Lesson 10 (09/16/2026): Equilibrium Solutions and Stability (2.2)

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Lesson 10 (09/16/2026): Equilibrium Solutions and Stability (2.2)

Warmup:

  1. \( P'(t) = P(2-P), \quad P(0) = 3 \)
  2. \( P'(t) = -P(2-P), \quad P(0) = 3 \)

Suppose \( P_1(t) \) and \( P_2(t) \) are solutions to (1) & (2) respectively. Determine \( \lim_{t \to \infty} P_1(t) \) and \( \lim_{t \to \infty} P_2(t) \).

\( P = 0, P = 2 \) are Equilibrium Solutions

Phase plot and solution curves for the autonomous differential equation P' = P(2-P). Horizontal equilibrium lines are drawn at P = 0 and P = 2. In the region P > 2, the derivative is negative (labeled P' = P(2-P) < 0), with solution curves decreasing asymptotically toward P = 2 with directional arrows pointing down and right. In the region 0 < P < 2, the derivative is positive (labeled P' = P(2-P) > 0), with solution curves increasing asymptotically toward P = 2 with directional arrows pointing up and right. The labeled equation below the plot is P' = P(2-P).
Visual Description: Phase plot and solution curves for the autonomous differential equation P' = P(2-P). Horizontal equilibrium lines are drawn at P = 0 and P = 2. In the region P > 2, the derivative is negative (labeled P' = P(2-P) < 0), with solution curves decreasing asymptotically toward P = 2 with directional arrows pointing down and right. In the region 0 < P < 2, the derivative is positive (labeled P' = P(2-P) > 0), with solution curves increasing asymptotically toward P = 2 with directional arrows pointing up and right. The labeled equation below the plot is P' = P(2-P).

\[ P' = P(2-P) \]

Phase plot and solution curves for the autonomous differential equation P' = -P(2-P). Horizontal equilibrium lines are shown at P = 2 and P = 0. In the region P > 2, the derivative is positive (labeled P > 2, P' > 0), with solution curves moving upward away from P = 2. In the region 0 < P < 2, the derivative is negative (labeled 0 < P < 2, P' < 0), with solution curves decreasing away from P = 2 and approaching P = 0 with downward directional arrows. The labeled equation below the plot is P' = -P(2-P).
Visual Description: Phase plot and solution curves for the autonomous differential equation P' = -P(2-P). Horizontal equilibrium lines are shown at P = 2 and P = 0. In the region P > 2, the derivative is positive (labeled P > 2, P' > 0), with solution curves moving upward away from P = 2. In the region 0 < P < 2, the derivative is negative (labeled 0 < P < 2, P' < 0), with solution curves decreasing away from P = 2 and approaching P = 0 with downward directional arrows. The labeled equation below the plot is P' = -P(2-P).

\[ P' = -P(2-P) \]

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Autonomous ODEs and Critical points

time independent ODE

\[ \frac{dy}{dt} = f(y) \]

"Another name for equilibrium solution"

\( y = c \) such that \( f(c) = 0 \)

\[ \leadsto \frac{dy}{dt} = \frac{dc}{dt} = 0 \]

"\( y = c \)" is a solution

Examples:

1.

\( p = 0, p = 2 \) are two critical points

2.

\( y = 1, y = 0, y = -1 \) are three critical points

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Stability of Critical Points

e.g.:
Graph showing solution trajectories for the autonomous differential equation p'(t) = p(2-p) plotted against time t. The vertical axis represents p and the horizontal axis represents t. Two horizontal lines mark the critical points at p = 2 and p = 0. A solution curve beginning above p = 2 curves downwards towards the asymptote p = 2 with a right-pointing arrow indicating the direction of time. A second curve starting just above the line p = 0 curves upward asymptotically towards p = 2, also marked with an arrow pointing along the trajectory towards p = 2.
Visual Description: Graph showing solution trajectories for the autonomous differential equation p'(t) = p(2-p) plotted against time t. The vertical axis represents p and the horizontal axis represents t. Two horizontal lines mark the critical points at p = 2 and p = 0. A solution curve beginning above p = 2 curves downwards towards the asymptote p = 2 with a right-pointing arrow indicating the direction of time. A second curve starting just above the line p = 0 curves upward asymptotically towards p = 2, also marked with an arrow pointing along the trajectory towards p = 2.

Start with any initial value close to \( p = 2 \)

\( p(t) \) is close to \( 2 \) for all \( t > 0 \)

Call \( p = 2 \) as stable critical point or stable equilibrium

\( p = 0 \) is "unstable" bcz if you start close to \( 0 \), you going away as \( t > 0 \)

Def 1.3: \( y' = f(y) \quad f(c) = 0 \leadsto y = c \) is critical point

stable if \( \forall\ \varepsilon > 0 \), \( \exists\ \delta > 0 \) such that

\[ |y(0) - c| < \varepsilon \implies |y(t) - c| < \delta \quad \forall\ t > 0. \]

\( y = c \) is unstable if it is not stable.

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Example: \( y' = (y - 1)(y - 3) \)

Critical points: \( y = 1 \), \( y = 3 \)

Phase portrait and solution curves in the t-y plane for y' = (y - 1)(y - 3). The vertical axis is labeled y and horizontal axis is labeled t. Two horizontal equilibrium lines are plotted at y = 3 and y = 1. For y > 3, trajectories curve upwards with positive slope since y' > 0, indicating y = 3 is unstable. For 1 < y < 3, trajectories curve downwards toward y = 1 since y' < 0. For y < 1, trajectories curve upwards toward y = 1 since y' > 0. The equilibrium at y = 1 is stable as nearby solutions approach it.
Visual Description: Phase portrait and solution curves in the t-y plane for y' = (y - 1)(y - 3). The vertical axis is labeled y and horizontal axis is labeled t. Two horizontal equilibrium lines are plotted at y = 3 and y = 1. For y > 3, trajectories curve upwards with positive slope since y' > 0, indicating y = 3 is unstable. For 1 < y < 3, trajectories curve downwards toward y = 1 since y' < 0. For y < 1, trajectories curve upwards toward y = 1 since y' > 0. The equilibrium at y = 1 is stable as nearby solutions approach it.
  • \( y > 3 \implies y' > 0 \implies y = 3 \) is unstable
  • \( y = 3 \implies y' = 0 \)
  • \( 1 < y < 3 \implies y' < 0 \)
  • \( y = 1 \implies y' = 0 \implies y = 1 \) is stable
  • \( y < 1 \implies y' > 0 \)

OR Look only in \( y \)-direction

One-dimensional phase line representing dynamics in the y-direction for y' = (y - 1)(y - 3). Critical points are marked at y = 1 with y' = 0 and y = 3 with y' = 0. For y < 1, there is a circled plus sign and an arrow pointing right toward y = 1, labeled y' > 0. For 1 < y < 3, there is a circled minus sign and an arrow pointing left toward y = 1, labeled y' < 0. For y > 3, there is a circled plus sign and an arrow pointing right away from y = 3, labeled y' > 0.
Visual Description: One-dimensional phase line representing dynamics in the y-direction for y' = (y - 1)(y - 3). Critical points are marked at y = 1 with y' = 0 and y = 3 with y' = 0. For y < 1, there is a circled plus sign and an arrow pointing right toward y = 1, labeled y' > 0. For 1 < y < 3, there is a circled minus sign and an arrow pointing left toward y = 1, labeled y' < 0. For y > 3, there is a circled plus sign and an arrow pointing right away from y = 3, labeled y' > 0.

\( y = 1 \implies \) solution going towards it \( \implies \) Stable

\( y = 3 \implies \) going away, so unstable

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eg: \( y' = (y+3)(y)(y-4) \)

Critical Points: \( (y+3)y(y-4) = 0 \)

\[ y = -3, \quad y = 0, \quad y = 4 \]

Stability:

Phase line diagram determining stability for y' = (y+3)y(y-4). A horizontal real number line marks three critical points where y' = 0: y = -3, y = 0, and y = 4. - For y < -3: the sign analysis is (-)(-)(-) resulting in y' < 0, with a blue arrow pointing to the left. - For -3 < y < 0: the sign analysis is (+)(-)(-) resulting in y' > 0, with a blue arrow pointing to the right. - At y = -3: arrows point away in both directions, labeled "Repelling" and "y = -3 Unstable". - For 0 < y < 4: the sign analysis is (+)(+)(-) resulting in y' < 0, with a blue arrow pointing to the left. - At y = 0: arrows from left and right both point toward y = 0, labeled "Attracting" and "y = 0 is stable". - For y > 4: the sign analysis is (+)(+)(+) resulting in y' > 0, with a blue arrow pointing to the right. - At y = 4: arrows point away, labeled "Repelling" and "y = 4 unstable".
Visual Description: Phase line diagram determining stability for y' = (y+3)y(y-4). A horizontal real number line marks three critical points where y' = 0: y = -3, y = 0, and y = 4. - For y < -3: the sign analysis is (-)(-)(-) resulting in y' < 0, with a blue arrow pointing to the left. - For -3 < y < 0: the sign analysis is (+)(-)(-) resulting in y' > 0, with a blue arrow pointing to the right. - At y = -3: arrows point away in both directions, labeled "Repelling" and "y = -3 Unstable". - For 0 < y < 4: the sign analysis is (+)(+)(-) resulting in y' < 0, with a blue arrow pointing to the left. - At y = 0: arrows from left and right both point toward y = 0, labeled "Attracting" and "y = 0 is stable". - For y > 4: the sign analysis is (+)(+)(+) resulting in y' > 0, with a blue arrow pointing to the right. - At y = 4: arrows point away, labeled "Repelling" and "y = 4 unstable".

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Another way!

Phase portrait sketch of an autonomous first-order differential equation showing equilibrium solutions and solution curves. A vertical black reference axis crosses three horizontal purple lines representing the equilibrium solutions: the top line at \(y = 4\) with \(y' = 0\), the middle line at \(y = 0\) with \(y' = 0\), and the bottom line at \(y = -3\) with \(y' = 0\). Blue arrows illustrate the behavior of solution trajectories across the regions: for \(y > 4\), \(y' > 0\) with solutions curving upward; for \(0 < y < 4\), \(y' < 0\) with solutions decreasing toward \(y = 0\); for \(-3 < y < 0\), \(y' > 0\) with solutions increasing away from \(y = -3\) toward \(y = 0\); and for \(y < -3\), \(y' < 0\) with solutions decreasing downward away from \(y = -3\).
Visual Description: Phase portrait sketch of an autonomous first-order differential equation showing equilibrium solutions and solution curves. A vertical black reference axis crosses three horizontal purple lines representing the equilibrium solutions: the top line at \(y = 4\) with \(y' = 0\), the middle line at \(y = 0\) with \(y' = 0\), and the bottom line at \(y = -3\) with \(y' = 0\). Blue arrows illustrate the behavior of solution trajectories across the regions: for \(y > 4\), \(y' > 0\) with solutions curving upward; for \(0 < y < 4\), \(y' < 0\) with solutions decreasing toward \(y = 0\); for \(-3 < y < 0\), \(y' > 0\) with solutions increasing away from \(y = -3\) toward \(y = 0\); and for \(y < -3\), \(y' < 0\) with solutions decreasing downward away from \(y = -3\).

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Example: Logistic Model with Harvesting

\[ \frac{dP}{dt} = 2P - P^2 - h \]

Logistic model with “harvesting”, \( h \) is constant

Critical points depend on \( h \):

to find:

\[ 2P - P^2 - h = 0 \implies P^2 - 2P + h = 0 \]

Quadratic formula

\[ ax^2 + bx + c = 0 \] \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

\( a = 1 \), \( b = -2 \), \( c = h \)

\[ P = \frac{2 \pm \sqrt{4 - 4h}}{2} \] \[ P = 1 \pm \sqrt{1 - h} \]

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\(h > 1\):

\[ 1 - h < 0 \]

\[ ax^2 + bx + c = 0 \implies b^2 - 4ac < 0 \implies \text{No solutions} \implies \begin{aligned} &a < 0 \\ &\implies ax^2 + bx + c < 0 \end{aligned} \]

\[ \implies P' = 2P - P^2 - h \text{ has no critical points} \]

\[ \implies P' < 0 \text{ for any value of } P. \]

Irrespective of \(P(0)\), all fish are eventually harvested or dead.

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\( h = 1 \):

\[ P' = 2P - P^2 - 1 = -[P - 1]^2 \]

\( P = 1 \) is critical point

A one-dimensional phase line diagram for the autonomous differential equation P' = -[P-1]^2. A horizontal line shows a critical point at P = 1. To the right of P = 1 (where P > 1), a blue arrow points to the left towards P = 1, labeled below with P' < 0. To the left of P = 1 (where P < 1), a blue arrow also points to the left away from P = 1 towards 0, labeled below with P' < 0. Below the phase line, the equilibrium is labeled as 'semi stable'.
Visual Description: A one-dimensional phase line diagram for the autonomous differential equation P' = -[P-1]^2. A horizontal line shows a critical point at P = 1. To the right of P = 1 (where P > 1), a blue arrow points to the left towards P = 1, labeled below with P' < 0. To the left of P = 1 (where P < 1), a blue arrow also points to the left away from P = 1 towards 0, labeled below with P' < 0. Below the phase line, the equilibrium is labeled as 'semi stable'.

"semi stable"

Qualitatively:

if \( P(0) > 1 \)
eventually fish population stabilize at \( P = 1 \)

if \( P(0) < 1 \)
eventually all fish are harvested, or dead.

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\( h < 1 \)

\[ p' = 2p - p^2 - h \]

\(\rightsquigarrow\) two critical points:

\[ 1 + \sqrt{1-h}, \quad 1 - \sqrt{1-h} \]

eg: \( h = 0 \)
Phase line diagram along the horizontal axis with critical points at p = 0 and p = 2. For p < 0, p' < 0. In the interval between 0 and 2, p' > 0 with an arrow pointing right toward 2. For p > 2, p' < 0 with an arrow pointing left toward 2, showing that p = 2 is a stable equilibrium point.
Visual Description: Phase line diagram along the horizontal axis with critical points at p = 0 and p = 2. For p < 0, p' < 0. In the interval between 0 and 2, p' > 0 with an arrow pointing right toward 2. For p > 2, p' < 0 with an arrow pointing left toward 2, showing that p = 2 is a stable equilibrium point.

\( p = 2 \) is stable \( \Rightarrow \) whatever population you start with, population stabilizes at \( p = 2 \) eventually.

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Similar behaviour for any \( h < 1 \)

two critical points

\[ c_1 = 1 - \sqrt{1-h}, \qquad c_2 = 1 + \sqrt{1-h} \] \[ p' = 2p - p^2 - h \]

Phase line diagram along a horizontal line representing the differential equation p' = 2p - p^2 - h. Two critical points c_1 and c_2 are marked on the line, with c_1 to the left of c_2. To the left of c_1, a circled minus sign indicates a negative rate of change (p' < 0). Between c_1 and c_2, a circled plus sign and an arrow pointing right toward c_2 indicate that p increases toward c_2. To the right of c_2, a circled minus sign and an arrow pointing left toward c_2 indicate that p decreases toward c_2, demonstrating that c_2 is a stable equilibrium.
Visual Description: Phase line diagram along a horizontal line representing the differential equation p' = 2p - p^2 - h. Two critical points c_1 and c_2 are marked on the line, with c_1 to the left of c_2. To the left of c_1, a circled minus sign indicates a negative rate of change (p' < 0). Between c_1 and c_2, a circled plus sign and an arrow pointing right toward c_2 indicate that p increases toward c_2. To the right of c_2, a circled minus sign and an arrow pointing left toward c_2 indicate that p decreases toward c_2, demonstrating that c_2 is a stable equilibrium.

population eventually stabilizes at \( c_2 \)