As an alternative to the popular Hutchinson population model, which introduces a delay in the per capita growth rate, one can introduce a delay solely in the growth contribution and consider a probability, $P(T, N(t-T))$ that individuals survive that delay. The differential equation reads then as $$ N^{\prime}(t)=b P(T, N(t-T)) N(t-T)-(d+c N(t)) N(t), $$ where $b>0$ is the growth rate, $d>0$ the death rate, and $c>0$ the competition factor. Suppose that $$ P(T, N(t-T))=\frac{d}{\left(e^{d T}-1\right) c N(t-T)+d e^{d T}} . $$
a) Show that the model can be reduced to $$ \frac{d u(\tau)}{d \tau}=\frac{\beta u(\tau-\tilde{T})}{\left(\delta^{\tilde{T}}-1\right) u(\tau-\tilde{T})+\delta^{\tilde{T}}}-u(\tau)(1+u(\tau)) . $$
b) Show that there exists a critical threshold $\tilde{T}_c$ such that a positive equilibrium exists only if $0<\tilde{T}<\tilde{T}_c$.
c) Show that if $0<\tilde{T}_c$, the unique positive equilibrium $u^*(\tilde{T})$ decreases as the delay grows larger.
d) Plot solutions for the case of $\tilde{T}>\tilde{T}_c>0$. What does that imply for the species?
e) Discuss very briefly the differences in the dynamics to the Hutchinson model.
f) Based on the structure of $P(T, N(t-T))$, what is an implicit assumption in the survival probability $P(T, N(t-T))$ ? Argue its biological applicability and propose a modification.
These are the series of problems where I am stuck and not getting any clue so any help in parts will be useful. I know Mathematica, Python, MATLAB
Here the Hutchinson model is $$ N^{\prime}={r N\left(1-\frac{N(t-T)}{K}\right)} $$ with growth rate $r>0$, carrying capacity $K>0$.
I will also try to edit if I figure out something more.