Dynamical Behaviors of a Stochastic Single-Species Model with Allee Effects

In this letter, we test a scalar stochastic nonlinear equation used to portray the growth of a population with Allee effects. We first testify that there is a unique dynamical bifurcation point Lambda to the equation, and the sign of A determines the dynamical properties of the equation: if Lambda is negative, then the equation has a unique invariant measure - the Dirac measure concentrated at zero; if Lambda is positive, the equation has a unique invariant measure concentrated on (0, +infinity), and the density function of the invariant measure can be expressed explicitly. Then we probe the lower-growth rate and the continuity of the solution. Finally, we apply the theoretical results to research the growth of African hunting dogs.