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

The Verhulst‑Pearl logistic growth model describes how a population  grows over time while being limited by a maximum sustainable size called the carrying capacity .

Key components of the model:

1. Intrinsic growth rate r: the rate at which the population would grow without any constraints.

2. Current population N: the number of individuals at time .

3. Limiting factor $\frac{K-N}{K}$ which measures how close the population is to the carrying capacity. When  is small, this factor is near 1, allowing near‑exponential growth. As  approaches , the factor approaches 0, slowing growth.

Combining these elements yields the differential equation:

$\frac{dN}{dt}=rN\left(\frac{K-N}{K}\right)$

Explanation of each term:

• $rN$ represents the unrestricted exponential growth (the classic “Malthusian” term).

• Multiplying by $\frac{K-N}{K}$ introduces a negative feedback that reduces growth as  gets closer to .

This equation can be rearranged to the more familiar form $dN/dt=\; rN\; \backslash left(1\; -\; \backslash frac\{N\}\{K\}\backslash right)$, which is algebraically identical because:

$\frac{K-N}{K}=\; 1\; -\; \backslash frac\{N\}\{K\}$

Therefore, the equation listed above precisely matches the Verhulst‑Pearl logistic growth model.