Question

A feral house mouse population can increase at \(r=\) 0.0246 per day. At this rate of increase, how many days are needed for the population to double?

Short answer

The population will double in approximately 28.17 days.

Step-by-step solution

Understand Doubling Time Formula

In continuous growth scenarios, doubling time can be calculated using the formula \( T = \frac{\ln(2)}{r} \), where \( r \) is the growth rate.

Substitute given growth rate

Here, the growth rate is given as \( r = 0.0246 \) per day. Substitute this value into the formula: \[ T = \frac{\ln(2)}{0.0246} \]

Calculate Natural Logarithm of 2

Compute \( \ln(2) \). This is a constant and is approximately equal to 0.693.

Calculate Doubling Time

Now substitute \( \ln(2) \approx 0.693 \) into the formula: \[ T = \frac{0.693}{0.0246} \]. Divide 0.693 by 0.0246 to find the doubling time.

Perform Division

Perform the division \( \frac{0.693}{0.0246} \) using a calculator. The result is approximately 28.17.

Conclusion

The calculation shows that it takes approximately 28.17 days for the population to double, given a growth rate of 0.0246 per day.

Key concepts

Doubling Time

Doubling time is a key concept in population dynamics and refers to the time it takes for a population to double in size at a constant growth rate. It is especially relevant in scenarios of exponential growth, where populations increase rapidly. The formula to calculate doubling time when growth is continuous is given by

• \( T = \frac{\ln(2)}{r} \),

where \( T \) is the doubling time and \( r \) is the growth rate. The natural logarithm of 2, \( \ln(2) \), is approximately 0.693.
For instance, if we have a feral house mouse population growing at a rate of 0.0246 per day, we can determine the time required for the population to double.
Substituting the growth rate into the formula gives:
\[ T = \frac{0.693}{0.0246} \].Performing this calculation reveals that the doubling time is approximately 28.17 days.
This concept helps us understand how quickly biological populations can expand under ideal conditions.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases with passing time, forming a curve on a graph. In population dynamics, it occurs when the growth rate of the value of a mathematical function is proportional to the function’s current value.
This situation often results in populations expanding rapidly.
For example, if a population of feral house mice grows continuously at a certain rate per day, each individual mouse contributes to the creation of more mice, hence compounding the overall population.
Exponential growth can be characterized by the equation

• \( P(t) = P_0 e^{rt} \),

Where:

• \( P(t) \) is the future population size,
• \( P_0 \) is the initial population size,
• \( r \) is the growth rate,
• and \( t \) is time in consistent measurement units with \( r \) (e.g., days).

A critical trait of exponential growth is its accelerating growth rate; the larger the population, the faster it grows. Understandably, this makes exponential growth unsustainable in the long term unless checked by limiting factors such as resource availability or space constraints.

Growth Rate Calculation

Calculating the growth rate is essential for understanding the rate at which a population or any quantifiable entity increases over time. The growth rate is principally used to forecast future sizes or states of populations. In population dynamics, it can be referred to in terms of continuous compounding.
To find the growth rate per time unit (often a day or a year for biological populations), the following formula is used:

• \( r = \frac{(P_t - P_0)}{P_0} \times \frac{1}{t} \),

where \( P_t \) is the population at time \( t \), \( P_0 \) is the initial population, and \( t \) is the time elapsed.
In this context, when the growth rate \( r \) was stated as 0.0246 per day, it implies that each day, the population grows by approximately 2.46% of its current size, highlighting its capacity for swift expansion.
Understanding how to compute and apply growth rates allows researchers and planners to handle population challenges and sustainability issues effectively.