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Chapter 2: Problem 20

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). Find the exponential growth equation for a population that triples in size every unit of time and that has 72 individuals at time 0 .

Short Answer

Expert verified
The formula is \(N_t = 72 \cdot 3^t\).

Step by step solution

01

Understand the Problem

We need to find a formula of population size \(N_t\) as a function of time \(t\). The population triples every unit time and starts with 72 individuals at \(t = 0\).
02

Exponential Growth Formula

The general formula for exponential growth is \(N_t = N_0 \cdot a^t\), where \(N_0\) is the initial population, \(a\) is the growth factor, and \(t\) is time.
03

Determine Growth Factor

Since the population triples every unit of time, the growth factor \(a = 3\). Therefore, the proper growth factor in our formula is \(3\).
04

Write the Specific Formula

Substitute the initial population \(N_0 = 72\) and the growth factor \(a = 3\) into the formula from Step 2: \[N_t = 72 \cdot 3^t\].
05

Verify the Formula

Check that \(N_0 = 72\) at \(t = 0\). When \(t = 0\), \(N_t = 72 \cdot 3^0 = 72 \cdot 1 = 72\), confirming the initial population is correct.

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Key Concepts

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

Population Size Formula
When dealing with exponential growth scenarios, the population size formula is fundamental. This formula helps us model how populations change over time. It goes like this: the population at time \( t \), noted as \( N_t \), is equal to the initial population \( N_0 \) multiplied by the growth factor \( a \), raised to the power of the time \( t \). It is written as:\[N_t = N_0 \cdot a^t\]
  • \( N_0 \): the initial population size, which is where it all begins, at time \( t = 0 \).
  • \( a \): the growth factor that influences how much the population increases every time interval.
  • \( t \): the time that has passed since the beginning of the observation.
Understanding this structure is key, as it tells us how populations grow exponentially, with each unit of time causing the population to increase by the same percentage.
Growth Factor
The growth factor is crucial to understanding exponential growth. It represents how many times the population multiplies over a single time unit. In our problem, we've learned that the population triples every unit of time. This means if you have 72 individuals to start, one time unit later you will have three times that amount.In formula terms, the growth factor \( a \) is 3. This means at each time step:
  • The population isn't just increasing by the same number each time; it's multiplying.
  • Because the growth is exponential, each passing time unit sees the population grow based on its current size, multiplying by 3.
The growth factor is the lever that dictates just how rapid or modest the growth rate is.
Initial Population
The initial population \( N_0 \) is the starting size of the population we're examining. It's the number of individuals you have before any time has passed, specifically at \( t = 0 \). This value serves as the baseline in our exponential growth formula.In the given exercise, we're informed that the initial population is 72 individuals. This number is pivotal because it gives us a starting point to apply the exponential growth process.
  • Without knowing \( N_0 \), we couldn't calculate the future population sizes accurately.
  • Each subsequent population size is a reflection of not only the growth factor but also this initial figure.
In our case, all future calculations of \( N_t \) depend on this initial population being correctly accounted for, highlighting its importance in long-term projections.

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Most popular questions from this chapter

Assume that the population growth is described by the Beverton-Holt recruitment curve with parameters \(R_{0}\) and a. Find the population sizes for \(t=1,2, \ldots, 5\) and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0} .\) \(R_{0}=3, a=1 / 20, N_{0}=7\)

Mountain Gorilla Conservation You are trying to build a mathematical model for the size of the population of mountain gorillas in a national park in Uganda. The data in this question are taken from Robbins et al. (2009). (a) You start by writing a word equation relating the population of gorillas \(t\) years after the study begins, \(N_{t}\), to the population \(N_{t+1}\) in the next year: \(N_{t+1}=N_{t}+\begin{array}{l}\text { number of gorillas } \\ \text { born in one year }\end{array}-\begin{array}{l}\text { number of gorillas } \\\ \text { that die in one yeai }\end{array}\) We will derive together formulas for the number of births and the number of deaths. (i) Around half of gorillas are female, \(75 \%\) of females are of reproductive age, and in a given year \(22 \%\) of the females of reproductive age will give birth. Explain why the number of births is equal to: \(\quad 0.5 \cdot 0.75 \cdot 0.22 \cdot N_{t}=0.0825 \cdot N_{t}\) (ii) In a given year \(4.5 \%\) of gorillas will die. Write down a formula for the number of deaths. (iii) Write down a recurrence equation for the number of gorillas in the national park. Assuming that there are 300 gorillas initially (that is \(N_{0}=300\) ), derive an explicit formula for the number of gorillas after \(t\) years. (iv) Calculate the population size after \(1,2,5\), and 10 years. (v) According to the model, how long will it take for population size to double to 600 gorillas? (b) In reality the population size is almost totally stagnant (i.e., \(N_{t}\) changes very little from year to year). Robbins et al. (2009) consider three different explanations for this effect: (i) Increased mortality: Gorillas are dying sooner than was thought. What percentage of gorillas would have to die each year for the population size to not change from year to year? (ii) Decreased female fecundity: Gorillas are having fewer offspring than was thought. Calculate the female birth rate (percentage of reproductive age females that give birth) that would lead to the population size not changing from year to year. Assume that all other values used in part (a) are correct. (iii) Emigration: Gorillas are leaving the national park. What number of gorillas would have to leave the national park each year for the population to not change from year to year?

Write each sum in expanded form. $$ \sum_{k=1}^{3} \frac{k^{2}}{k^{2}+1} $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\log \left(1+\frac{2}{n^{2}}\right), \epsilon=0.05 $$

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