Problem 69 - College Algebra and Calculus: An Applied Approach

Question

Population of Deer The Game Commission introduces 100 deer into newly acquired state game lands. The population \(N\) of the herd is given by \(N=\frac{25(4+2 t)}{1+0.02 t}, \quad t \geq 0\) 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

The populations when \(t = 5, 10, 25\) respectively, and the population will approach a limit as time progresses, which is the limiting size of the herd.

Step 1: Calculate Population for Given Times

Substitute \(t = 5\), \(10\), and \(25\) into the function to find the population at these times. \n\nFor \(t = 5\), \(N=\frac{25(4+2(5))}{1+0.02(5)}\), \n\nfor \(t = 10\), \(N=\frac{25(4+2(10))}{1+0.02(10)}\), \n\nand for \(t = 25\), \(N=\frac{25(4+2(25))}{1+0.02(25)}\).

Step 2: Calculate Limiting Size of Herd

The limiting size of the herd as time progresses can be found by taking the limit of the function as \(t\) approaches infinity \(\lim_{t \to \infty} \frac{25(4+2t)}{1+0.02t}\). By using L'Hospital's rule (which says that as long as the limit of the ratio of the derivatives exists, the original limit equals that value) or by long division, get the final answer.

Step 3: Conclusion

After performing the calculations, derive the population when \(t = 5, 10, 25\) respectively and the limiting size of the herd. The population will change over time, but it will gradually converge to a particular value.

Key Concepts

Limiting Behavior

Limiting behavior refers to what happens to a function or a variable as its input grows indefinitely. In the context of population modeling, it helps us understand the eventual size or value a population will settle toward as time continues.
When we evaluate the deer population function, we are interested in what happens when time, or \(t\), becomes very large—essentially, we want to know the population size as \(t\) approaches infinity.
This limiting behavior tells us the "carrying capacity" of the environment, or the maximum population that the area can sustain over a long period.
To find this limit mathematically, we take the limit of the function \(N(t)\) as \(t\) approaches infinity using tools like L'Hospital's rule or simplification techniques such as long division.

Deer Population

The deer population problem is modeled by the equation \(N=\frac{25(4+2 t)}{1+0.02 t}\), representing the population \(N\) at time \(t\).
This equation models how the number of deer grows over time on the new game lands.
Initially, 100 deer are introduced. The purpose of this model is to predict how this number changes over time with respect to the given parameters.

• The numerator \(25(4+2t)\) reflects growth potential driven by reproduction and immigration of deer.
• The denominator \(1+0.02t\) introduces a limiting factor impeding growth as time progresses.

These elements combine to show that while the deer population will grow initially, the rate of growth will decrease, eventually reaching a stable size—its limiting behavior.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario. In this case, we use an equation to capture the dynamics of the deer population over time.
This model incorporates both biological and environmental aspects, such as breeding rates and environmental capacity to sustain deer. Here’s how the model helps:

• It predicts population changes over time through parameters and variables like growth rates and time.
• Provides insights into the sustainable population size that the environment can support due to limiting factors incorporated in the model.
• Allows for calculations that show not just immediate changes but also long-term forecasts, crucial for managing wildlife and ecological balance.

Through mathematical modeling, complex biological processes can be simplified into understandable and predictable patterns.

Calculus

Calculus is essential in analyzing how quantities change over time, which is very useful in population models like the deer population example.
In this scenario, calculus allows us to find the limit of \(N(t)\) as \(t\) approaches infinity, a concept crucial for understanding long-term behavior.
Here are some calculus techniques applied to this model:

• Limits: By evaluating \(\lim_{t \to \infty} \frac{25(4+2t)}{1+0.02t}\), we determine the deer herd's limiting size.
• L'Hospital's Rule: Used when evaluating limits, especially when direct substitution results in indeterminate forms, like \(\frac{\infty}{\infty}\).
• Derivatives: While not directly used in this simple model, derivatives could help analyze the growth rate of the population.

Calculus provides the mathematical framework necessary to derive meaningful insights about how populations evolve over time.