Question

Predator Population In predator-prey relationships, the populations of the predator and prey are often cyclical. In a conservation area, rangers monitor the red fox population and have determined that the population can be modeled by the function $$ P(t)=40 \cos \left(\frac{\pi t}{6}\right)+110 $$ where \(t\) is the number of months from the time monitoring began. Use the model to estimate the population of red foxes in the conservation area after 10 months, 20 months, and 30 months.

Short answer

After 10 months: 130 red foxes; after 20 months: 90 red foxes; after 30 months: 70 red foxes.

Step-by-step solution

- Identify the given function

The population of red foxes is modeled by the function: \[ P(t) = 40 \, \cos \left(\frac{\pi t}{6}\right) + 110 \] where \(t\) is the number of months from the time monitoring began.

- Substitute 10 months into the function

To find the population after 10 months, substitute \(t = 10\) into the function: \[ P(10) = 40 \, \cos \left(\frac{\pi \times 10}{6}\right) + 110 \] Simplify the argument of the cosine function: \[ P(10) = 40 \, \cos \left(\frac{10\pi}{6}\right) + 110 \] Reduce the fraction: \[ P(10) = 40 \, \cos \left(\frac{5\pi}{3}\right) + 110 \]

- Calculate the cosine value

The cosine of \(\frac{5\pi}{3}\): \[ \cos \left(\frac{5\pi}{3}\right) = \cos \left(2\pi - \frac{\pi}{3}\right) = \cos \left(- \frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \] So, substitute that into the function: \[ P(10) = 40 \times \frac{1}{2} + 110 = 20 + 110 = 130 \] Therefore, the population after 10 months is 130 red foxes.

- Substitute 20 months into the function

Similarly, to find the population after 20 months, substitute \(t = 20\) into the function: \[ P(20) = 40 \, \cos \left(\frac{\pi \times 20}{6}\right) + 110 \] Simplify the argument: \[ P(20) = 40 \, \cos \left(\frac{20\pi}{6}\right) + 110 \] Reduce the fraction: \[ P(20) = 40 \, \cos \left(\frac{10\pi}{3}\right) + 110 \]

- Calculate the cosine value

The cosine of \(\frac{10\pi}{3}\) is equivalent to \(\frac{4\pi}{3}\): \[ \cos \left(\frac{4\pi}{3}\right) = -\frac{1}{2} \] So, substitute that into the function: \[ P(20) = 40 \times -\frac{1}{2} + 110 = -20 + 110 = 90 \] Therefore, the population after 20 months is 90 red foxes.

- Substitute 30 months into the function

Finally, to find the population after 30 months, substitute \(t = 30\) into the function: \[ P(30) = 40 \, \cos \left(\frac{\pi \times 30}{6}\right) + 110 \] Simplify the argument: \[ P(30) = 40 \, \cos \left(\frac{30\pi}{6}\right) + 110 \] Reduce the fraction: \[ P(30) = 40 \, \cos \left(5\pi\right) + 110 \]

- Calculate the cosine value

The cosine of \(5\pi\): \[ \cos(5\pi) = \cos(\pi) = -1 \] So, substitute that into the function: \[ P(30) = 40 \times -1 + 110 = -40 + 110 = 70 \] Therefore, the population after 30 months is 70 red foxes.

Key concepts

Population Modeling

Population modeling is a key concept in biology that helps us understand how populations of species change over time. In predator-prey relationships, like that of red foxes and their prey, these models can be particularly insightful.
Here, the red fox population is modeled using the function:
\[ P(t) = 40 \, \cos \left( \frac{\pi t}{6} \right) + 110 \]
This function allows us to estimate the population at different times (\(t\)).
By substituting various time values (measured in months) into the function, we can predict the population size. These predictions can help rangers make decisions about conservation efforts.
This type of population modeling is crucial for maintaining balanced ecosystems.

Cosine Function

The cosine function, represented as \( \cos(x) \), is a trigonometric function that varies between -1 and 1. In population models, it can capture cyclical patterns, like the rise and fall of a predator's population over time.
In our red fox population model, the term \( 40 \, \cos \left( \frac{\pi t}{6} \right) \) depicts this cyclical behavior.

• The coefficient (40) stretches the function vertically, affecting the amplitude.
• The fraction inside the cosine function (\( \frac{\pi t}{6} \)) determines the periodicity, or how often the cycles repeat.
  By manipulating these variables, the cosine function enables accurate representation of population shifts in a natural setting.

Cyclical Populations

Cyclical populations refer to species whose population sizes rise and fall in predictable patterns over time. This behavior is often seen in predator-prey relationships.
In our example, the red fox population shows cyclical patterns. This can happen due to various ecological factors like availability of food, disease, and seasonal changes.
Here’s a simplified breakdown of the cyclical population model:

• The cosine function captures these natural rises and falls.
• By understanding these cycles, ecologists can better manage wildlife and predict future population trends.
  Understanding cyclical populations helps in making informed decisions about conservation and resource management.

Trigonometric Functions in Population Biology

Trigonometric functions like sine and cosine are powerful tools in population biology. They help model periodic phenomena in nature. In the context of population modeling:

• They capture the regular increases and decreases in populations over time.
• They allow for the prediction of future population sizes based on current trends.
In our red fox example, the cosine function was crucial for estimating population sizes at different time intervals.
Using this approach helps establish a deeper understanding of ecological dynamics, supports conservation efforts, and ensures stable predator-prey relationships.
Ultimately, trigonometric functions provide a mathematical foundation for studying and managing the complexities of natural ecosystems.