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Chapter 1: Problem 41

Imagine that you have become a biologist, studying rats in Indonesia. Most of the time, Indonesian rats maintain a constant population. Every half century, however, these rats suddenly begin to multiply exponentially! Then the population crashes back to the constant level. Sketch a graph that shows the rat population over two of these episodes.

Short Answer

Expert verified
Draw two peaks for exponential growth and crashes at 50 and 100 years, with constant population lines in between.

Step by step solution

01

- Understand the Population Dynamics

The rat population in Indonesia mostly remains constant, but it experiences exponential growth and then a crash every half-century. This pattern repeats, so we need to illustrate this over two cycles.
02

- Determine the Time Period

Each cycle lasts 50 years, and since there are two cycles, our graph will cover 100 years.
03

- Plot the Constant Population Line

On the graph, draw a horizontal line representing the constant rat population. This line should be at a certain height representing the population level during non-growth periods.
04

- Plot Exponential Growth

For the first half-century (0 to 50 years), include an exponentially increasing curve starting at the constant population line, reaching a peak before 50 years.
05

- Plot the Population Crash

Immediately after the peak, draw a steep decline back to the constant population level just before the 50-year mark.
06

- Repeat for the Second Cycle

For the second half-century (50 to 100 years), repeat steps 4 and 5. Plot another exponential growth starting at year 50, reaching a peak and then crashing back to the constant level before year 100.
07

- Label the Axes

Label the x-axis as 'Time (years)' ranging from 0 to 100, and the y-axis as 'Rat Population'. Include markers to show the 50-year intervals and peaks.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Growth
Exponential growth is a process where the quantity of something increases at a constant percentage rate over time. In the context of population dynamics, this means that the population size will double, triple, or even increase tenfold within a certain period depending on the growth rate.

This kind of growth often occurs under ideal conditions where resources are abundant, and there are few predators or diseases to limit population size. Mathematically, exponential growth can be expressed as:
\[ P(t) = P_0 e^{rt} \]
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