Problem 4 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Draw the graph of \(F\) and the graph of \(y=x\) on a single axes with \(0 \leq x \leq 1\) and \(0 \leq y \leq 1 .\) The points of intersection of the graphs of \(F\) and \(y=x\) are listed with \(F\), correct to 3 decimal places. They are the equilibrium points of the iteration \(x_{n+1}=F\left(x_{n}\right) .\) For each such point, determine whether it is a locally stable equilibrium or an unstable equilibrium. a. \(F(x)=0.7 x+0.2\) (0.667,0.667) b. \(F(x)=1.1 x-0.05 \quad(0.500,0.500)\) c. \(F(x)=x^{2}+0.1 \quad(0.113,0.113) \quad(0.887,0.887)\) d. \(F(x)=\sqrt{x}-0.2 \quad(0.076,0.076) \quad(0.524,0.524)\) e. \(F(x)=-0.9 x^{2}+2 x-0.2 \quad(0.262,0.262) \quad(0.850,0.850)\) f. \(F(x)=-0.9 x^{2}+2 x-0.1 \quad(0.111,0.111) \quad(1.000,1.000)\) g. \(F(x)=x^{3}+0.2\) (0.879,0.879) h. \(F(x)=8 x^{3}-12 x^{2}+6 x-1 / 2 \quad(0.146,0.146) \quad(0.500,0.500)\) (0.854,0.854)

Step-by-Step Solution

Answer

Examples a, e, f, h (0.146 and 0.854) are stable; b, c (0.887), d (0.076), g, h (0.500) are unstable.

Step 1: Understand the problem

We need to graph the function \( F(x) \) and the line \( y = x \) within the domain \( 0 \leq x \leq 1 \) and the range \( 0 \leq y \leq 1 \). We are given intersections that are potential equilibrium points. To determine stability, we analyze the derivative of \( F \) at each point.

Step 2: Understand Equilibrium Points

Equilibrium points occur at \( F(x) = x \). These are points where the graphs of \( F(x) \) and \( y = x \) intersect. Stability requires \( |F'(x)| < 1 \), meaning small deviations will return to the equilibrium. If \( |F'(x)| > 1 \), the equilibrium is unstable.

Step 3: Work on Example a

For \( F(x) = 0.7x + 0.2 \), the derivative \( F'(x) = 0.7 \). At the equilibrium point \( (0.667, 0.667) \), \(|0.7| < 1\), thus it is a stable equilibrium.

Step 4: Work on Example b

For \( F(x) = 1.1x - 0.05 \), the derivative \( F'(x) = 1.1 \). At \( (0.5, 0.5) \), \(|1.1| > 1\), making it an unstable equilibrium.

Step 5: Work on Example c

For \( F(x) = x^2 + 0.1 \), the derivative \( F'(x) = 2x \). At \( (0.113, 0.113) \), \( |2(0.113)| = 0.226 < 1 \), stable. At \( (0.887, 0.887) \), \(|2(0.887)| = 1.774 > 1 \), unstable.

Step 6: Work on Example d

For \( F(x) = \sqrt{x} - 0.2 \), the derivative \( F'(x) = \frac{1}{2\sqrt{x}} \). At \( (0.076, 0.076) \), \( \frac{1}{2\sqrt{0.076}} > 1 \), unstable. At \( (0.524, 0.524) \), \( \frac{1}{2\sqrt{0.524}} < 1 \), stable.

Step 7: Work on Example e

For \( F(x) = -0.9x^2 + 2x - 0.2 \), the derivative \( F'(x) = -1.8x + 2 \). At \( (0.262, 0.262) \), \( |-1.8(0.262) + 2| < 1 \), stable. At \( (0.850, 0.850) \), \( |-1.8(0.850) + 2| < 1 \), stable.

Step 8: Work on Example f

For \( F(x) = -0.9x^2 + 2x - 0.1 \), the derivative \( F'(x) = -1.8x + 2 \). At \( (0.111, 0.111) \), \( |-1.8(0.111) + 2| < 1 \), stable. At \( (1.000, 1.000) \), \( |-1.8(1.000) + 2| = 0.2 < 1 \), stable.

Step 9: Work on Example g

For \( F(x) = x^3 + 0.2 \), the derivative \( F'(x) = 3x^2 \). At \( (0.879, 0.879) \), \( |3(0.879)^2| > 1 \), unstable.

Step 10: Work on Example h

For \( F(x) = 8x^3 - 12x^2 + 6x - \frac{1}{2} \), the derivative \( F'(x) = 24x^2 - 24x + 6 \). Calculate this at equilibrium points: at \( (0.146, 0.146) \), \( |-24(0.146)^2 + 24(0.146) - 6| < 1 \), stable; at \( (0.500, 0.500) \), unstable; at \( (0.854, 0.854) \), stable.

Key Concepts

Stability Analysis

When dealing with dynamical systems, understanding whether equilibrium points are stable or unstable is crucial. An equilibrium point is where the system can remain constant if undisturbed. To determine the stability of these points, we rely on the derivative of the function at these points.
Key insights include:

• If the absolute value of the derivative, say \(|F'(x)|\), at the equilibrium point is less than 1, the point is stable. This means that small disturbances away from this equilibrium will not grow and will instead shrink, guiding the system back to equilibrium.
• If \(|F'(x)|\) is greater than 1, the point is unstable. Here, disturbances or deviations will magnify, pushing the system further away from equilibrium over time.
• If \(|F'(x)|\) equals 1, the analysis might become more intricate, often requiring a deeper look into the specific system dynamics or additional methods.

By applying these principles, we effectively predict how a system behaves near its critical points.

Graphical Representation

Visualizing functions and intersections is a powerful tool in understanding dynamical systems. In problems like these, graphing the function \(F(x)\) alongside the line \(y = x\) can provide intuitive insights.
By graphing:

• The intersections of \(F(x)\) and \(y = x\) represent the equilibrium points. These are the places where the system can settle without change (equilibrium).
• Graphical analysis helps us see immediately where these points occur and often suggests where stability or instability might be present. For instance, noticing how steep or flat a function is at these intersection points can give clues about the derivative's magnitude.
• The range and domain, usually given as \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\), limit our view to the relevant portion of the graph, highlighting just the section where equilibrium points are possible within the constraints.

Thus, plotting serves not only as verification of mathematical results but also as a medium for deeper engagement with the problem.

Derivative Analysis

The derivative is a fundamental mathematical tool that informs us about the behavior of a function. In the context of determining stability in dynamical systems, the derivative plays a critical role.
Understanding Derivative Analysis for Stability:

• By taking the derivative of a given function \(F(x)\), noted as \(F'(x)\), we calculate how the function changes with small perturbations around a point.
• The key to determining the stability of the equilibrium is examining \(F'(x)\) at the equilibrium points. A derivative smaller than one in magnitude (\(|F'(x)| < 1\)) generally points to a damping effect where deviations lessen, leading to stability.
• Conversely, a higher derivative magnitude (\(|F'(x)| > 1\)) implies any small perturbation is amplified, marking instability.

This step of taking and analyzing the derivative underpins proper stability analysis, allowing predictions about how things evolve near equilibrium points.