K br we obtain br K br Furthermore br K
when αβ ≤ 1. If αβ > 1, we use simple algebraic computations to obtain
Clearly, the sign of D is determined by
Since αβ > 1, the preceding arguments for Theorem 3.1 implies that X2∗ is the smaller positive root of (A.4). Thus, 0 < X2∗ < −λ2/(2λ1 ). It follows from (B.3) that D1 > 0. Then the Routh–Hurwitz criteria means that the eigenvalues of J have negative real parts. Therefore the equilibrium point E2 is locally asymptotically stable. By similar discussions to above, one can verify that the determinant of the Jacobian matrix at E1 is positive and the determinant of the Jacobian matrix at E2 is negative. Therefore, we conclude that E1 = (X11, X21 ) is asymptotically stable and E2 = (X12, X22 ) is a saddle. The proof of Theorem 3.2 is thus completed.
Appendix C. Proof of Theorem 3.3
It follows from Dulac’s criterion that there is no closed orbit in (3.1). Therefore, by Poincaré–Bendixson Theorem and the local Atezolizumab of the equilibria imply that E1 is globally stable if (3.4) is satisfied and E2 is globally asymptotically stable under (3.2). The proof is completed.
 Baez J, Kuang Y. Mathematical models of androgen resistance in prostate cancer patients under intermittent androgen suppression therapy. J Appl Sci 2016;6:352.
 Botrel TEA, Clark O, Reis RBD, Pompeo ACL, Ferreira U, Sadi MV, Bretas FFH. Intermittent versus continuous androgen deprivation for locally advanced, recurrent or metastatic prostate cancer: A systematic review and meta-analysis. BMC Urol 2014;14:p9.
 Bruchovsky N, Koltz L, Crook J, Malone S, Ludgte C, Morries WJ, Gleave ME, Goldenberg SL. Final results of the Canadian prosepective phae II trial of intermittent androgen suppression for men in biochemical recurrence after radiotherapy for locally advanced prostate cancer: clinical parameters. Cancer 2006;107:389–95.
 Burton G, Giribaldi M, Munoz A, Halvorsen K, Patel A, Jorda M, Perez-Stable C, Rai P. Androgen deprivation-induced senescence promotes outgrowth of androgen-refractory prostate cancer cells. PLoS ONE 2013;8:e68003.
 Dason S, Allard CB, Wang JG, Hoogenes J, Shayegan B. Intermittent androgen deprivation therapy for prostate cancer: translating randomized controlled into trials clinical pratice. Can J Urol 2014;21:28–36.
 Duan J. An introduction to stochastic dynamics. Cambridge texts in applied mathematics. New York: Cambridge University Press; 2015.
 Everett RA, Packer AM, Kuang Y. Can mathematical models predict the outcomes of prostate cancer patients undergoing intermittent androgen depri-vation therapy? Biophys Rev Lett 2014;9:173–91.
 Gregory CW, Johnson RT, Mohler JL, French FS, Wilson EM. Androgen receptor stabilization in recurrent prostate cancer is associated with hypersen-sitivity to low androgen. Cancer Res 2001;61:2892–8.
 Gulley JL, Madan RA. Developing immunotherapy strategies in the treatment of prostate cancer. Asian J Urol 2016;3:278–85.
 Hatano T, Hirata Y, Suzuki H, Aihara K. Comparison between mathematical models of intermittent androgen suppression for prostate cancer. J Theor Biol 2015;366:33–45.
 Hirata Y, Bruchovsky N, Aihara K. Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer. J Theor Biol 2010;264:517–27.
 Hua TZ, Bing YY. Preliminary experimental study of drug resistance mechanisms on androgen independent prostate cancer cells. Phd thesis Chongqing: Chongqing medical university; 2010.
 Ideta AM, Tanaka G, Takeuchi T, Aihara K. A mathematical model of intermittent androgen suppression for prostate cancer. J Nonlinear Sci 2008;18:593–614.
 Ikeda N, Watanabe S. A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J Math 1977;14:619–33.
 Jackson TL. A mathematical investigation of the multiple pathways to recurrent prostate cancer: comparison with experimental data. Neoplasia 2014;6:697–704.
 Jackson TL. A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete Contin Dyn Syst Ser B 2004;4:187–201.
 Jain HV, Friedman A. Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy. Discrete Contin Dyn Syst Ser B 2013;18:945–67.
 Ji C, Jiang D, Shi N. Analysis of a predator-prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. J Math Anal Appl 2009;359:482–98.
 Koralov L, Sinai Y. Theory of probability and random processes. second edition. Berlin: Universitext. Springer; 2007.