Is the allee effect relevant to stochastic cancer model?

This paper delineates the evolution of a tumor cell population with Allee effect through a system of stochastic differential equations. A stochastic extension of the deterministic model is examined to encapsulate the uncertainty or variation observed in the tumor evolution using parametric perturbation method. We have discussed the existence, uniqueness, stochastically ultimate bounded, stochastically permanence and asymptotic stability of the solutions to the stochastic tumor cell population with the aid of constructing Lyapunov function. Then we have investigated that the model has a unique dynamical bifurcation point Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta $$\end{document} with the following conditions: if Θ<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta < 0$$\end{document}, then the model has a unique invariant measure, the Dirac measure concentrated at zero, and it is stable. If Θ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta > 0$$\end{document} then a stable unique invariant measure on R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}_{+}$$\end{document} occurs, and the Dirac measure concentrated at zero is unstable. Numerical results are performed using first-order Ito-Wiener stochastic scheme to exhibit the theoretical analysis.