br Supported in part by the National Science Fund of
Supported in part by the National Science Fund of China (11571284). ∗ Corresponding author.
E-mail address: firstname.lastname@example.org (W. Wang).
the level of PSA. The treatment is activated when the PSA level is above a certain value L1 and it turns off (patient takes a vacation period) when the PSA is below a threshold value L2 < L1. Intermittent treatment could be effective if the off-treatment period begins before the androgen-dependent Taxol become resistant to the therapy. Therefore, tumor cells may take longer time to adapt and mutate because of the periodic changes in the environment. These changes could lead to prolong the survival of AD cells (because in off-treatment period AD cells still can proliferate) and delay the dominance of resistance cells (relapse). Nevertheless, there is no definite conclusion that IAS therapy is superior than CAS therapy in all cases in the medical community .
Mathematical modeling and analysis have greatly contributed to the understanding of mechanisms of tumor cell pro-gression under CAS, IAS and immunotherapy response when it is combined with androgen deprivation therapy (see, e.g, [1,8,9,13–15,17,20,21,35,37,41–44,47]). Ideta et al.  proposed a mathematical model to compare CAS and IAS, and discuss the benefit of IAS. Rutter and Kuang  formulated a population model for androgen deprivation therapy with immunother-apy. In their model, the growth rates of AD cells and AI cells are described by logistic functions and AI cells might convert back to AD cells in an androgen-rich environment. A similar model was proposed and studied in Portz and Kuang .
The models in the references [17,35,37] are deterministic. But tumor growth is sensitive to certain fluctuations such as temperature, radiation and chemical products, oxygen supply and nutrients [27–29,41,42], and these environmental pertur-bations are inevitable. Hence, the model of Ideta et al.  was extended to a stochastic model by Tanaka et al.  where stochastic white noises are introduced to the ODE model to represent the intrinsic fluctuations of tumor dynamics. Based on the IAS protocol (on and off-treatment periods schedule) which was applied in clinical trials by Bruchovsky et al. , the authors used numerical simulations to show that their stochastic model is able to describe some features and characteristics observed in clinical trials. Moreover, the noises could be responsible for the variability of responses to the therapy from one patient to another.
Competition between AD cells and AI cells may alter the fate of tumor cells. Their competitive effects have been consid-ered in [35,37,39] with identical competition coe cient. However, the recent oncology researches [16,48] have shown that AD cells and AI cells have distinct shapes and functions. More specifically, AD cells appear as a triangle or spindle form, whereas AI cells look rounder and flatter. In addition, AD cells and AI cells also have different gene levels so that AD cells and AI cells have different functions such as cell proliferation, apoptosis, adhesion and cell motility, which plays an impor-tant role in cell migration and cancer metastasis . Therefore, AD cells and AI cells may compete for nutrient and oxygen with different intensities.
In this paper, we introduce stochastic noises into the model of continuous androgen deprivation therapy to study how white noises and the different competitive intensities of cells affect the dynamics of tumor and treatment strategy. In Section 2, the mathematical model is formulated. The mathematical results for the global convergence of the model without noise perturbation is stated in Section 3. Section 4 focuses on deriving the threshold conditions between the extinction and the persistence of AD cells and AI cells, which reveal how stochastic noise and the competitive coe cients influence the relapse of tumor. The su cient conditions for the existence of stationary distribution are also derived. In Section 5, numer-ical simulations are carried out to compare the treatment results under different values of e cacy of hormone therapy and investigate the influences of stochastic noises on the development of tumor cells and treatment strategy. Section 6 presents the discussions of this paper.