How to solve this DDE: \begin{align} \frac{1}{N_t} \frac{dN_t}{dt}=r\left(1-\frac{N_{t-\tau}}{K}\right) \end{align} where $N_0,K,r,\tau$ are constant?
This differential equation is based on logistic model, which adds a delay $\tau$ to its equation, namely delayed logistic model.
To simplify, denote $y(t)=\log N_t$. Then we have a simple differential-difference equation $$y'(t)=f(y(t-\tau)),\quad f(y)=r(1-y/K).$$ Let $\phi$ be an arbitrary continuous function on $[0,\tau]$ which has one sided derivative at $\tau$ satisfying $\phi'(\tau-)=\phi(0)$. Then the formulas $Y(t)=\phi(t), 0\leq t\leq \tau$, and $$y(t)=y(n\tau)+\int_{n\tau}^t f(y(t-\tau))dt, \quad n\tau\leq t\leq (n+1)\tau,\quad n\geq 1$$ defines a continuous solution which is everywhere continuous and continuously differentiable for $t\geq \tau$. Then $N_t=\exp(y(t))$ defines a solution of your problem.
