Verhulst pearl logistic population growth is represented by the equat…

The correct answer is Option (3) → \(\frac{dN}{dt}\) = rN(\(\frac{K-N}{K}\))

In logistic growth the population grows fast at first, then slows down and levels off at the carrying capacity (K), the largest number the habitat can support. The term (K − N)/K does this: when N is small it is close to 1, and when N reaches K it becomes 0, so growth stops.

Option (1) \(\frac{dt}{dN}\) = rN(\(\frac{K-N}{K}\)) — Incorrect. Growth rate is the change in number with time, dN/dt; this option has it upside down.

Option (2) \(\frac{dN}{dt}\) = rN(\(\frac{K}{K-N}\)) — Incorrect. Here the term grows larger as N nears K, so growth would speed up instead of stopping.

Option (3) \(\frac{dN}{dt}\) = rN(\(\frac{K-N}{K}\)) — Correct. This is the Verhulst-Pearl logistic equation; it gives an S-shaped (sigmoid) curve.

Option (4) \(\frac{dt}{dN}\) = rN(\(\frac{K}{K-N}\)) — Incorrect. Both the rate and the bracket are upside down.