Question

Which of the following population growth equation correctly represents the Verhulst-Pearl Logistic growth equation? [Where N =population density at time t; r = intrinsic rate of natural increase; K= carrying capacity]

• A. \( \frac{dN}{dt} = rN \frac{(N-K)}{K} \)
• B. \( \frac{dN}{dt} = rN \frac{(K-N)}{K} \)
• C. \( \frac{dN}{dt} = rN \frac{(K-N)}{N} \)
• D. \( \frac{dN}{dt} = rN \frac{(N-K)}{N} \)

Hint:

To remember the logistic growth term, think of it as "how much room is left for growth". \( (K-N) \) is the remaining capacity, and dividing by K, \( \frac{(K-N)}{K} \), turns it into a proportion. This proportion multiplies the exponential growth rate to slow it down.

Answer 1

The Correct Option is B

Step 1: Conceptual Understanding: The Logistic Growth Equation models population expansion under resource constraints. Initial growth is exponential but decelerates as the population size (N) approaches the environment's carrying capacity (K).

Step 2: Core Formula/Methodology: The equation originates from the exponential growth model, \( \frac{dN}{dt} = rN \). Environmental resistance is incorporated by multiplying this by a factor that diminishes towards zero as N approaches K. This factor is \( \frac{(K-N)}{K} \).

Step 3: Detailed Elaboration: The complete equation is: \[ \frac{dN}{dt} = rN \left( \frac{K-N}{K} \right) \] Analyzing the term \( \left( \frac{K-N}{K} \right) \): \[\begin{array}{rl} \bullet & \text{When N is significantly smaller than K, \( \frac{(K-N)}{K} \) is approximately 1, resulting in near-exponential growth (\( \frac{dN}{dt} \approx rN \)).} \\ \bullet & \text{As N increases towards K, \( \frac{(K-N)}{K} \) approaches 0, thereby reducing the population growth rate.} \\ \bullet & \text{When N equals K, \( \frac{(K-N)}{K} \) becomes 0, halting population growth (\( \frac{dN}{dt} = 0 \)).} \\ \end{array}\] Option (B) accurately reflects this dynamic.

Step 4: Conclusion: The correct Verhulst-Pearl Logistic growth equation is \( \frac{dN}{dt} = rN \frac{(K-N)}{K} \).