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A population of deer: When a breeding group of animals is introduced into a restricted area such as a wildlife reserve, the population can be expected to grow rapidly at first but to level out when the population grows to near the maximum that the environment can support. Such growth is known as logistic population growth, and ecologists sometimes use a formula to describe it. The number \(N\) of deer present at time \(t\) (measured in years since the herd was introduced) on a certain wildlife reserve has been determined by ecologists to be given by the function $$ N=\frac{12.36}{0.03+0.55^{t}} $$ a. How many deer were initially on the reserve? b. Calculate \(N(10)\) and explain the meaning of the number you have calculated. c. Express the number of deer present after 15 years using functional notation, and then calculate it. d. How much increase in the deer population do you expect from the 10 th to the 15 th year?

Step by step solution

01

Calculate Initial Population

To find out how many deer were initially on the reserve, substitute \(t = 0\) into the given formula. The function is \(N = \frac{12.36}{0.03 + 0.55^t}\). Substituting \(t = 0\) gives:\[N = \frac{12.36}{0.03 + 0.55^0} = \frac{12.36}{0.03 + 1} = \frac{12.36}{1.03}\]Calculating the value results in:\[N \approx 12\]

02

Calculate Deer Population after 10 years

We need to calculate \(N(10)\). Substitute \(t = 10\) into the formula.\[N = \frac{12.36}{0.03 + 0.55^{10}}\]Calculate \(0.55^{10} \approx 0.00000475\). Therefore:\[N = \frac{12.36}{0.03 + 0.00000475} \approx \frac{12.36}{0.03000475} \approx 411.98\]This means there are approximately 412 deer on the reserve after 10 years.

03

Calculate Deer Population after 15 years

First, express the number of deer present after 15 years using functional notation: \(N(15)\).Substitute \(t = 15\) into the formula:\[N = \frac{12.36}{0.03 + 0.55^{15}}\]Calculate \(0.55^{15} \approx 0.00000093\). Therefore:\[N = \frac{12.36}{0.03 + 0.00000093} \approx \frac{12.36}{0.03000093} \approx 412\]Hence, the population is approximately 412 deer after 15 years.

04

Calculate Population Increase from 10 to 15 years

To find the increase in population from the 10th to the 15th year, subtract \(N(10)\) from \(N(15)\).From Steps 2 and 3, we have \(N(10) \approx 412\) and \(N(15) \approx 412\).\[412 - 412 = 0\]The expected increase in deer population is 0. This shows that the population has stabilized by this time.

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Key Concepts

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

Deer Population Modeling

Modeling the population of deer in a restricted environment like a wildlife reserve is essential to understanding how populations grow and stabilize over time. Scientists often use mathematical functions, such as the logistic growth function, to predict population changes. This model is particularly useful because it shows how populations initially grow rapidly and then begin to stabilize as the carrying capacity of the environment is reached. For instance, in our case, the number of deer at any time \( t \) is given by the function \( N = \frac{12.36}{0.03 + 0.55^t} \). This equation illustrates how the population behaves over time.

Wildlife Reserve Ecology

Wildlife reserves are areas set aside specifically to protect certain wildlife species. Understanding ecology within these environments is crucial for maintaining a balanced ecosystem. On a wildlife reserve, there is a limited amount of resources such as food, space, and water, which determines the carrying capacity – the maximum population size an environment can support sustainably. In the context of our deer population model, the reserve's ecology plays a pivotal role in determining how quickly the population can grow and to what extent it can be maintained. The logistic growth model we are using helps to illustrate this ecological balance.

Population Stabilization

Population stabilization refers to the point at which a population stops growing and remains relatively constant. In ecological terms, this is often due to reaching the carrying capacity of the environment. In the deer population example, we calculated that the population increased rapidly in the initial years. However, between the 10th and 15th years, there was no increase in population, meaning it had stabilized at around 412 deer. This stabilization is a result of the limiting factors within the environment, such as available resources and space, effectively balancing out the birth and death rates of the species.

Functional Notation

Functional notation is a way of representing functions in mathematics that helps us easily express and compute values. In our deer population problem, we use the function \( N = \frac{12.36}{0.03 + 0.55^t} \) to model the number of deer over time. When we write \( N(10) \), we are describing the number of deer projected to be on the reserve at time \( t = 10 \) years. This makes it simple to substitute different time values to calculate the corresponding population size. Functional notation allows us to succinctly express these relationships and compute them as needed with accuracy.