Suppose that the increase in a population can be modelled as

where N is the size of the population, K is the carrying capacity, r is the per capita growth rate and t
is time. Which of the following statements is correct?

Correct option is A

Given:

The population growth is modeled as:
dN/dt = rN (K - N) / K,
where:

• N is the size of the population,
• K is the carrying capacity,
• r is the per capita growth rate,
• t is time.

Solution:
The equation describes the rate of change of a population, considering both its size and the carrying capacity (K). Let us analyze each statement:

1. When N is close to 0:
• The term (K - N) / K approaches 1 because N / K is almost 0.
• Substituting this into the equation, dN/dt ≈ rN, which represents exponential growth.
2. When N equals K:
• The term (K - N) / K becomes 0, so dN/dt = 0.
• This means the population stops growing but does not go extinct.
3. When N is close to 0:
• The growth rate dN/dt is not maximum here; instead, it grows exponentially as explained earlier.
4. When N is around K/4:
• The population growth rate is maximum when N is approximately K/4. This is because the growth term (K - N) balances with N to maximize dN/dt.

The correct statement is: (a) When N is close to 0, the change in population N is nearly exponential.