A student used the mark-recapture method to assess the population size of grasshoppers in a field. The student was asked to repeat the recapture procedure once on three consecutive days. The procedure followed by the student and the observations made are as follows:

• A. On day one, 40 grasshoppers were captured, marked, and released back in the field.
• B. On day two, 60 grasshoppers were re-captured, of which 4 were marked. He marked the unmarked ones and released all 60 in the field.
• C. On day three, 50 grasshoppers were re-captured, of which 7 were marked. He marked the unmarked ones and released all 50 in the field.
• D. On day four, 25 grasshoppers were re-captured, of which 6 were marked.

The student was asked to calculate the population size based on the mean of the three observations. The estimated population size is:

Correct option is B

Solution:

​The mark-recapture method is used to estimate the population size of grasshoppers. This method is based on the assumption that the ratio of marked to total individuals in the sample is the same as the ratio of marked to total individuals in the entire population.

The population size ($N$) can be estimated using the Lincoln-Petersen formula for each day:

N=(M×C)RN = \frac{(M \times C)}{R}N= $$\frac{M \times C}{R}$$​

N=RM×CN = R M \times 

Where:

• $M$ = Total number of marked individuals in the population (before the recapture on that day),
• $C$ = Total number of individuals captured on the day,
• $R$ = Number of marked individuals in the recaptured sample.

Day 2:

• $M=40$ (from day 1, all marked)
• $C=60$ (recaptured grasshoppers)
• $R=4$ (marked in the recapture sample)

Using the formula:

​$$N_2$$= $$\frac{(40\times60)}{4}$$

=600

N2N_2N2​​​

Day 3:

• $M=40+(60-4)=96$ (40 already marked, plus 56 newly marked grasshoppers)
• $C=50$ (recaptured grasshoppers)
• $R=7$ (marked in the recapture sample)

Using the formula:

​$$N_3$$ = $$\frac{96\times50}{7}$$

=685.71 ~686

Day 4:

• $M=96+(50-7)=139$ (96 already marked, plus 43 newly marked grasshoppers)
• $C=25$ (recaptured grasshoppers)
• $R=6$ (marked in the recapture sample)

Using the formula:

$$N_4$$ = $$\frac{(139\times 25)}{6}$$ =579.17 ~579​​

Average Population Size:

Now, the average population size can be calculated from the three estimates (N2,N3,N4N_2, N_3, N_4N2​,N3​,N4​):

Mean=(600+686+579)3=18653≈621.67Mean = \frac{(600 + 686 + 579)}{3} = \frac{1865}{3} \approx 621.67Mean= $$\frac{600 + 686 + 579}{3}$$=​ $$\frac{1865}{3}$$​≈621.67

Thus, the estimated population size based on the mean of the three observations is approximately 622 grasshoppers.