Which of the following equations depicts Verhulst-Pearl logistic population growth ?

The correct equation that represents the Verhulst‑Pearl logistic population growth is

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

Here, N denotes the population size at time t, r is the intrinsic growth rate, and K is the environmental carrying capacity—the maximum population that the environment can sustain.

To understand why this form is the logistic model, start with the simple exponential growth law:

\frac{dN}{dt}=rN

This equation assumes unlimited resources, so the growth rate is proportional to the current population.

In reality, resources become scarce as the population approaches the carrying capacity K. Verhulst introduced a limiting factor that reduces the effective growth rate when N gets close to K. The factor is:

1-\frac{N}{K}

Multiplying the exponential term by this factor yields the logistic differential equation:

\frac{dN}{dt}=rN(1-\frac{N}{K})

Re‑arranging the term 1 − N/K gives the equivalent expression K−NK, which matches the provided answer exactly.

This form captures three essential behaviours:

• When N ≪ K, the fraction K−NK ≈ 1, so the population grows nearly exponentially.
• When N = K, the fraction becomes zero, halting growth—population stabilises at the carrying capacity.
• When N > K, the fraction becomes negative, causing a decline back toward K.

Thus, the equation \frac{dN}{dt}=rN\left(\frac{K-N}{K}\right) correctly represents the Verhulst‑Pearl logistic population growth model.