Question

Consider a population that shows logistic growth of the form 
\[ \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right) \] where \(\frac{dN}{dt}\) is the population growth rate, \(r\) is the instantaneous rate of increase, \(K\) is the carrying capacity and \(N\) is the population size.
For such a population \((N > 0)\), which one of the following graphs shows the correct relationship between per capita growth rate \((\frac{1}{N} \frac{dN}{dt})\) on the y-axis, and population size (\(N\)) on the x-axis? 

• A. (i)
• B. (ii)
• C. (iii)
• D. (iv)

Hint:

In logistic growth, per capita growth rate is highest when the population is small and declines as the population approaches the carrying capacity.

Answer 1

The Correct Option is A

The logistic growth equation models population growth where the rate of growth is proportional to both the current population size and the amount of resources available (which are limited by the carrying capacity, \(K\)).
To determine the relationship between per capita growth rate \(\frac{1}{N} \frac{dN}{dt}\) and population size \(N\), we start by simplifying the logistic growth equation:
\[ \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right) \] The per capita growth rate is defined as:
\[ \frac{1}{N} \frac{dN}{dt} = r \left( 1 - \frac{N}{K} \right) \] This equation shows that the per capita growth rate is high when \(N\) is small (far from the carrying capacity \(K\)) and decreases as the population size \(N\) approaches the carrying capacity \(K\). At \(N = K\), the per capita growth rate becomes zero because the population is at equilibrium, and no further growth occurs.
This relationship is represented in graph (i), which shows a curve where the per capita growth rate is highest at low population sizes, gradually decreases, and becomes zero as the population size approaches the carrying capacity.
Thus, the correct answer is (A), graph (i).