Problem 41 - Calculus for Biology and Medicine

Question

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 \(t .\) \(R_{0}=2, x_{0}=0.9\)

Step-by-Step Solution

Answer

Compute \( x_t \) iteratively using the given equation and parameters for \( t=0 \) to 20, then graph the results.

Step 1: Initialize Parameters

Set the parameters as given: \( R_0 = 2 \) and initial condition \( x_0 = 0.9 \). These values will be used to compute the successive values of \( x_t \).

Step 2: Compute Next Values

Using the formula \( x_{t+1} = R_0 x_t (1-x_t) \), compute the value of \( x_1 \) as:\[ x_1 = 2 imes 0.9 imes (1 - 0.9) = 0.18 \]

Step 3: Iterate the Process

Continue iterating using the equation for each \( t \) from \( t=2 \) to \( t=20 \). Each subsequent \( x_t \) is calculated based on the previous value of \( x_{t-1} \).

Step 4: Record the Values

As each value of \( x_t \) is calculated, record it. For example:\[ x_2 = 2 imes 0.18 imes (1 - 0.18) = 0.2952 \]Continue until \( x_{20} \) is obtained.

Step 5: Graph the Results

Once all values from \( x_0 \) to \( x_{20} \) are computed, graph these values with \( t \) on the x-axis and \( x_t \) on the y-axis. This plot will show how \( x_t \) evolves over time.

Key Concepts

Iterative Computation

Iterative computation is a process used extensively in mathematical modeling to understand how a sequence changes over time. When dealing with the discrete logistic equation, it involves repeatedly applying a formula to compute the sequence of values. Starting with an initial value, the equation is used over and over to produce a list of numbers, each dependent on the one before it. To illustrate, consider the discrete logistic equation that begins with given parameters like the growth rate \(R_0\) and an initial population size \(x_0\). The sequence of values \(x_t\) is generated using the formula \(x_{t+1} = R_0 x_t (1 - x_t)\).

• Begin by setting initial conditions, like \(R_0 = 2\) and \(x_0 = 0.9\).
• Calculate \(x_1\) by substituting into the formula: \(x_1 = 2 \times 0.9 \times (1 - 0.9)\).
• Compute subsequent \(x_t\) values through iteration until the desired endpoint, such as \(t=20\).

To effectively use iterative computation, maintain a systematic approach to applying the formula repeatedly while keeping track of each result. Iteration is a powerful tool in capturing the dynamics of systems that evolve step-by-step.

Graphing Functions

Graphing functions is a fundamental technique to visualize the behavior of a mathematical model over time. By plotting the values obtained through iterative computation, you can see how the system behaves and identify patterns or trends. For the discrete logistic equation, you graph the values of \(x_t\) against time \(t\), helping to shed light on the series generated.

• Once all iterations have been computed, list the ordered pairs \((t, x_t)\).
• On a graph, place time \(t\) on the horizontal axis (x-axis) and \(x_t\) on the vertical axis (y-axis).
• Plot each point on the graph and connect them to see the trajectory of the population size over time.

This visual representation allows for easy identification of steady states, periodic cycles, or chaotic behavior inherent in the system. By analyzing the graph, students can gain intuitive insights into the dynamic changes occurring within the model.

Mathematical Modeling

Mathematical modeling is a valuable tool for simulating real-world systems to predict their behavior under various conditions. The discrete logistic equation is an example of such a model, often used in population dynamics to study how a population changes over time with respect to specified parameters. In this context, mathematical modeling involves:

• Defining initial conditions and parameters like reproduction rate \(R_0\) and initial population size \(x_0\).
• Using the equation \(x_{t+1} = R_0 x_t (1-x_t)\) to simulate each step forward in time, which allows the model to mimic natural processes.
• Interpreting the results to understand the long-term behavior of the population, such as growth stabilization or oscillations.

Mathematical models like the discrete logistic equation provide a framework for testing hypotheses about population growth. As learners delve into the topic, they discover how altering conditions can influence outcomes and gain insights into the interplay between population size and growth parameters. This helps build a deeper understanding of complex systems and their predictive power.