Problem 50 - 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 \(t .\) \(R_{0}=3.8, x_{0}=0\)

Step-by-Step Solution

Answer

For \( R_{0}=3.8 \) and \( x_{0}=0 \), \( x_t = 0 \) for all \( t \). The plot is a horizontal line at \( x=0 \).

Step 1: Understanding the Discrete Logistic Equation

The discrete logistic equation is formulated as \( x_{t+1}=R_{0} x_{t}(1-x_{t}) \). For this problem, we have \( R_{0}=3.8 \) and an initial condition \( x_{0}=0 \). We will use this equation to compute \( x_{t} \) for \( t=0,1,2,\ldots, 20 \).

Step 2: Apply Initial Conditions

We start with \( x_0 = 0 \). Substitute this into the equation to find \( x_1 \):\[ x_1 = R_0 \cdot x_0 \cdot (1 - x_0) = 3.8 \cdot 0 \cdot (1 - 0) = 0 \]Thus, \( x_1 = 0 \).

Step 3: Iterate to Compute Subsequent Values

Since \( x_1 = 0 \), we apply the equation again to find \( x_2 \):\[ x_2 = 3.8 \cdot x_1 \cdot (1-x_1) = 3.8 \cdot 0 \cdot 1 = 0 \]Continue this process for each \( t \) from 1 to 20. Due to the initial value being zero, all iterations yield \( x_t = 0 \).

Step 4: Conclusion of Iterations

Since every iteration results in \( x_t = 0 \), this pattern will continue for every \( t \geq 0 \). No matter how many times the logistic map equation is applied, each \( x_t \) remains 0.

Step 5: Graphing the Results

Plot \( x_t \) against \( t \). Since all values of \( x_t \) are 0, the graph is simply a horizontal line at \( x=0 \) from \( t=0 \) to \( t=20 \).

Key Concepts

Iteration Process

Let's delve into the iteration process of the discrete logistic equation. This equation is a fascinating tool to model population growth where each generation is affected by its current size. It's defined as follows:\[ x_{t+1} = R_0 x_t (1 - x_t) \]In this formula, \( R_0 \) is the growth rate, and \( x_t \) represents the population size (or proportion from the carrying capacity) at time \( t \). In the context of our problem, we're asked to evaluate this equation when the growth rate \( R_0 \) is set to 3.8, and the initial condition \( x_0 \) is 0.The beauty and challenge of this iteration process lie in computing successive values of \( x_t \) by feeding the previous value back into the equation. The specified range, from \( t=0 \) to \( t=20 \), means we keep applying the equation continually to generate the next value of \( x \) over time.The iteration shows that when \( x_0 = 0 \), the equation simplifies drastically:

• \( x_1 = 3.8 \times 0 \times (1 - 0) = 0 \)
• \( x_2 = 3.8 \times 0 \times 1 = 0 \)
• Continuing this pattern yields \( x_t = 0 \) for all \( t \).

As we can see, the iteration here doesn't develop into complexity due to the initial condition.

Initial Conditions

Initial conditions are the starting point of any iterative process. In many equations like the logistic map, they have a profound impact on the trajectory of the results.For our specific example with the discrete logistic equation, the initial condition is set as \( x_0 = 0 \). What this means is that at time \( t = 0 \), the population, or the value we are tracking, is completely absent or at its minimum starting point.This initial starting value is crucial because it dictates the entire behavior of the equation upon iteration. Here, substituting \( x_0 = 0 \) into the logistic equation leads directly to:

• \( x_1 = 0 \), which then causes every subsequent \( x_t \), from \( x_2 \) onward, to also be 0.

The pattern persists because any multiplier with zero remains zero, leading to a constant and unchanging sequence. Testing different initial values, other than zero, could result in vastly different outcomes, potentially involving cycles or chaotic behavior.

Graphing Results

Graphing is a powerful means of visualizing the output of the discrete logistic equation.After performing the iterations based on the initial condition \( x_0 = 0 \), we observe that every computed \( x_t \) is 0.When graphing these results, plotting \( x_t \) against \( t \) gives us a clear and straightforward visual: a horizontal line at the value 0, which spans from \( t = 0 \) to \( t = 20 \). This graph unequivocally shows that no change occurs in the value of \( x_t \) as \( t \) evolves in this specific scenario.Graphs help in understanding:

• The nature of the logistic map outputs.
• How different initial conditions might drastically alter the graph's shape.
• The simplicity or complexity of the model behavior over time.

By seeing this line, the conclusion is immediately evident: irrespective of how many iterations we perform, the output remains fixed at zero under these conditions. Contrast this with different initial values where graphs could show oscillations or chaos resulting from the same logistic equation.