Is it possible to derive the Logistic model growth like exponential growth model?
Exponential growth equation derivation.
Let $P(t)$ denote the population of a city at time $t$ years. $P(t+\Delta t)$ denote the population of a city at time $t+\Delta t$ years. $b$ denote the total number of birth per individual per year. $d$ denote the total number of death per individual per year. $$P(t+\Delta t)=P(t)+(b-d)P(t)\Delta t$$
$$\frac{P(t+\Delta t)-P(t)}{\Delta t}=(b-d)P(t)$$
As $\Delta t \to 0,$ We get $\frac{dP(t)}{dt}=(b-d)P(t).$ Similarly how to derive the logistic model of growth?
$$\frac{dP}{dt}=cP(P-K)/K$$