Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Population of Elk The Game Commission introduces 40 elk into newly acquired state game lands. The population \(N\) of the herd is given by \(N=\frac{10(4+2 t)}{1+0.03 t}, \quad t \geq 0\) where \(t\) is time (in years). (a) Find the populations when \(t\) is 5,10, and 25 . (b) What is the limiting size of the herd as time progresses?

Step-by-Step Solution

Answer

Evaluating the population function, we find: \(N(5) = 120\), \(N(10) = 180\), and \(N(25) = 300\). The limit as \(t\) goes to infinity yields a constant value of \(67\). This indicates that the population size will eventually stabilize over time at 67.

Step 1: Evaluate the population at different time points

To find the population at \(t = 5, 10, 25\), simply substitute these values into the equation \(N=\frac{10(4+2 t)}{1+0.03 t}\). Here's how to do it: \[N(5) = \frac{10(4+2(5))}{1+0.03(5)}\] \[N(10) = \frac{10(4+2(10))}{1+0.03(10)}\] \[N(25) = \frac{10(4+2(25))}{1+0.03(25)}\]

Step 2: Calculate the limiting value of the function

We need to find the limiting value of the population size as time \(t\) tends to infinity. This is done by calculating the following limit: \[ \lim_{{t}\to\infty}\frac{10(4+2 t)}{1+0.03 t} \] To compute this limit, divide both numerator and denominator by the highest power of \(t\) that occurs in the denominator. After reducing, we get \[ \lim_{{t}\to\infty}\frac{2}{0.03} \] which simplifies to a constant value.

Key Concepts

Limiting Value

In any population model, the concept of a limiting value, or carrying capacity, is crucial. It represents the maximum population size that an environment can sustainably support over an extended period. For the Elk population, as time progresses toward infinity, the equation that governs their population changes attains a stable limit.

In our particular exercise, the formula given is: \[ N = \frac{10(4 + 2t)}{1 + 0.03t} \]We are tasked with finding the limiting size of the elk population. To do this, we evaluate the limit of the equation as time \(t\) approaches infinity:\[ \lim_{{t}\to\infty}\frac{10(4+2t)}{1+0.03t} \]
By dividing both the numerator and the denominator by \(t\), the highest power in the denominator, we simplify the expression to:\[ \lim_{{t}\to\infty}\frac{2}{0.03} \]
Upon calculation, this results in a limiting population size of approximately 66.67 elk. This suggests that despite the introduction of more elk over time, resource constraints stabilize the population at this figure.

Time Progression

Understanding how things change over time is key to analyzing population dynamics. In this scenario, time \(t\) represents the progression of years. As time increases, certain factors such as birth rates, death rates, and external conditions like available resources affect how the population needs to be modeled.

In our elk population exercise, the mathematical model captures how the elk population grows based on the equation given. When we look at different years such as \(t = 5\), \(t = 10\), and \(t = 25\), we substitute these values into the equation to predict the population at those specific times.

• At \(t = 5\), by substituting \(t\) in the formula, the population is calculated as \(N(5) = \frac{10(4+2(5))}{1+0.03(5)}\).
• Similarly, at \(t = 10\), \(N(10) = \frac{10(4+2(10))}{1+0.03(10)}\).
• Lastly, at \(t = 25\), \(N(25) = \frac{10(4+2(25))}{1+0.03(25)}\).

As these values show, the population grows larger with time, but the increase slows, approaching the limiting value we found earlier.

Elk Population

The concept of elk population dynamics is fascinating and illustrative of broader ecological principles. When modeling animal populations, it's crucial to consider various factors, such as habitat space, food availability, and environmental conditions.

In the case of the elk introduced to the state game lands, mathematical modeling helps predict how the population will evolve. Initially, the population grows rapidly as the elk adapt to the new environment and the resources available. However, as time goes on, resources may become limiting, or natural checks like predation or disease might slow further growth.

The model employed here, with its equation,\[ N = \frac{10(4 + 2t)}{1 + 0.03t} \]
highlights how starting conditions and environmental factors converge to create a predictable population size limit. Such analyses are vital for wildlife management, helping ensure sustainable population levels and ecological balance.