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?
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In population studies, consistent birth rates simplify the analysis of death rates, highlighting changes due solely to mortality.
The death rate during April–June is equal to that during October–December.
The death rate during July–September is lower than that during January–March.
The death rate during July–September is higher than that during January–March.
The death rate during April–June is higher than that during October–December.
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The Correct Option isA, C
Solution and Explanation
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.
