(i) A variant of the classic logistic population model is given by the HutchinsonWright equation

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

where $\alpha, T>0$. Determine the condition on $\alpha$ (in terms of $T$ ) for the constant solution $x(t)=1$ to be stable.

(ii) Another variant of the logistic model is given by the equation

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

where $\alpha, T>0$. Give a brief interpretation of what this model represents.

Determine the condition on $\alpha$ (in terms of $T$ ) for the constant solution $x(t)=1$ to be stable in this model.