Lab · food chain

What happens if two systems start in almost exactly the same way?

Same rules. Almost the same start. No added randomness.

This is an abstract three-level food-chain model, not a real ecosystem, and not a model of Global Polycrisis.

Prey Consumer Top predator

A — reference

Reference start. It follows the model’s rules, with no noise.

B — almost identical

Same start, plus one millionth on prey only. Same rules, still no noise.

The two systems follow exactly the same rules and differ at the start by only one millionth on the prey population. What do you think will happen? They will stay almost identical. Their paths will eventually become very different. I don’t know.

Relative population levels

These are not headcounts of animals. x, y and z are dimensionless variables. A and B share the same scale, from the same model variables.

The three cards share the same time window: the three levels’ peaks need not fall at the same instant.

Prey (x)

0.622

Recent evolution — same time window

Consumer (y)

0.124

Recent evolution — same time window

Top predator (z)

8.979

Recent evolution — same time window

For now, A and B remain almost indistinguishable.

Measure the gap

You have just seen the populations. This curve measures the gap between the two simulations. Because that gap is tiny at first, the logarithmic scale acts as a magnifying glass: the data themselves are not amplified.

gap on prey (solid) on the consumer (dashes) on the predator (dots)

Prey gap: 0.000001 — about 1 times the initial millionth.

Even so, the overall shape remains

The two histories no longer match point for point, yet they stay in the same cage. That cage is sometimes called an attractor; here it is simply the model’s global structure.

This view shows the global structure in which the paths remain constrained.

It is not used to measure the gap between A and B. The magnifying-glass curve does that.

At this initial difference, A and B still look glued in the overview: that is expected. Read the gap on the curve above, not by splitting the ribbons.

A name for what you have just seen

No randomness was added, and yet the paths eventually become very different.

This kind of behaviour has a name: deterministic chaos.

This shows

Same rules ≠ the same long-run paths
A tiny initial difference can matter
Diverging still allows a shared overall structure

This does not show

That the system is random
That this model can predict a named ecosystem
That this model stands for Global Polycrisis

Why is this experiment on the Observatory?

This experiment does not model Polycrisis.

It lets us explore a general property of some complex systems: knowing a system’s rules does not always mean we can forecast its long-run path precisely.

It helps separate three things: knowing the mechanisms, forecasting a trajectory, and understanding a system’s structure.

Read the in-depth briefing (French)

Explore the three populations over time

A is a solid line, B is dashed. You can move time, change speed, or read the current state.

Current state (numbers)

time0.0000

A · prey0.6223

A · consumer0.1241

A · predator8.9791

B · prey0.6223

B · consumer0.1241

B · predator8.9791

Method, sources, advanced options

The only initial difference is 0.000001 (one millionth) on B’s prey. No random draw is added.

Hastings and Powell (1991) model, tea-cup parameters. RK4 integration, time step 0.01. Gaps are computed on scientific coordinates, before any 3D drawing.

The gauges show a relative population level: x, y and z are dimensionless variables, not animal counts. The graphic scale is the same for A and B (prey 0–1, consumer 0–0.5, predator 7.5–10.5) and does not change the data.

Flip the 3D vertical axis (a figure convention, not a change of the model)

The 3D drawing may flip the vertical axis and pack height to match a published figure. That does not change the model.

Sources

Hastings, A., Powell, T. (1991). Chaos in a Three-Species Food Chain. Ecology 72(3), 896-903. DOI 10.2307/1940591.
Dansereau et al. (2020). Replication ReScience C. Conditions initiales des figures non données chez Hastings–Powell 1991.
Parameters: a1=5, b1=3, a2=0.1, b2=2, d1=0.4, d2=0.01.
Companion briefing (French): model, limits and references.