Problem 38 - Algebra 2

Question

For Exercises \(36-38,\) use the following information. The population of predators and prey in a closed ecological system tends to vary periodically over time. In a certain system, the population of owls \(O\) can be represented by \(O=150+30 \sin \left(\frac{\pi}{10} t\right)\) where \(t\) is the time in years since January \(1,2001 .\) In that same system, the population of mice \(M\) can be represented by \(M=600+300 \sin \left(\frac{\pi}{10} t+\frac{\pi}{20}\right)\) Why would the maximum owl population follow behind the population of mice?

Step-by-Step Solution

Answer

The phase shift causes mice to reach peak population before owls, leading to owls' peak lagging behind.

Step 1: Understanding the Sine Functions

Both the populations of owls and mice are represented by sinusoidal functions of time. The general form of a sine function, \(y = A \sin(Bx + C) + D\), indicates periodic behavior. The coefficients present in each function's equation provide details regarding amplitude, frequency, phase shift, and vertical shift.

Step 2: Comparing Phase Shifts

The owl population is given by \(O = 150 + 30 \sin\left(\frac{\pi}{10}t\right)\) and the mouse population is given by \(M = 600 + 300 \sin\left(\frac{\pi}{10}t + \frac{\pi}{20}\right)\). The key to understanding the relationship is in the phase shift. The owl function has no additional phase shift \((\frac{\pi}{10}t)\), while the mouse function has a phase shift of \(+\frac{\pi}{20}\).

Step 3: Calculating the Phase Shift Impact

The phase shift in the mouse population function affects its peak times compared to the owl function which has no phase shift. The term \(\frac{\pi}{20}\) indicates that the mice reach their maximum population \(\frac{\pi}{20}\) radians or \(\frac{1}{20}\) of the period earlier than the owls. Given \(B = \frac{\pi}{10}\), the period of both functions is \(\frac{2\pi}{\frac{\pi}{10}} = 20\) years.

Step 4: Understanding the Population Relationship

Since the problem describes a predator-prey relationship, increases in the prey population (mice) will naturally be followed by increases in the predator population (owls). The phase shift indicates that the mice reach their peak population first, providing more food for the owls, which subsequently increases the owl population.

Key Concepts

Sinusoidal Functions

Sinusoidal functions are a type of mathematical function that graphically represent waves. They are commonly used to model periodic behaviors like sound waves, light waves, and in this case, animal populations. The general form of a sinusoidal function is given by:

• \( y = A \, \sin(Bx + C) + D \)

Here,

• \( A \) is the amplitude, indicating the height of the wave from its midline.
• \( B \) affects the frequency, showing how many cycles occur in a given interval.
• \( C \) is the phase shift, determining the horizontal shift of the wave.
• \( D \) is the vertical shift, moving the wave up or down on the graph.

In our exercise, both owl and mouse populations are described using sinusoidal functions. These functions convey crucial information through their structure, such as population peaks and troughs, and overall trends over time.

Phase Shift

Phase shift in sinusoidal functions dictates the horizontal shift of the wave along the x-axis. This term plays a significant role in determining the timing of wave peaks and troughs. In the context of this exercise,
the phase shift helps us understand the time difference between peak populations of predators and prey.

• The owl population function, \( O = 150 + 30 \, \sin \left(\frac{\pi}{10}t\right) \), has no additional phase shift. This means its peaks occur at integer multiples of the period.
• The mouse population function, \( M = 600 + 300 \, \sin \left(\frac{\pi}{10}t + \frac{\pi}{20}\right) \), includes a phase shift of \( +\frac{\pi}{20} \), advancing its peaks earlier relative to the owl population.

This advance is quantified as \( \frac{\pi}{20} \) radians, causing the timing of the mouse population peaks to precede those of the owls by \( \frac{1}{20} \) of their period.
This phased aspect ensures that peaks in prey availability happen before predator peaks, explaining predator-prey dynamics.

Predator-Prey Relationship

In ecology, predator-prey relationships describe natural interactions where one organism (the predator) hunts another organism (the prey). This cyclical interaction helps maintain ecological balance through self-regulating dynamics. In our exercise:

• The mice serve as prey. Their population cycles influence the owls, who rely on mice as a food source.
• Owls are predators, and their population dynamics follow those of the mice, but slightly staggered due to reproductive and hunting lag time.

The modeled functions show how prey availability dictates predator success. Peak mouse populations result in ample food, enhancing owl population growth post-peak. This delay, captured as phase shifts, is critical as it aligns the reproduction and consumption cycles.

Periodic Behavior

Periodic behavior refers to phenomena that repeat at regular intervals. Such behavior is a hallmark of many natural systems, including ecological cycles.
In the context of this exercise:

• The populations of owls and mice both exhibit periodic behavior, captured by their sinusoidal representations.
• The periodicity stems from environmental factors that cyclically influence food availability, reproduction patterns, and environmental conditions.

Both the owl and mouse population functions have a calculated period of 20 years. This means every 20 years,
the full cycle of rise and fall in population numbers repeats. Periodicity helps researchers and ecologists predict future population dynamics based on past data.
Understanding periodic behavior is vital for managing ecosystems sustainably, ensuring that predator-prey balances stay healthy and intact.