4.23) is valid for any ω
∈
0. Let us choose ε1 such that

K

R

T

where

T

v

K

Using the boundedness of X2 and applying the Kolmogrov Theorem [26], we see that there exists a positive constant M0 such that

lim sup

K

K

Thus,

Without loss of generality, for any ε2 > we can select T3 > T large enough such that

t
s

where

K

L

Hence,

and

L

This is a contradiction. Consequently, X2 is persistent in mean.
(ii) We prove the persistence of X1 by contradiction. Assume that X1 is not persistent in mean. Then there exists 1 =

Then

r

K

t

s

Obviously,

K

t

t

K

K

t

t
s

K

T

where

T

r
β

K

By similar discussions to above, there exists an M4 > such that

lim sup

Hence, there exists a positive constant η such that

K

t

t

By a similar procedure to above, we obtain

r

r2β
X
t

t

t
t

t

t

t

t

We are led to a contradiction. Consequently, X1 is persistent in mean.

For the second case, from the second equation of (4.1) we have

where
I2 =
t

t

For any ω ∈ 1 and all t ≥ T , we use (4.32) and the continuous path of Brownian motion to deduce that there exists a positive constant S1 such that

t

T

T

r

B

r

T

Reversing the order of integration in I3 and using (4.40), we get

t

t

s

τ

T

T

B

s

τ

T

t

s

τ

t

t

B

s

τ

T

Consequently, we arrive at

t

t

t

t

X

m
u

which contradicts the fact that X2 is persistent in mean due to the assertion (i). Therefore, X1 is persistent in mean. (iii) We first show that androgen-dependent cells X1 go to extinction. From (4.36) and (4.37) we get

K
t
r2

t

If the conditions in (a) are satisfied, then

t

Taking the superior limit on both sides of (4.46), we obtain
lim sup
t−1 ln

On the other hand, by virtue of the comparison principle and Lemma 4.6, we get
X1 t∗ ≤

Furthermore, if the conditions in (b) are satisfied, then
lim sup
t−1 ln

r

Note that X2(t) has a positive upper bound M which follows from Theorem 4.5. As a result,
0 ≤

Taking the superior limit on both sides of (4.49), we obtain

Consequently,

(4.19) is established by similar arguments to those for assertion (ii). This completes the proof.

The results of Theorem 4.9 indicate that the persistence of AI cells and extinction of AD cells favor the larger α and the smaller β. Thus, if the treatment could not eradicate AD cells, reducing the competitive coe cient α of AI cells or increasing the competitive coe cient β of AD cells may prevent a relapse. This could be achieved by finding some drugs that reduce the activity of resistant cells.

4.3. Stationary distribution and ergodicity for the system

In this subsection, we investigate the existence and uniqueness of an ergodic stationary distribution for system (4.1). To this end, we follow the same method as in [23,30,34,46], where we need to prove that

(i) There exists a bounded domain E ∈ IntR2+ with regular boundary such that its closure E ⊂ IntR2+ and a non-negative C2- function V(x) such that for any x ∈ IntR2+ \ E, LV is negative.
(ii) For any bounded domain Eˆ ⊂ IntR2+ , there exists a positive number η such that the diffusion matrix for system

satisfies
2

Then system (4.1) has a unique ergodic stationary distribution.

Proof. Let

An application of Itô’s formula yields

where

K
K

K

K

r

X

K

K

It is easy to get

where

K

K

K

Note that the leading coe cients of the quadratic functions ϕi are negative. Thus ϕ1 has an upper bound M1 = sup ϕ1 (X1 ).