Stability

Global stability

The stability of an ecosystem is defined by its inter-annual variability when forced by a scenario with a stationary annual cycle. Virtual Ecosystems are globally stable. That is a benefit of individual-based modelling. They do not suffer from the chaotic instability found in simulations derived from population-based models. (That is an artefact of how biofeedback is parametrized in such modelling.)

Annual cycle from a 16-year virtual ecosystem in a mesocosm held at a fixed location off the Azores with stationary ambient climate

Three emergent properties: the number of phytoplankton and zooplankton per square metre, and the concentration of dissolved nitrogen in the mixed layer.
Each variable is represented by two curves with daily values: the upper curve is the inter-annual mean; the lower curve is the inter-annual standard deviation.
Demographic noise is defined as the st.dev. divided by the mean each day.

Ensemble statistics

Ensemble mean and inter-instance standard deviation for 1st March on each of 16 years.
The ensemble demographic noise is defi ned as the st.dev. divided by the mean. Here it is about 3%.

Emergent ergodicity

A comparison of the annual cycle of demographic noise (standard deviation divided by mean for each day) as derived from

1. inter-annual variation over 16 years,
2. inter-instance variation in an ensemble of 16 instance for one year,
3. inter-instance variation in an ensemble of 16 instance for 16 years.

The difference measures the deviation from ergodicity. In phtyoplankton it is less than 1%, while in zooplankton it reaches 5% after reproduction.

This is due to an unexpected emergent property , namely inheritence of zooplankton winter body weight.

34-year Virtual Ecosystem

The numbers of phytoplankton and zooplankton on 1st March each year

Insensivity to initial conditions

Poincaré maps of the demographic attractor on 1st March each year for four initial conditions.
This is the crucial proof of stability.

Annual demographic attractor

Poincaré maps with logarithmic scales. Each dot represents the mean value for one day of the year:
(Upper panel) Number of zooplankton versus number of phytoplankton
(Lower panel) Mixed layer nitrogen concentration versus number of phytoplankton.