Application of ergodic theory in ecosystem modeling

Terrestrial ecosystem models are designed to assess the flux of energy, water, carbon and nitrogen corresponding to a given vegetation type. Ecosystem model validations use classical statistical methods (regression analysis of predicted versus observed values, paired t-statistics and error assessment procedures). These validations concentrate on static aspects of the model but fail to describe model dynamics. Recent research revealed that inconclusive outputs of large scale BGC-models resulted from instabilities in model dynamics. Ecosystem models are designed as diagnostic tools to study the behaviour of complex ecosystem processes. In this research we present ergodic theory as an excellent tool to provide us with estimates on the instabilities of ecosystem models. In our approach we will gain sound scientific knowledge by alternating the use of two principal descriptive views of reality; (i) the statistical view uses a set of observations on particular properties of individuals to extrapolate the behavior of the system from observations; (ii) the dynamic view explains the particular properties of observed system behavior using mechanistic descriptions of the interactions among the participating forces. We will use statistically generated climate data to run a dynamic ecosystem model. Model uncertainty will be assessed using statistical validation techniques, while model dynamics will be captured by ergodic theory. This flip-flop use of the two principal descriptive views is a major strength of our approach. The ergodic theory will enable us to assess five descriptive measures of system dynamics (i.e. Lyapunov exponents, entropy and three estimates for attractor dimension). Our hypothesis is that these five measures indicate the stability status of a given model simulation. If the model behaves stable then the accuracy and precision of model outputs will remain valid. If the model behaves unstable it may indicate either a reduced predictability or a reduction in the stability of a real world ecosystem. Outputs from this research project may have fundamental practical implications allowing us to compare the stability of different ecosystem types or the change in the stability according to a change in driving forces like climate change.

In the course of this project methods from higher mathematics (i.e. of the theory of dynamic systems) were applied to ecosystem modelling. As application case the ecosystems of the Congo basin were chosen, because, compared to European ecosystems, they were and are still facing low human impact, but nevertheless exhibited phases of rain forest regression and expansion in the past. The most important results of this research project can be summarized as follows: Modelling of the dynamics of the ecosystems of the Congo basin is possible with known accuracy and precision. The simulations for the rain forest exhibited a stable dynamics for areas where rain forest is the dominating ecosystem today, but unstable dynamics in regions that are dominated by savannas. The length of the annual dry season as well as the amount in annual precipitation determine the carbon storage capacity of the rain forest. However, they do not decide upon the stability of model dynamics. Unstable dynamics appear when the inter-annual variation in precipitation exceeds a certain threshold value, e.g., 27.5% for 1000 mms, 32.5% for 1500 mms, 37.5% for 2100 mms and 45% for 5000 mms of annual precipitation. Furthermore, a pronounced effect of Hysteresis, i.e. the stability of the rain forest versus climate change depends on whether the rain forest is already fully established or is just establishing after stand replacing disturbances, whereby the established rainforest exhibits a more stable dynamic than the forest that establishes from the scratch. In addition, this project made a first step to change the resilience of an ecosystem (i.e. the ability to recover after disturbances) from a theoretical concept into a measurable quantity. The correlation dimension of the modelled ecosystem seems to be a suitable measure for changes of the resilience of ecological systems. The correlation dimension changes measurably (from >8 to <7), before the rain forest ecosystem becomes unstable and exhibits forest breakdown.