DE-BOOK Parameter Analysis

Section Parameter Analysis

[provisional cross-reference: IN PROGRESS]

Many mathematical models include one or more parameters—constants that represent things like birth rates, drug dosage, resource limits, or physical constants. These parameters aren’t just placeholders—they often control the qualitative behavior of the system. Small changes in a parameter’s value can cause major shifts in the solution’s behavior. Understanding those changes is the goal of parameter analysis.

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In this section, we focus on the most important modeling insight: how equilibria and their stability depend on parameters. When a parameter crosses a critical threshold, the model’s behavior can shift dramatically—this is called a bifurcation.

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Subsection Bifurcation Analysis

A bifurcation is a qualitative change in the system’s dynamics caused by varying a parameter. In the context of differential equations, it usually means that the number or stability of equilibrium solutions changes at certain critical values.

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Consider the one-parameter system:

\begin{equation*} \frac{dx}{dt} = f(x, \mu), \end{equation*}

where \(\mu\) is a parameter. As \(\mu\) changes:

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Existing equilibria may change from stable to unstable (or vice versa).

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New equilibria may appear or disappear altogether.

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In some cases, the long-term behavior of the entire system changes.

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The basic workflow for bifurcation analysis is:

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Find equilibria: Solve \(f(x, \mu) = 0\) for \(x\) in terms of the parameter \(\mu\text{.}\)

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Classify stability: Compute \(\frac{\partial f}{\partial x}\) at each equilibrium. The sign determines whether the equilibrium is a sink (stable) or a source (unstable).

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Track changes: See how the equilibria and their stability change as \(\mu\) varies. Identify critical values where the behavior “flips.”

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Consider the parameterized system:

\begin{equation*} \frac{dx}{dt} = \mu - x^2\text{.} \end{equation*}

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First, we find the equilibria:

\begin{equation*} \mu - x^2 = 0 \quad \Rightarrow \quad x = \pm \sqrt{\mu}\text{.} \end{equation*}

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To see how the equilibria depend on \(\mu\text{,}\) solve for equilibrium:

For \(\mu > 0\text{:}\) two equilibria exist (\(x = \pm \sqrt{\mu}\)).

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For \(\mu = 0\text{:}\) a single equilibrium (\(x = 0\)).

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For \(\mu < 0\text{:}\) no real equilibria.

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Stability analysis shows that one branch is stable and the other is unstable when \(\mu > 0\text{.}\) The bifurcation diagram reveals a “collision” of equilibria at \(\mu=0\text{,}\) called a saddle-node bifurcation.

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The results are often summarized in a bifurcation diagram—a picture showing:

The equilibria plotted against \(\mu\text{.}\)

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Solid lines for stable equilibria, dashed lines for unstable ones.

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Critical parameter values where the diagram changes shape.

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Figure 106 shows the bifurcation diagram for the previous example.

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Figure 106. Bifurcation diagram for \(\frac{dx}{dt} = \mu - x^2\)
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Bifurcation diagrams are a cornerstone of modeling—they show, at a glance, how a system’s behavior shifts as conditions change.

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Subsection Why Parameter Analysis Matters in Modeling

Why should we care about bifurcations? Because in applied problems, parameters represent real things: a harvest rate in an ecology model, a dosage in a drug model, or an investment threshold in an economics model. Changing those parameters changes the system.

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The most important modeling insight:

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Stable equilibria correspond to long-term states the system will settle into.

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Bifurcation points are thresholds where those long-term states shift—suddenly and sometimes irreversibly.

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For example:

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In epidemiology, there’s often a threshold vaccination rate: below it, disease spreads (unstable equilibrium); above it, the disease dies out (stable equilibrium).

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In ecology, a critical harvest rate might push a fish population from stable sustainability to collapse.

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📤 Wrap-Up.

🗝️ Key Takeaways...

Parameters control behavior: Changing them can add, remove, or flip the stability of equilibria.

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Bifurcation diagrams show where those changes occur and summarize the system’s response to parameter variation.

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Critical parameter values in a model often correspond to real-world thresholds—places where “just a little more” or “just a little less” has huge consequences.

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Check Your Understanding.

Checkpoint 107. 📖❓ Parameter Analysis.

(a) Spot the Bifurcation.

The model \(\frac{dx}{dt} = \mu - x^2\) changes behavior as \(\mu\) varies. What happens when \(\mu\) crosses zero?

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Two equilibria collide and disappear.
Correct—this is the hallmark of a saddle-node bifurcation.
The system gains an oscillation.
This model is one-dimensional; it cannot generate oscillations.
The system’s slope changes sign but equilibria stay the same.
No—the set of equilibria itself changes.

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