In many stochastic predator-prey models I have seen that as we increase the strength of multiplicative noise the extinction of species happens quickly (with some threshold condition). For example, lets take the case: \begin{align} dx(t)&=x(t) \left( a-bx(t)-\frac{cy(t)}{m+x(t)} \right) dt + \alpha~ x(t) dB_1(t)\\ dy(t)&= y(t) \left(r-\frac{f y(t)}{m+x(t)} \right) dt+ \beta~ y(t) dB_2(t) \end{align}
So here if $a < \frac{\alpha ^2}{2}$, then the extinction of species $x$ happens. Now if we increase the noise further satisfying the condition $a < \frac{\alpha ^2}{2}$, we get earlier extinction. For example if $a=0.4$ and $\alpha=0.9$, the extinction of species $x$ occurs in around 12 days and if $\alpha=1.2$ the extinction of species $x$ occurs in around 7 days. I was trying to understand why the extinction occurs quicker in time.
My thinking: I was thinking that as the noise intensity becomes higher the environmental variability becomes higher and as the noise is multiplicative, it changes the state of the species dynamics quickly. Is there any way by which I can analytically show that as the noise intensity increases extinction time decreases or any intuitive explanation? Please help me.