Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Plot graphs of solutions to $$ \begin{array}{lll} \text { a. } & w_{0}=2 & w_{t+1}=1.2 \times \frac{w_{t}}{0.5+w_{t}} \\ \text { b. } & w_{0}=0.2 & w_{t+1}=1.2 \times \frac{w_{t}}{0.5+w_{t}} \end{array} $$ c. \(\quad w_{0}=2 \quad w_{t+1}=1.2 \times w_{t} \times e^{-w_{t} / 10}\) d. \(\quad w_{0}=0.1 \quad w_{t+1}=1.2 w_{t} \times \cos \left(w_{t}\right)\) e. \(w_{0}=0.001 \quad w_{t+1}=w_{t}+\sin w_{t}\) f. \(\quad w_{0}=0 \quad w_{t+1}=w_{t}+\sin w_{t}\) g. \(\quad w_{0}=0 \quad w_{t+1}=w_{t}+1\) h. \(\begin{aligned} w_{0} &=0 \\ w_{1} &=1 \end{aligned} \quad w_{t+2}=w_{t+1}-w_{t}\)

Step-by-Step Solution

Answer

Generate and plot graphs for each recurrence relation to observe distinct dynamics such as growth, oscillation, or decay.

Step 1: Understand the task

We need to plot graphs of the solutions for eight different discrete dynamic systems defined by recurrence relations. Each system has its own initial condition and specific update rule for the variable \( w_t \).

Step 2: Analyze system a

For part a, the recurrence relation is \( w_{t+1} = 1.2 \times \frac{w_t}{0.5+w_t} \) with initial condition \( w_0 = 2 \). We need to calculate subsequent terms using this formula.

Step 3: Calculate terms for system a

With \( w_0 = 2 \), compute \( w_1 = 1.2 \times \frac{2}{0.5+2} = 1.28 \), then use \( w_1 \) to find \( w_2 \), and so on for a few more terms to see the trend.

Step 4: Plot graph for system a

Plot points \((0, 2), (1, 1.28), ...\) for several iterations to observe how \( w_t \) changes over time.

Step 5: Analyze system b

For part b, the rule is the same as a: \( w_{t+1} = 1.2 \times \frac{w_t}{0.5+w_t} \) but with \( w_0 = 0.2 \).

Step 6: Calculate terms for system b

With \( w_0 = 0.2 \), compute \( w_1 = 1.2 \times \frac{0.2}{0.5+0.2} = 0.343 \), and continue the process to see the trend.

Step 7: Plot graph for system b

Plot points \((0, 0.2), (1, 0.343), ...\) similarly to system a, noting the growth pattern.

Step 8: Analyze system c

For part c, the recurrence relation is \( w_{t+1} = 1.2 \times w_t \times e^{-w_t / 10} \) with \( w_0 = 2 \).

Step 9: Iterate terms for system c

Using \( w_0 = 2 \), calculate \( w_1 \) and subsequent terms using the given formula to capture the behavior of \( w_t \).

Step 10: Plot graph for system c

Graph the solution to showcase how \( w_t \) evolves according to the nonlinear dependence in the update rule.

Step 11: Analyze system d

For part d, the equation is \( w_{t+1} = 1.2 \times w_t \times \cos(w_t) \) starting with \( w_0 = 0.1 \).

Step 12: Iterate terms for system d

Start from \( w_0 = 0.1 \) to calculate a few terms, noting how each term oscillates with each iteration.

Step 13: Plot graph for system d

Graph the iterative terms to observe the cosine-modulated evolution.

Step 14: Analyze system e

For part e, the update rule is \( w_{t+1} = w_t + \sin(w_t) \) with \( w_0 = 0.001 \).

Step 15: Iterate terms for system e

Use \( w_0 = 0.001 \) to find \( w_1 = 0.001 + \sin(0.001) \approx 0.002 \), and proceed to observe the trend.

Step 16: Plot graph for system e

Draw how \( w_t \) possibly grows with a slight sinusoidal increment.

Step 17: Analyze system f

For part f, the update rule is unchanged with \( w_0 = 0 \), similar to system e.

Step 18: Iterate terms for system f

Use \( w_0 = 0 \) to begin the sequence, computing further points to see the persistence of the sinusoidal growth.

Step 19: Plot graph for system f

Showcase how \( w_t \) progresses starting from zero but still affected by \( \sin(w_t) \).

Step 20: Analyze system g

The update rule for g is simpler: \( w_{t+1} = w_t + 1 \) with \( w_0 = 0 \).

Step 21: Iterate terms for system g

This linear system shows that with each step, \( w_t \) simply increases by 1. Graph these points for clarity.

Step 22: Plot graph for system g

Display a linear graph that ascends at a constant rate due to the constant additive step.

Step 23: Analyze system h

This is a Fibonacci-like sequence with different initial conditions; \( w_0 = 0, w_1 = 1 \) and \( w_{t+2} = w_{t+1} - w_t \).

Step 24: Compute terms for system h

Generate terms like \( w_2 = 1 - 0 = 1, w_3 = 1 - 1 = 0, w_4 = 0 - 1 = -1 \), and continue.

Step 25: Plot graph for system h

Visualize how the sequence oscillates between values showing a pattern of increments and decrements.

Key Concepts

Recurrence Relations

When we discuss recurrence relations in discrete dynamic systems, we are talking about equations that express how each term in a sequence, denoted as \(w_t\), depends on the previous term \(w_{t-1}\). For instance, in exercise (a), the recurrence relation is given by \(w_{t+1} = 1.2 \times \frac{w_t}{0.5+w_t}\). This shows how the next value, \(w_{t+1}\), can be computed using the current value, \(w_t\).
Recurrence relations are a powerful mathematical tool for modeling time-dependent systems, such as population growth or financial investments. They help in determining the future behavior of the system based on its current state. Understanding the recursive nature of such relations is crucial, as it can reveal insights into long-term trends or stability of the system.
The key parameters within the relations often dictate this behavior. For example, in the given equation, parameters like the multiplier (1.2) or the divisor (0.5) can greatly influence how quickly the sequence converges to stability or diverges. Different initial conditions, like \(w_0=2\) and \(w_0 = 0.2\) in exercises a and b, can also lead to varying patterns and rates of growth.

Graph Plotting

Graph plotting is a visual tool that helps represent the outcomes of recurrence relations over time. By plotting each computed term \(w_t\) on a graph, we can visually assess the behavior and tendencies of a discrete system.
For each part of the exercise, graphing allows us to observe whether the sequences are converging, diverging, or oscillating. In systems a and b, plotting points such as \((0, 2), (1, 1.28)\) for system a helps illustrate how the values change step-by-step.
Graphs should be generated by iterating the formula until finding a visible pattern or behavior over several points. These visual representations are essential because they can immediately highlight features like periodicity, stability, or chaos, which might not be easily deduced from tables of numbers alone. They serve as a bridge between numeric computation and intuitive understanding of the dynamics displayed by the recurrence relations.

Iterative Methods

Iterative methods are crucial when dealing with recurrence relations because they allow us to compute terms of a sequence step-by-step. For each given initial condition, you use the prescribed formula to find successive values like \(w_1, w_2, w_3\), and so forth.
In discrete dynamic systems, iterative computation is an essential technique for studying the evolution of the system over time. At each step, you apply the same update rule, such as the one in exercise (d) \(w_{t+1} = 1.2 \times w_t \times \cos(w_t)\), repeatedly to generate new terms.
Iterative methods help in approximating solutions for complex systems where finding a closed-form expression might be challenging or impossible. They provide flexibility and handle nonlinear dynamics effectively, as seen in the various recurrence relationships in the exercise. By following a consistent approach to calculating each subsequent term, iterative methods illuminate how a system progresses with each iteration, highlighting trends or patterns that are vital for deeper analysis.