The Mathematical Structure of an Aquatic Ecosystem Model

As ecological models become more complex, the ques tion of whether they are stable becomes more diffi cult to answer. Some ecologists believe that a com plex system is more stable than a simple system because it is less dependent on any single path through it. Mathematicians, on the other hand, have shown that systems of equations are more likely to become unstable as the number of equations increases. Neither tendency may apply to ecological systems that include time-varying components and in which the number of significant interactions does not de pend strongly on the number of components.

In the case of the aquatic ecosystem model MS. CLEANER, solutions of the governing differential equations are stable even though the model contains up to 40 state variables and simulates many biologi cal, biochemical, microbial, and chemical processes. The stability is the result of feedback in the system and the sparseness of the interaction matrix. A qualitative analysis of the model shows that the functions used to represent predation have signifi cantly different forms at high, low, and intermediate nutrient levels. The different forms of this func tion cause unique, bounded solutions for high and low nutrient conditions and two stable, periodic solutions for intermediate conditions. The predic tion of a unique solution independent of the initial conditions in one case and more than one stable periodic solution in another case agrees with the published results of simulation.