Problem 44 - Calculus for Biology and Medicine

Question

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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 \(t\). \(r=3.8, x_{0}=0.1\)

Step-by-Step Solution

Answer

Calculate \(x_t\) recursively using the logistic equation for \(t=0\) to \(t=20\) and plot the values.

Step 1: Understanding the Equation

The discrete logistic equation is given by \(x_{t+1} = r x_{t} (1 - x_{t})\). This recursive equation models population growth where \(x_t\) is the population at time \(t\), \(r\) is the growth rate, and \(1 - x_t\) represents the limitation due to environmental factors.

Step 2: Initial Condition Setup

We start with the initial condition \(x_0 = 0.1\). This value is given and will be used to compute subsequent population values \(x_t\).

Step 3: Recursive Calculation

Using \(r = 3.8\) and the initial condition, calculate \(x_1, x_2, \ldots, x_{20}\) using the equation:1. Calculate \(x_1 = r x_0 (1 - x_0) = 3.8 \times 0.1 \times (1 - 0.1)\).2. Use the result to compute \(x_2 = r x_1 (1 - x_1)\), and continue this process up to \(x_{20}\).

Step 4: Perform Calculations

Perform the actual calculations for \(t = 0\) to \(t = 20\):- \(x_0 = 0.1\)- \(x_1 = 3.8 \times 0.1 \times 0.9 = 0.342\)- \(x_2 = 3.8 \times 0.342 \times (1 - 0.342)\), and continue calculating each subsequent \(x_t\).

Step 5: Graphical Representation

Plot \(x_t\) as a function of \(t\) to visualize the behavior of the population over time. Use \(t\) on the x-axis and \(x_t\) on the y-axis. This can be done using graphing software or manually if values are calculated.

Key Concepts

Population Growth

Population growth is a key concept when studying models such as the discrete logistic equation. This equation is used to simulate the changes in a population over time. It considers both the growth potential and the environmental limitations that might curb unlimited growth. In our equation, \( x_t \) represents the population at a given time \( t \), while \( r \) is the intrinsic growth rate, indicating how fast the population might grow without limitations. The term \( 1 - x_t \) reflects factors like finite resources or other constraints that naturally limit population size.

For example, if \( r = 3.8 \), as in the exercise, this high rate suggests a rapid potential increase in population. However, as the population nears its carrying capacity, the limiting term \( 1 - x_t \) becomes significant, slowing down further growth. This captures the essence of realistic population dynamics, where limitless growth is rarely observed due to environmental constraints.

Recursive Equation

A recursive equation is one that builds each term based on the previous one. In the context of the discrete logistic equation, it helps us calculate \( x_{t+1} \), the population at time \( t+1 \), by using \( x_t \), the population at the current time \( t \). This relationship is written as \( x_{t+1} = r x_t (1 - x_t) \).

Starting from an initial population \( x_0 \), the recursive nature allows us to calculate the next population value. For instance, with \( x_0 = 0.1 \) and \( r = 3.8 \), the first population value \( x_1 \) becomes \( 3.8 \times 0.1 \times (1 - 0.1) = 0.342 \).

We can continue this step-by-step calculation to determine the values for \( x_2, x_3, \ldots x_{20} \), progressively using each result to compute the next, ensuring a sequence that evolves and reflects changes in population dynamics over time.

Graphical Representation

Graphical representation is a powerful way to visualize complex processes described by equations, such as population dynamics in the discrete logistic equation. By plotting \( x_t \) as a function of time \( t \), we can observe how the population evolves.

To create this graph, the x-axis represents time steps \( t \), ranging from 0 to 20 in our example. The y-axis shows the population size \( x_t \). As you calculate each \( x_t \), you plot it based on its corresponding time \( t \). This visual distribution reveals patterns, whether they be steady growth, stabilizing population, or chaotic fluctuations, especially at higher values of \( r \).

The plot gives a clear, immediate picture of how the population responds to different conditions and parameters set by the discrete logistic equation, making it easier to interpret the behavior of the equation without solely relying on numerical output.