## Concluding Remarks

Commented above, the persistent metapopulations (with their adequate mathematics) represent a counterexample to those model findings which were originally called for to uncrown the ecological paradigm of stability due to complexity. Other counterexamples of this kind appeared soon after the stability-versus-complexity theme had been coined into the field of population and community models. It was argued, for instance, that a few constraints imposed on the randomly constructed webs to create a statistical 'universe' of all possible webs with a fixed set of complexity parameters were still too loose to generate a sample of ecosystem structures. For example, a randomly constructed web of n species may happen to have n trophic levels, whereas real ecosystems, as was noted in the section entitled 'Qualitative stability and sign-stable community matrices', cannot have more than five to six trophic levels. Also, pure randomness may result in random formation of food loops where the top consumer is eaten by a species two levels lower in that food chain, which is certainly unrealistic. Moreover, an ecosystem should form on a primary resource species, that is, at least one row of the random matrix should not contain any pluses. All these additional constraints reduce the set of ecosystem prototypes to a very low fraction of the universe of randomly constructed webs, and Monte Carlo simulations within that fraction bet rather on the stability-due-to-complexity paradigm.

However, theoretical results of community matrix stability analyses, which can hardly be disproved as mathematical facts, still tend to support the opposite view. It can however not be stated that mathematics always proves complexity to counteract stability. For instance, the effect of predator-mediated coexistence, that is, stabilization of unstable competitors by the complication of the trophic structure, was both observed empirically and proved mathematically in the proper models.

Even indisputable mathematical facts can hardly put an end to any stability-versus-complexity dispute for some fundamental reasons. First, ecological stability has no stable definition in mathematical terms. While quantitative persistence gives the most general analog, the local Lyapunov stability suggests only one of various modes of persistence. Second, it is only a restricted class of systems (the dissipative ones of the general n and some particular cases of low n < 3) where local stability analysis comprehends the global dynamic behavior. Local stability can generally not guarantee global stability, while persistence does generally not require global stability. Third, various studies within the stability-versus-complexity theme usually differ in their stability or/and complexity definitions and particular setups. Different answers are therefore logical consequences from differences in the formal questions imposed.

Although stability-versus-complexity controversy has little chance to be ever resolved, its provoking and stimulating role is indisputable for theoretical and experimental studies in several related environments -fields, laboratories, mathematics, and computers.

See also: Ecological Niche; Metapopulation Models; Stability; Trophic Structure.

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