The mathematical expression that represents the logistic growth in a population is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\frac{dN}{dt}=rN(\frac{K-N}{K})\)
In logistic growth the growth rate slows as the population nears the carrying capacity K: \(\frac{dN}{dt}=rN\left(\frac{K-N}{K}\right)\).
Option (1) \(\frac{dN}{dt}=rN(\frac{K}{K-N})\) — Incorrect. The fraction is upside down; growth would rise without limit as N nears K.
Option (2) \(\frac{dN}{dt}=rN(\frac{K-N}{K})\) — Correct. The Verhulst-Pearl logistic growth is written dN/dt = rN((K − N)/K), where r is the intrinsic rate of natural increase and K the carrying capacity. Growth slows as N nears K, giving an S-shaped curve.
Option (3) \(\frac{dN}{dt}=rN\) — Incorrect. \(\frac{dN}{dt}=rN\) is exponential growth.
Option (4) \(\frac{dN}{dt}=rN(K-N)\) — Incorrect. Without dividing by K the term is not a fraction of the carrying capacity.