Question

The population of whirligig beetles in a lake grows or declines exponentially, i.e., \( N(t) = N(0)e^{rt} \), where \( N(t) \) is the population size at time \( t \), \( N(0) \) is the initial population size, and \( r \) is the per capita rate of population change, occurring only due to birth and death.
A researcher tracks population sizes for a year and finds the following:

Assuming that the individual birth rates remain constant throughout the year and only death rates are affected, which one or more of the following statements is/are true?

• A. The death rate during April–June is equal to that during October–December.
• B. The death rate during July–September is lower than that during January–March.
• C. The death rate during July–September is higher than that during January–March.
• D. The death rate during April–June is higher than that during October–December.

Hint:

In population studies, consistent birth rates simplify the analysis of death rates, highlighting changes due solely to mortality.

Answer 1

The Correct Option is A, C

Step 1: Calculate the growth rate (\( r \)) for each interval by rearranging the exponential growth formula to \( r = \frac{1}{t} \log\left(\frac{N(t)}{N(0)}\right) \). 
Step 2: Apply the formula for each quarter to determine \( r \) values, noting that a negative \( r \) indicates a net loss in population, thus a higher death rate. 
Step 3: Comparison of \( r \) values shows that the death rates in July–September are notably higher than in January–March, indicating more significant population decline during these months. 
Step 4: Observe that the periods April–June and October–December have similar \( r \) values, indicating similar death rates assuming constant birth rates.