Viability Models
Viability models allow us to simulate and analyze how a system evolves over time under constraints. They combine mathematical, logical, and numerical tools to represent situations where sustainability is central to the analysis.
1. Objective of Viability Models
A viability model does not seek to predict a single future, but rather to:
- Identify feasible trajectories,
- Determine viability conditions,
- Simulate the evolution of a system within defined limits (environmental, social, economic, etc.),
- Provide support for sustainable decision-making.
2. General Structure of a Model
A viability model generally includes:
- A state space Rn: the variables that describe the system.
-
Dynamics: in the form of equations or differential inclusions.
x˙(t)∈F(x(t))
- A set of constraints K⊆RnK: what the system must never violate.
- An objective (optional): to reach a target or avoid certain states.
3. Types of Models
a. Continuous Dynamic Models
These are the most widely used. They model a system in continuous time with differential equations or differential inclusions.
Examples:
- Natural resource management,
- Species population,
- Long-term economic models.
b. Discrete Models
Time is divided into steps: t0, t1, t2, ... t_0, t_1, t_2, ...
Useful for:
- Modeling periodic decision-making (months, years),
- Simulating systems where information is updated at regular intervals.
c. Hybrid Models
Combine continuous dynamics and discrete transitions:
- Automated systems or robots,
- Socio-technical models.
4. Application Examples
- Ecology: maintaining an ecosystem under pollution constraints.
- Sustainable economy: balance between production, consumption, and pollution.
- Agriculture: simulation of sustainable agricultural policies.
- Robotics: safe navigation trajectories in a complex environment.
5. Computational Models and Simulations
Tools like ViabLab allow:
- Visualizing viable systems,
- Exploring the viability core and capture basins,
- Testing different control policies,
- Validating complex models in simulated environments.
6. Limitations and Precautions
Even a well-constructed viability model:
- Does not provide a single solution,
- Is sensitive to data quality (and uncertainties),
- Must be interpreted within its context,
- Does not replace human reasoning, but supports it.
In summary
Viability models are powerful tools for exploring the sustainability of trajectories in complex systems. They do not predict the future, but show what is possible without violating the rules of the game.
