Which one of the following equations represents the Verhulst-Pearl Logistic Growth of population?

The Verhulst‑Pearl logistic growth model describes how a population N changes over time t when there is a limiting “carrying capacity’’ K. It starts from the simple exponential growth law and adds a factor that slows growth as N approaches K.

1. Exponential growth assumes the per‑capita growth rate is constant r:

$\frac{dN}{dt}=\; r\; N$

2. Limiting factor – when resources become scarce, the growth rate should decline linearly with the fraction of carrying capacity that remains unused. This fraction is (K − N)/K.

3. Combine the two pieces: multiply the exponential term by the limiting factor:

$\frac{dN}{dt}=\; r\; N · \backslash frac\{K‑N\}\{K\}$

4. Rewrite for clarity: the right‑hand side can be written as r N ((K − N)/K), which is exactly the equation in the correct answer.

Thus the logistic equation captures two biological realities:

• The growth is proportional to the current population (r N).
• The growth is reduced by the factor (K − N)/K, which goes to zero as N approaches the carrying capacity K, preventing unlimited growth.

Therefore, the equation

$\frac{dN}{dt}=\; r\; N\; \backslash bigg(\; \backslash frac\{K‑N\}\{K\}\; \backslash bigg)$

is the correct representation of the Verhulst‑Pearl logistic growth model.