To understand logistic growth, (say for example population growth) we employ a Verhulst-Pearl logistic equation and introduce a time lag to account for the fact that the system under study takes some time to respond to any perturbation and hence must also respond to the increasing population after a time lag. This time lag depends on the nature of the system and predicts different outcomes.In case of population growth, it accounts for the time lag in environmental response to increasing population. Below is the Verhulst-Pearl logistic equation to include a time lag $\tau$:-
$$\frac{dN_t}{dt}=rN_t \frac{K-N_{t-\tau}}{K}$$
Where $N_t$ stands for the net population at time $t$, and $r$ and $K$ are constants specific to the system.
I know this is too complex to solve for in the general case or even for a predefined small $\tau$, but can someone help me visualise how this equation behaves and what outcomes does it predict for different values of $\tau$ and $K$. I used wolframalpha Mathematica but i couldn't get a plot on it (probably because I am not comfortable with plotting differential equations on it).
The source from where I read it also included a passing reference to what it's graph might look like. It ($N(t)$) rises approximately sigmoidally till it equals $K$ and then oscillates about depending on $\tau$. Moreover, for large values of $\tau$ the final oscillations become wild and therefore are also relevant in chaos theory.
Best Answer
In this link you can find actual solutions of the delayed logistic equation $$ y'=ay(1-y(t-1)) $$ depending on $a$. You can also learn there how to model this in Mathematica and modify to include the cases of interest for you.