Logistic population equation and exponential model In  Verhulst's model of population growth and control, Let $K$ represent the carrying capacity for a particular organism in a given environment, and let $r$ be a real number that represents the growth rate. The function $P(t)$ represents the population of this organism as a function of time $t$, and the constant $P_0$ represents the initial population (population of the organism at time $t=0$. Then the logistic differential equation is
$${dP}/{dt}=rP\left(1−\dfrac{P}{K}\right)$$.
I have understood the cause for doing this, where $K\gg P$ the value of parenthesis is 1 and follows the normal exponential equation: $${dP}/{dt}=rP$$
However, isn't the original exponential equation a 1st order LINEAR differential equation...whereas the logistic model equation seems to me to be a 1st order NON Linear differential equation. Is that okay because we are describing the same system in the end.
 A: Everything is "okay" as long as it describes what you observe. Posing ordinary differential equations is not an exercise in reproducing academic showcases, but rather serves the purpose of predicting the future from the current state.
In your case, the linear differential equation (exponential growth) is an approximation of the nonlinear differential equation (logistic growth) for the case $P \to 0$. The former is easier to solve (and is probably accurate enough in case of high capacity systems), while the latter is more accurate (with respect to biological systems - possibly), but also more difficult to solve.
A: When $P$ is small the growth can be approximated with exponential growth. Exponential growth means that after a while $P$ isn't small any more and you need the full logistic equation to accurately model the growth.
In this example you see the two cases compared. In both cases $P(0)=0.1$ and $r=1$, and the logistic growth has $K=5$. In the beginning they match up nicely but as $P$ gets larger the curves diverge.

A: The statement is just that the limiting case $K \gg P$ the non-linear logistic equation can be approximated by a linear equation. As long as you remain in the domain where $K \gg P$ the results given by the two equations agree.
Once you break this assumption the approximation with the linear equation will no longer hold, and the results given by the two equations no longer match. The logisitic equation is of course more realistic for a large population, as resources will be finite, so at some point the population can't grow anymore.
This is similar, to how you can approximate any analytic function with a linear function around some point. It is valid, as long as the assumptions are valid, but the approximation doesn't show the whole picture.
