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Chapter 2: Problem 44

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 \(t .\) \(R_{0}=3.1, x_{0}=0.1\)

Short Answer

Expert verified
Compute each \(x_t\) using the logistic formula iteratively, plot \(x_t\) against \(t\) to visualize population dynamics.

Step by step solution

01

Understand the Discrete Logistic Equation

The discrete logistic equation is given by \(x_{t+1} = R_0 x_t (1-x_t)\). Here, \(x_t\) represents the population at time \(t\), \(R_0\) is the growth rate, and \(x_0\) is the initial population. This equation is used to model population dynamics.
02

Initialize Variables

We are given \(R_0 = 3.1\) and \(x_0 = 0.1\). The task is to compute \(x_t\) for \(t = 0, 1, 2, \ldots, 20\). Start with \(x_0 = 0.1\).
03

Calculate Population for Each Time Step

Using the logistic equation, compute \(x_t\) iteratively for each time \(t\) up to 20:- \(x_{1} = 3.1 imes 0.1 imes (1-0.1) = 0.279\)- \(x_{2} = 3.1 imes 0.279 imes (1-0.279) \approx 0.62317\)- Continue this computation using the formula for further values of \(t\).
04

Iterate Through Time Steps

Continue the iterations:- \(x_{3} \approx 3.1 imes 0.62317 imes (1-0.62317) \approx 0.72856\)- Calculate similarly up to \(x_{20}\).- Make sure each step uses the last calculated \(x\) value.
05

Plot the Values

Once all computations from \(x_{0}\) to \(x_{20}\) are completed, plot \(x_t\) against \(t\). This graph will help in visualizing how the population evolves with time.

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Key Concepts

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

Understanding Population Dynamics
The discrete logistic equation plays a crucial role in understanding how populations change over time. In the equation \(x_{t+1} = R_0 x_t (1-x_t)\), \(x_t\) represents the population at time \(t\). This equation is widely used to model population dynamics because it captures the essence of how a population grows and eventually stabilizes or fluctuates based on the available resources and other environmental factors.

The concept of **population dynamics** includes several key aspects:
  • Initial Population: This is the starting size of the population, represented as \(x_0\). In our example, \(x_0 = 0.1\).
  • Carrying Capacity: The term \(1-x_t\) suggests that as the population grows, the space or resources available for each individual decrease, leading to a limit to the total population size.
  • Time Evolution: By calculating \(x_{t}\) through each time step, we can see the population's growth pattern over time.
Unlike models that assume infinite growth, the logistic equation integrates natural restrictions and limitations, making it more applicable to real-world scenarios.
The Role of Iterative Computation
Iterative computation is central to solving the discrete logistic equation over time. This involves repeatedly applying the logistic equation to find the population size at each time step \(t\).

**How Iterative Computation Works:**
  • Start with the Initial Condition: Begin with \(x_0\), the known initial population.
  • Repeated Calculations: Use the equation \(x_{t+1} = R_0 x_t (1-x_t)\) to calculate the population for the next time step. Each new \(x_t\) depends entirely on the previous value \(x_{t-1}\).
  • Prediction Over Time: Continue these calculations for every \(t\) up to 20, as required by the exercise.
Each iteration builds upon the last, illustrating how the population changes over time. This method highlights the dynamics of growth and stabilization as it approaches a steady state, showcasing the importance of computation in predicting future states.
The Impact of Growth Rate
The growth rate \(R_0\) is a core component of the logistic equation and significantly influences population behavior. In our case, \(R_0 = 3.1\), which indicates how aggressively the population is expected to grow before density or resources limit further expansion.

**Understanding Growth Rate Effects:**
  • High Growth Rate: With a higher \(R_0\), the population increases rapidly, potentially leading to overshooting the carrying capacity, which can cause cycles of boom and bust.
  • Stable Growth: If \(R_0\) is balanced, population growth proceeds smoothly toward a stable state.
  • Chaotic Dynamics: Certain values of \(R_0\) can result in complex dynamics, where the population sizes fluctuate unpredictably over time.
Growth rate determines the overall characteristic of population dynamics, influencing whether a population thrives, declines, or cycles through repeated patterns. Understanding \(R_0\) helps predict long-term trends, making it invaluable for ecological modeling.

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Most popular questions from this chapter

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 \(t .\) \(R_{0}=3.1, x_{0}=0.9\)

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{3}{a_{n}+2}, a_{0}=0 $$

Because of complex interactions with other drugs, some drugs have zeroth order elimination kinetics in some circumstances, and first order kinetics in other circumstances, depending on what other drugs are in the patient's system, as well as on age and preexisting medical conditions. Use the data on how concentration varies with time to determine whether the drug has zeroth or first order kinetics. Given the following sequence of measurements of drug concentration, determine whether the drug has zeroth or first order kinetics. $$ \begin{array}{lcccc} \hline \boldsymbol{t} \text { (Hours) } & 1 & 2 & 3 & 4 \\ \hline c_{t}(\mu \mathrm{g} / \mathrm{m} \mathrm{l}) & 20 & 18 & 16 & 14 \\ \hline \end{array} $$

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 \(n \rightarrow \infty\).

You do not know whether a drug has zeroth order or first order elimination kinetics. You will use data to determine which type of kinetics it has. You measure the concentration of the drug (in \(\mathrm{mg} / \mathrm{ml}\) ) at time \(t=0\) and at time \(t=1 .\) No drug is added to the blood between \(t=0\) and \(t=1\). You measure the following data: \begin{tabular}{ll} \hline \(\boldsymbol{t}\) & \(\boldsymbol{c}_{t}\) \\ \hline 0 & 40 \\ 1 & 32 \\ \hline \end{tabular} (a) Assume that the drug has zeroth order kinetics. What amount is eliminated from the blood each hour? (b) Assume that the drug has zeroth order kinetics and no more drug is added to the blood. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\). (c) Now assume the drug has first order elimination kinetics. What percentage of drug is eliminated from the blood each hour? (d) Assume that the drug has first order kinetics and no more drug is added. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\). (e) You measure the concentration at time \(t=2\) and find \(c_{2}=\) 25.6. By comparing with your predictions from (b) and (d), decide: Does the drug have zeroth or first order kinetics?
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