(a) A variant of the classic logistic population model is given by:

$$\frac{d x(t)}{d t}=\alpha\left[x(t)-x(t-T)^{2}\right]$$

where $\alpha, T>0$.

Show that for small $T$, the constant solution $x(t)=1$ is stable.

Allow $T$ to increase. Express in terms of $\alpha$ the value of $T$ at which the constant solution $x(t)=1$ loses stability.

(b) Another variant of the logistic model is given by this equation:

$$\frac{d x(t)}{d t}=\alpha x(t-T)[1-x(t)]$$

where $\alpha, T>0$. When is the constant solution $x(t)=1$ stable for this model?