Question

When does the growth rate of a population following the logistic model equal zero? The logistic model is given as dN/dt = rN(1-N/K) :

• A. When N nears the carrying capacity of the habitat
• B. When N/K equals zero.
• C. When death rate is greater than birth rate
• D. When N/K is exactly one.

Answer 1

The Correct Option is D

Logistic Growth Model 

The logistic growth model describes how a population grows with limited resources. The population growth is governed by the following equation:

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

Where:

• N = Population density at time \( t \)
• r = Intrinsic rate of natural increase
• K = Carrying capacity of the environment

In the logistic growth model, when the population density reaches the carrying capacity \( K \), the growth rate decreases and eventually reaches zero.

When \( N/K = 1 \):

At the point when the population density \( N \) reaches the carrying capacity \( K \), we have:

\(\frac{K - N}{K} = 0\)

Substituting this into the growth equation:

\(\frac{dN}{dt} = 0\)

Conclusion:

When the population reaches its carrying capacity (\( N = K \)), the growth rate becomes zero, meaning the population stops growing.