Question

Asymptote in a logistic growth curve is obtained when :

• A. K = N
• B. K > N
• C. K < N
• D. The value of 'r' approaches zero

Answer 1

The Correct Option is A

 To solve the given question about asymptotes in a logistic growth curve, we need to understand the basics of the logistic growth model in population biology.

The logistic growth model is depicted by the equation:

\(N(t) = \frac{K}{1 + Ae^{-rt}}\)

where:

• \(N(t)\) is the population size at time \(t\).
• \(K\) is the carrying capacity of the environment.
• \(r\) is the intrinsic growth rate.
• \(A\) is a constant determined by initial conditions.

 

The logistic growth curve is an S-shaped curve where the population grows rapidly at first, then slows as it approaches the carrying capacity \(K\). The asymptote of this curve is the value that the population size approaches, which is the carrying capacity \(K\).

Now, let’s examine the options:

• Option K = N: This is correct because when the population size \(N\) reaches the carrying capacity \(K\), growth rate approaches zero, and the population size approaches an asymptote.
• Option K > N: This implies the population size is less than the carrying capacity, meaning the population is still growing and has not yet reached the asymptote.
• Option K < N: This situation is not possible in a logical model as populations do not exceed the carrying capacity \(K\) sustainably.
• Option The value of 'r' approaches zero: While a decrease in 'r' lowers the growth rate, it does not define where the asymptote lies. It affects the speed of achieving the growth limit.

Therefore, the correct answer is K = N, as this is when the population has reached the carrying capacity, establishing the asymptote of the logistic growth curve.