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Chapter 2: Problem 47

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.8, x_{0}=0.5\)

Short Answer

Expert verified
Compute \( x_t \) using the logistic equation iteratively for 20 steps. Graph \( x_t \) against \( t \).

Step by step solution

01

Understand the Logistic Equation

The discrete logistic equation is given by \( x_{t+1} = R_0 x_t (1 - x_t) \). This recursive formula helps generate the next value \( x_{t+1} \) based on the current value \( x_t \) and the parameter \( R_0 \).
02

Initialize Values

Set the initial value \( x_0 = 0.5 \) as given. The parameter \( R_0 \) is also given as 3.8. We will use these to compute subsequent values of \( x_t \).
03

Compute Values Iteratively

We'll calculate \( x_t \) for \( t = 0, 1, 2, \ldots, 20 \) by iterating the logistic equation. The calculation for each \( t \) requires the previous value \( x_{t-1} \). Repeat the calculation for each time step up to \( t=20 \).
04

Calculate \( x_{t+1} \) for each \( t \)

Use the equation \( x_{t+1} = 3.8 x_t (1 - x_t) \). Starting with \( x_0 = 0.5 \):- \( x_1 = 3.8 \, (0.5) \, (1-0.5) = 0.95 \)- \( x_2 = 3.8 \, (0.95) \, (1-0.95) = 0.1805 \)- Continue this process iteratively up to \( t=20 \).
05

Record and Graph the Results

After calculating \( x_t \) for \( t=0,1,2,\ldots,20 \), record the results for each \( t \). Graph \( x_t \) on the y-axis against \( t \) on the x-axis to visualize the behavior of the system over time.

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Key Concepts

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

Discrete Mathematics in the Logistic Equation
The logistic equation is a classic topic in discrete mathematics, a branch that deals with distinct and separate values. Unlike calculus, which often deals with continuous functions, discrete mathematics focuses on sequences and iterative computations. The discrete logistic equation is particularly interesting because it models populations in which growth is not constant over time and can change rapidly based on external factors. This kind of equation is represented in a recursive format, allowing us to calculate future states based on current values.
  • The logistic equation is defined as \( x_{t+1} = R_0 x_t (1 - x_t) \), where each step is computed separately.
  • It models a population where growth slows as the population reaches its carrying capacity, controlled by the growth parameter \( R_0 \).
By iteratively applying this equation, we can observe how the population evolves over discrete time periods, offering insights into complex systems that cannot be captured by simple linear models.
Understanding Recursive Formulas
Recursive formulas are a foundational component of discrete mathematics and are crucial for understanding iterative processes like those in the logistic equation. A recursive formula allows you to determine the value of a sequence based on the preceding values. For the discrete logistic equation, the recursive formula \( x_{t+1} = R_0 x_t (1 - x_t) \) lets you calculate the next value \( x_{t+1} \) by applying a fixed rule to the current value \( x_t \).
  • Using \( x_0 \) as an initial condition, the recursive formula generates a sequence of values.
  • Each subsequent value in the sequence depends on its predecessor, creating a step-by-step process that can be continued indefinitely.
In the context of the logistic equation, this approach allows us to model dynamic systems over discrete time periods, thus providing a powerful way to simulate real-world phenomena such as population dynamics in biology, where conditions shift from one time to another.
Graphing Techniques for the Logistic Equation
Graphing is a potent tool to visually analyze complex equations and sequences, particularly in recursive formulas like the logistic equation. By plotting the values of \( x_t \) against the time step \( t \), we can observe how the sequence evolves and detect patterns or behaviors, such as convergence, periodicity, or chaos. This graphing technique helps understand how parameters affect the outcome.
  • For \( R_0 = 3.8 \) and \( x_0 = 0.5 \), plotting \( x_t \) over time provides a visual representation of the population dynamics.
  • The graph shows fluctuations and potential stability or chaos in the system, depending on the parameter values used.
By studying these visual patterns, students can gain a deeper understanding of the logistic equation and the effects of different initial conditions or parameters. This method also highlights how small changes can dramatically alter the behavior of the entire system, emphasizing the sensitivity inherent in recursive systems.

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Most popular questions from this chapter

Tylenol in the Body A patient is taking Tylenol (a painkiller that contains acetaminophen) to treat a fever. The data in this question is taken from Rawlins, Henderson, and Hijab (1977). At \(t=0\) the patient takes their first pill. One hour later the drug has been completely absorbed and the blood concentration, measured in \(\mu \mathrm{g} / \mathrm{ml}\), is 15 . Acetaminophen has first order elimination kinetics; in one hour, \(23 \%\) of the acetaminophen present in the blood is eliminated. (a) Write a recursion relation for the concentration \(c_{t}\) of drug in the patient's blood. For \(t \geq 1\) you may assume for now that no other pills are taken after the first one. (b) Find an explicit formula for \(c_{t}\) as a function of \(t\). (c) Suppose that the patient follows the directions on the pill box and takes another Tylenol pill 4 hours after the first (at time \(t=4\) ). What is the concentration at the time at which the second pill is taken? In others words, what is \(c_{4}\) ? (d) Over the next hour \(15 \mathrm{\mug} / \mathrm{ml}\) of drug enter the patient's bloodstream. So, \(c_{5}\) can be calculated from \(c_{4}\) using the word equation: $$ c_{5}=c_{4}+ $$ nt added \(\quad\) amount eliminated blo Given that the amount added is \(15 \mu \mathrm{g} / \mathrm{ml}\), and the amount eliminated is \(0.23 \cdot c_{4}\), calculate \(c_{5} .\) (e) For \(t=5,6,7,8\) the drug continues to be eliminated at a rate of \(23 \%\) per hour. No pills are taken and no extra drug enters the patient's blood. Compute \(c_{8}\). (f) At time \(t=8\), the patient takes another pill. Calculate \(c_{9} .\) Do not forget to include elimination of drug between \(t=8\) and \(t=9\). (g) We want to calculate the maximum concentration of drug in the patient's blood. We know that concentrations are highest in the hour after a pill is taken, namely at time \(t=1, t=5, t=\) \(9, \ldots\) Define a sequence \(C_{n}\) representing the concentration of the drug one hour after the \(n\) th pill is taken. (h) What terms of the original sequence \(\left\\{c_{r}: t=1,2, \ldots\right\\}\) are \(C_{1}\), \(C_{2}\), and \(C_{3} ?\) (i) Explain why $$ C_{n+1}=(0.77)^{4} \cdot C_{n}+15 $$ and \(c_{1}=15\) (j) From the recursion relation, assuming that the patient continues to take Tylenol pills at 4 -hour intervals, calculate \(C_{1}, C_{2}\), \(C_{3}, C_{4}, C_{5}\), and \(C_{6}\) (k) Does \(C_{n}\) increase indefinitely, or do you think that it converges? (1) By looking for fixing point of the recursion relation in (h), find the limit of \(C_{n}\) as \(n \rightarrow \infty\).

Model painkillers that are absorbed into the blood from a slow release pill. Ourmathematical model for the amount, \(a_{t}\), of drug in the blood t hours after the pill is taken must include the amount absorbed from the pill each hour. Our model starts with the word equation. $$ \begin{array}{c} a_{t+1}=a_{t}+\begin{array}{l} \text { amount absorbed } \\ \text { from the pill } \end{array}-\begin{array}{l} \text { amount eliminated } \\ \text { from the blood } \end{array} \end{array} $$ Assume the amount absorbed from the pill between time \(t\) and time \(t+1\) is \(10 \cdot(0.4)^{t}\). (a) The drug has first order elimination kinetics. \(10 \%\) of the drug is eliminated from the blood each hour. Write down the recursion relation for \(a_{t+1}\) in terms of \(a_{t}\) (b) Assuming that \(a_{0}=0\), meaning that no drug is present in the blood initially, calculate the amount of drug present at times \(t=1,2, \ldots, 6\) (c) What is the maximum amount of drug present at any time in this interval? At what time is this maximum amount reached? (d) Use a spreadsheet to calculate the amount of drug present in hourly intervals from \(t=0\) up to \(t=24\). (e) Show when \(t\) is large, the amount of drug present in the blood decreases approximately exponentially with \(t .\) Hint: Plot the values that you computed for \(a_{t}\) against \(t\) on semilogarithmic axes.

A drug has first order elimination kinetics. At time \(t=0\) an amount \(a_{0}=20 \mathrm{mg}\) is present in the blood. One hour later, at \(t=1\), an amount \(a_{1}=14 \mathrm{mg}\) is present. (a) Assuming that no drug is added to the blood between \(t=0\) and \(t=1\), calculate the percentage of drug that is removed each hour. (b) Write a recursion relation for the amount of drug \(a_{t}\) present at time \(t\). Assume no extra drug. (c) Find an explicit formula for \(a_{t}\) as a function of \(t\). (d) Will the amount of drug present ever drop to 0 according to vour model?

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{n^{2}}{n^{2}+4}\right) $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=2, x_{0}=0.1\)
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