# br dt br K br K br Because this

dt

K

K

Because this is a system of ordinary differential equations, the global convergence of its solutions can be achieved through qualitative analysis. Set

Q

and

The results concerning the existence of equilibria and their stability are stated in the following theorems. The proofs of these results are given in Appendix A, Appendix B and Appendix C.

KQ

are satisfied.

(i) The trivial equilibrium point E0 is unstable. This means that CAS cannot eliminate all of the tumor cells.
(ii) E1 is locally asymptotically stable if

(i) E1 is globally asymptotically stable if (3.4) is satisfied.
(ii) The unique interior equilibrium point E2 is globally asymptotically stable under (3.2).
(iii) The separatrix of the saddle E2 is the boundary between the attractive basin of E1 and the attractive basin of E1.

Theorem 3.3 means that each positive solution of (3.1) converges to an equilibrium, and the convergence depends on the initial data in case (iii) because of the bistability of E1 and E1.

4. Analysis of stochastic model

dX

K

dX
=
r

K

For an integrable function f(t) on [0, ∞), we give the following notations:

t

f

t

t

f

t

t

t

Furthermore, we state two definitions and a lemma from [33,45].

Definition 4.2. The WZB117 Xi(t) is said to be persistent in mean if

Lemma 4.3. Let M = M(t )
t≥0 be a real-valued continuous local martingale vanishing at time zero. If

then

t

4.1. Existence and uniqueness of the solution

k
k

An application of Itô’s formula gives

where

K

where M is a positive constant that follows from the negative sign of the leading coe cients of the quadratic functions above. Consequently,

The existence of the global solution makes us to study the ultimate boundedness of the solutions which is one of the most important properties in population dynamics.

p
R

X pdt

pet

K

et

pet

where M1 is a positive constant. Integrating the above inequality from to t and taking the expectation on both sides, we get

Then we obtain

Similarly,
d(et X p )

et

pet

K

et

p
r

pet

Integrating the above inequality and taking the expectation on both sides yield

Note that

An application of Chebychev inequality leads to the desired result. The proof is completed.

4.2. Persistence in mean and extinction

We start by considering the following one-dimensional stochastic differential equation

tlim

t

b

Now we are in the position to consider the effect of the white noises on the therapy of prostate cancer. To this end, we establish the threshold conditions for extinction and persistence in mean of tumor cells.

Then we have the following assertions:

(i) Androgen-dependent cells X1 go to extinction.

then androgen-independent cells X2 go to extinction.

then androgen-independent cells X2 are persistent in mean.

Let ϕ(t) be the solution of

with the initial value ϕ(0) = X1 (0). An application of the comparison principle of stochastic differential equations [18] gives

Note that

satisfies the linear SDE:

By Lemma 4.3 of the strong law of large numbers of local martingale, we see that for any ∈ (0, 1), there exists a large T2 such that

t
s

almost
surely. Without loss of generality, we assume that (4.15) is valid for any ω ∈

t

where

By applying the Kolmogrov Theorem [26], we see that there exists a positive constant M1 such that

lim sup
t,

m
u

The arbitrariness of ε leads to

X

Let ϕ be the solution of

K

t

K

βε

tlim

t

r

By virtue of the comparison principle, we get

t

K

βε

lim sup

t
r

K

t

K

βε

lim sup

t

r

K

which is the required assertion. This completes the proof.

From the previous theorem, we see that the stochastic noises have a great effect on the development of tumor cells. Indeed, (4.9) and (4.10) are satisfied for the larger σ 1 and σ 2. Thus, large noise intensities can lead to the extinction of both AD cells and AI cells. This is different from the deterministic case where CAS therapy cannot eliminate all of the tumor cells from Theorem 3.2. This suggests that a drug that interferes with the proliferation of tumor cells is beneficial for CAS therapy due to the similar effect of noise perturbations.