Galactose 1-phosphate Moreover by means of we
Moreover, by means of (4.51) we see that ϕ2 has a negative upper bound M2 = K /4r2.
Let us choose a bounded domain
Parameter values used for numerical simulations.
P Biological meaning Value Source
K cancer cell carrying capacity 11 billion/day 
a0 basal level androgen concentration 20 ng /mL 
γ androgen clearance and production rate 0.08/day 
Therefore, under (4.51) we conclude that
which implies that the required conditions are satisfied. Therefore, system (4.1) has a unique ergodic stationary distribution. This completes the proof.
5. Numerical simulations
In this section, we use numerical simulations for model (4.1) to show effects of treatment and white noise perturbations on tumor dynamics. Table 1 lists the values of the parameters used in our model and its resources.
In Fig. 1, we fix α = 0.9, β = 0.8, σ1 = 0.7, σ2 = 0.2 so that r2 − σ22/2 < 0. If CAS therapy eliminates AD cells, then the results obtained in the Theorem 4.8 mean that the treatment could lead to a cure. We run a series of simulations by increasing the value of u until reaching the lowest value that could eliminate AD cells. From the simulations, we see that when u is small, the treatment could not suppress AD cells, while the value u = 0.5 could eliminate the tumor within 5 years. This means that reducing 50% of androgen level could inhibit the proliferation and induce opoptosis of AD Galactose 1-phosphate which lead to their extinction. As shown in Fig. 1, AI cells become also extinct.
In Fig. 2, we fix σ1 = 0.04, σ2 = 0.02 so that r2 − σ22/2 > 0. According to the results given in (iii) of Theorem 4.8 and Theorem 4.9, at least AI cells are persistent. Thus, CAS therapy in this case cannot eliminate the tumor. Then we are in-terested to the value of u which decreases the severity of the tumor. When 0.6 ≤ u ≤ 0.9, we get the extinction of AD cells and a very rapid increase of AI cells, which means that prostate cancer is more fatal because it has become an androgen-independent cancer. On the other hand, when 0.1 ≤ u ≤ 0.5, we observe the persistence of both AD cells and AI cells. How-ever, AI cells stay in a low level due to the stronger competitors of AD cells, and therefore, tumor cells are still controllable in treatment. Indeed, the simulations indicate that AI cells cannot turn over AD cells until more than 10 years when u ≤ 0.5. Consequently, the treatment where u ≤ 0.5 is better than it was in the previous case (u > 0.5). Therefore, u = 0.5 is the approximate optimal value to control both AD cells and AI cells.
In conclusion, the above simulations demonstrate that the optimal on-treatment period u exists when there is a possi-bility that the elimination of tumor cells can be achieved by CAS. Furthermore, the reduction of androgen may give a better result in the case where AD cells may exclude AI cells for the higher u.
In this paper, we propose a stochastic competition model for prostate cancer with androgen deprivation therapy. Firstly, we introduce different competition intensities for AD cells and AI cells because they have different functions and pathways [16,48]. Secondly, since the growth of tumors is sensitive to certain fluctuations such as temperature, radiation and chemical products, oxygen supply and nutrients [27–29,41,42], we incorporate the stochastic noises to find their influences on tumor dynamics. By using Lyapunov functions and the comparison principle for SDEs, the threshold conditions for extinction and persistence in mean for AD and AI cells are established and su cient conditions for the existence of an ergodic stationary distribution of system (2.2) are derived.
Based on the analytical results of system (2.2), we conclude that there are two possible outcomes of using androgen deprivation therapy. The first one is that under the larger noise σ 2, prostate cancer could be successfully treated with androgen deprivation therapy as shown in Theorem 4.8 (when u and σ 1 satisfy (4.9), AD cells go to extinction and AI
cells become extinct also). Secondly, in case that σ 2 is small or (4.9) is not satisfied, the treatment is unable to elimi-nate the tumor where the resistance cells (AI) persist, as shown in Theorems 4.8 and 4.9. Then, the competition inten-sities and the value of androgen suppression u determine the persistence and extinction of AD cells besides the persis-tence of AI cells as Theorem 4.9 indicates. Extinction of AD cells in this case (relapse) is more dangerous than their per-sistence because the tumor becomes resistant (dominance of AI cells) and does not respond to treatment in a short pe-riod of time. If the mutation rate from AI cells to AD cells is ignored in the model of Rutter and Kuang  (very small in the on-treatment period), we conclude that using only CAS therapy leads to relapse (special case for our model when