Static vs Dynamic Mathematical Models

Comparing Time-Invariant and Time-Dependent Modeling Approaches

Model Overview

This framework explores the fundamental differences between static and dynamic mathematical models as applied to sport science, building upon the concepts introduced in Module 3 (Static Models) and Module 4 (Dynamic Systems).

💡Prerequisites

It is highly recommended to review Module 3 (Static Models) and Module 4 (Dynamic Systems) before exploring this comparative framework.

Interactive Comparison

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Key Concepts

Static vs Dynamic Models

Static Models: Time-invariant, equilibrium-based representations assuming steady-state conditions
Dynamic Models: Time-dependent models that account for transient responses and system evolution

As introduced in Module 4: A dynamic model accounts for time-dependent changes in the state of a system, while a static (or steady-state) model assumes the system is in equilibrium and does not vary over time.

Understanding the Visualization

The interactive chart above demonstrates the key differences between static and dynamic modeling approaches using real experimental data from an incremental exercise test.

Left Panel: Static Model (Steady-State Analysis)

The scatter plot shows the steady-state relationship between exercise load and VO₂. Each point represents:

X-axis: Exercise load (watts) at each step
Y-axis: Average VO₂ during the final 60 seconds of each step (steady state)
Red dashed line: Linear regression fit

The linear equation reveals:

Intercept: Resting VO₂ (y-value when load = 0 watts)
Slope: Metabolic efficiency (mL O₂ consumed per watt of power)
R²: How well the linear model explains the steady-state data

Right Panel: Dynamic Response (Time-Series Analysis)

The time-series plot shows the real-time system behavior:

Blue line: Actual VO₂ measurements over time
Black line: Exercise load profile (step protocol)
Dual y-axes: VO₂ (left) and Load (right)

Key Insight: Static Predicts Dynamic Equilibrium

When you hover over any steady-state point (left panel), notice how the horizontal line appears in the time-series (right panel). This demonstrates that:

The static model predicts where the system will settle
The dynamic response shows how the system approaches that equilibrium
The transient behavior reveals system inertia and response characteristics

VO₂ Kinetics: Static vs Dynamic Formulations

Static Approach

The steady-state relationship can be expressed as:

\mathrm{VO}_{2,steady} = \mathrm{VO}_{2,rest} + \epsilon \cdot Load

Where $\epsilon$ is the metabolic efficiency (slope) and $\mathrm{VO}_{2,rest}$ is the intercept.

Dynamic Approach (from Module 4)

The dynamic formulation uses differential equations:

\tau \frac{d}{dt} VO_2(t) + VO_2(t) = \mathrm{VO}_{2,rest} + \epsilon \cdot Load(t)

This accounts for:

Real-time system response with time constant $\tau$
Rate of change considerations
Adaptation to changing inputs over time

Model Parameters Comparison

[PLACEHOLDER: Interactive parameter controls will be implemented here]

Static Model Parameters:

$A$ : Baseline value
$\Delta$ : Response amplitude
$\tau$ : Time constant (fitted parameter)

Dynamic Model Parameters:

$\tau$ : System time constant (inherent property)
$\epsilon$ : Efficiency coefficient
$VO_{2RES}$ : Resting consumption

Implementation Approaches

Static Model Implementation

=A + Delta*(1-EXP(-t/tau))

Dynamic Model Implementation (Finite Difference)

From Module 4, the recursive equation:

VO_{2,k+1} = VO_{2,k} + \frac{t_{k+1} - t_k}{\tau} \left( P_k \cdot \epsilon + VO_{2RES} - VO_{2,k} \right)

=VO2_k + (dt/tau)*(P_k*epsilon + VO2_RES - VO2_k)

When to Use Each Approach

Static Models Are Ideal For:

Steady-state analysis
Simple parameter fitting
Educational demonstrations
Quick approximations

Dynamic Models Are Essential For:

Transient analysis
Real-time simulations
System design and control
Realistic behavior prediction

🚨Model Selection

The choice between static and dynamic models should be based on your specific application. Static models sacrifice temporal accuracy for simplicity, while dynamic models provide realistic system behavior at the cost of complexity.

Excel Implementation Guide

[PLACEHOLDER: Detailed step-by-step comparison implementation]

Comparative Setup:

Column A: Time (seconds)
Column B: Power input P(t)
Column C: Static model output
Column D: Dynamic model output
Column E: Difference analysis

Key Formulas:

Static: Reference Module 3 equations
Dynamic: Reference Module 4 finite difference approach

Relevant in this course in Sport Science

Static Model Applications:

VO₂ kinetics analysis (Module 3)
Lactate kinetics clearance (Module 3)
Supercompensation principle visualization (Module 3)

Dynamic Model Applications:

Real-time VO₂ response simulation (Module 4)
Mountain bike suspension modeling (Module 4)
Locomotion dynamics analysis (Module 4)

Research and Practical Implications

[PLACEHOLDER: Extended applications and case studies]

This comparative framework demonstrates:

Model complexity trade-offs
Temporal resolution requirements
Parameter interpretation differences
Validation methodology considerations

📚Further Reading

For detailed mathematical foundations, see the differential equations treatment in Module 4 and the static model applications in Module 3. The mathematical modeling importance is well discussed by Clarke and Skiba, 2013.