Understanding Intraspecific Competition

The question asks us to identify the value of the term \(\frac{(K-N)}{K}\) in the logistic growth equation where intraspecific competition is likely to be the highest. Intraspecific competition refers to the competition between individuals of the same species for limited resources such as food, water, shelter, and mates. The given equation is the logistic growth model: \[ \rm \frac{dn}{dt}=rN\frac{(K-N)}{K} \] Here:

• \(\rm \frac{dn}{dt}\) represents the rate of population growth.
• \(\rm r\) is the intrinsic rate of natural increase, which is the maximum potential growth rate per individual.
• \(\rm N\) is the current population size.
• \(\rm K\) is the carrying capacity, which is the maximum population size that the environment can sustain.
• The term \(\frac{(K-N)}{K}\) is the density-dependent factor or environmental resistance factor. It shows how much of the carrying capacity is still available for population growth.

Competition and the (K-N)/K Term

The logistic growth equation shows that the population growth rate depends on the intrinsic growth rate, the current population size, and the density-dependent factor \(\frac{(K-N)}{K}\). Let's analyze the term \(\frac{(K-N)}{K}\):

• When the population size \(\rm N\) is much smaller than the carrying capacity \(\rm K\), \(\rm (K-N)\) is close to \(\rm K\), so \(\frac{(K-N)}{K}\) is close to 1. In this situation, resources are relatively abundant per individual, and intraspecific competition is low. The population grows almost exponentially.
• As the population size \(\rm N\) approaches the carrying capacity \(\rm K\), \(\rm (K-N)\) becomes small, and \(\frac{(K-N)}{K}\) approaches 0. When \(\rm N\) is close to \(\rm K\), resources are scarce relative to the population size. Individuals compete intensely for these limited resources, leading to high intraspecific competition. The population growth rate slows down.
• When \(\rm N\) equals \(\rm K\), \(\rm (K-N) = 0\), so \(\frac{(K-N)}{K} = 0\). The population growth rate becomes zero, and the population size stabilizes at the carrying capacity. Competition is very high at or near the carrying capacity.

Therefore, intraspecific competition is highest when the population size \(\rm N\) is closest to the carrying capacity \(\rm K\). This happens when the term \(\frac{(K-N)}{K}\) is closest to zero.

Comparing Values for Highest Competition

We are given several possible values for the term \(\frac{(K-N)}{K}\):

• 0.001
• 0.009
• 0.15
• 0.015

To find where intraspecific competition is likely to be the highest, we need to find the value of \(\frac{(K-N)}{K}\) that is closest to 0. Let's list the values and compare them:

• 0.15
• 0.015
• 0.009
• 0.001

The value closest to zero among the options is 0.001. A value of 0.001 for \(\frac{(K-N)}{K}\) implies that \(K-N\) is very small compared to \(K\), meaning \(\rm N\) is very close to \(\rm K\). This condition corresponds to the highest level of intraspecific competition among the given options, as the population is near the carrying capacity and resources are most limited. Thus, intraspecific competition is likely to be highest when \(\frac{(K-N)}{K} = 0.001\).