Problem 12 - Thomas Calculus

Question

The autonomous differential equations represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0) .\) Which equilibria are stable, and which are unstable? \(\frac{d P}{d t}=3 P(1-P)\left(P-\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

Equilibria: \( P=0 \) (unstable), \( P=\frac{1}{2} \) (stable), \( P=1 \) (unstable).

Step 1: Identify Equilibria

Equilibrium points occur where \( \frac{d P}{d t} = 0 \). For the equation \( \frac{d P}{d t}=3 P(1-P)\left(P-\frac{1}{2}\right) \), set the right-hand side equal to zero: \( 3P(1-P)\left(P-\frac{1}{2}\right) = 0 \). This gives the equilibria at \( P = 0 \), \( P = \frac{1}{2} \), and \( P = 1 \).

Step 2: Determine Stability of Equilibria

To determine stability, analyze the sign changes in \( \frac{dP}{dt} \) around the equilibria. For each interval of \( P \), substitute a test point into \( 3 P(1-P)\left(P-\frac{1}{2}\right) \):- \( P < 0 \): Expression is negative- \( P = 0 \): Equilibrium- \( 0 < P < \frac{1}{2} \): Expression is positive- \( P = \frac{1}{2} \): Equilibrium- \( \frac{1}{2} < P < 1 \): Expression is negative- \( P = 1 \): Equilibrium- \( P > 1 \): Expression is positive

Step 3: Interpret Stability

Based on the phase line analysis: \- \( P = 0 \) is unstable (flows away for \( P > 0 \)), \- \( P = \frac{1}{2} \) is stable (flows toward for both \( P < \frac{1}{2} \) and \( P > \frac{1}{2} \)), \- \( P = 1 \) is unstable (flows away for \( P < 1 \)).

Step 4: Phase Line Diagram

Draw a phase line with equilibria marked at \( P = 0, \frac{1}{2}, \) and \( 1 \). Indicate the direction of flow with arrows showing \(\) - Left of \( P = 0 \): Arrow left (down),\(\) - Between \( P = 0 \) and \( P = \frac{1}{2} \): Arrow right (up),\(\) - Between \( P = \frac{1}{2} \) and \( P = 1 \): Arrow left (down),\(\) - Right of \( P = 1 \): Arrow right (up).

Step 5: Sketch Solution Curves

Using the phase line, sketch the solution curves in the \( P(t) \) plane. For starting points \( P(0) > 0 \), \( P(t) \) increases toward 1 if \( P(0) < \frac{1}{2} \), decreases toward \( \frac{1}{2} \) if \( P(0) > \frac{1}{2} \), and converges to 0 if \( P(0) = \frac{1}{2} \). While for \( P(0) > 1 \), \( P(t) \) increases indefinitely.

Key Concepts

Population Growth Models

When we talk about population growth models, we are essentially discussing mathematical representations that describe how populations change over time. A popular model used in such contexts is the *autonomous differential equation*, which expresses the rate of change of a population as a function of the current population size itself.

Let's consider the equation given in the exercise: \[ \frac{dP}{dt} = 3P(1-P)\left(P-\frac{1}{2}\right) \]In this model, the rate of change of the population \( P \) over time \( t \) depends on the current population size \( P \). This means that how fast or slow the population grows hinges directly on how many individuals are present at any given time.

This particular model accounts for factors like:

• Decrease in growth as population approaches capacity (represented by terms like \((1-P)\)).
• Critical threshold values where the population might stabilize or behave differently. In this equation, these thresholds appear at \( P = 0 \), \( P = \frac{1}{2} \), and \( P = 1 \).

Phase Line Analysis

Phase line analysis is a graphical tool that helps us visualize the behavior of solutions to autonomous differential equations, like the one in the population model. It provides insight into how different initial populations influence the system's evolution over time.

To carry out a phase line analysis:

• Identify equilibrium points by setting the derivative \( \frac{dP}{dt} \) to zero and solve for \( P \). In our scenario, these points are \( P = 0 \), \( P = \frac{1}{2} \), and \( P = 1 \).
• Analyze the sign of \( \frac{dP}{dt} \) in intervals around those equilibrium points to understand how the population behaves. For instance, if the derivative is positive in a region, it means population \( P \) will increase in that region; if negative, \( P \) will decrease.

With this analysis, you can determine whether equilibria act as attractors (stabilizing points) or repellers (unstable points). By marking these behaviors on a phase line, you get a clear picture of how populations shift and change direction over time.

Stability of Equilibria

Stability refers to whether a population will remain at or return to equilibrium points if slightly disturbed. Understanding the stability of equilibria helps predict long-term population trends.

In our exercise, we characterized equilibrium points as:

• **Unstable** at \( P = 0 \): If the initial population is slightly greater than zero, \( P(t) \) will increase and move away from this point.
• **Stable** at \( P = \frac{1}{2} \): Populations near this value tend to return, making it an attractive balance point.
• **Unstable** at \( P = 1 \): If disturbed, populations will move away, either increasing indefinitely or decreasing towards \( P = \frac{1}{2} \).

Recognizing the stability helps in predicting which population sizes are likely to be maintained naturally over time, and which sizes are prone to changing unless managed or controlled with external influences.

Solution Curves

Solution curves portray how population sizes change over time based on initial conditions. After performing phase line analysis, sketching solution curves helps in visualizing these dynamics in a more intuitive way.

For the specific problem given:

• If \( P(0) < \frac{1}{2} \), the solution curve indicates that \( P(t) \) will increase and approach the stable equilibrium at \( P = \frac{1}{2} \).
• For \( P(0) = \frac{1}{2} \), the population remains stable over time.
• If \( \frac{1}{2} < P(0) < 1 \), the population will decrease toward \( P = \frac{1}{2} \).
• For \( P(0) > 1 \), the population increases indefinitely.

These sketches of curves highlight transient behavior and equilibrium convergence of population sizes, enabling deeper insights into how long-term results align with initial population values.