Extended input-output flow analysis of ecosystems

Abstract

The compartmental model of ecosystems open to their environments is extended by including self-loop ‘flows’ at compartments. According to this extension, the Markovian model corresponding to the conventional compartmental model is naturally generalized into the Markovian model with self-loops. By means of this generalized Markovian model, it is proved that the self-loop flow at each compartment in the extended compartmental model represents the storage at the compartment, which, together with the throughflow at the same compartment, constitutes the inclusive throughflow at that compartment. It is shown, on the basis of the maximum entropy principle, that the amount of storage (and thus, of inclusive throughflow) can be explicitly determined in terms of the turnover rate (thus, existing energy and throughflow) at each compartment. This makes it possible to generalize the flow analysis into the one which can analyze the cause-effect (causation) relationship among compartments in terms of not only throughflows but inclusive throughflows. Since the inclusive throughflow represents not only the throughflow but storage, and, in effect, the total energy existing at each compartment, the generalized flow analysis should reveal a more substantial aspect of the total interaction among subsystems (compartments). This account is applied to two significant issues in ecology: the issue of energy (or matter) cycling in ecosystems and the issue on direct influences versus indirect ones. A general measure of cycling efficiency of ecosystems with respect to both throughflow and inclusive throughflow is derived based on the generalized Markovian model, and studied in terms of natural ecosystem examples. The explicit expressions for the direct and indirect influences from one subsystem (compartment) to another are derived on the basis of the classification of paths into direct influence paths and indirect influence ones. It is shown that the relationship between direct and indirect influences is invariant for the throughflow and inclusive throughflow cases, and it is suggested that indirect influences tend to dominate over direct ones as the system connectance increases. Finally, a general method for evaluating the effects of storages on the cycling efficiency of ecosystems and the relationship between direct and indirect influences is developed, and these effects are studied analytically as well as in terms of natural ecosystem examples.