When is a complex ecosystem stable?

This work represents an attempt to formalize and partially resolve in the framework of Liapunov's stability theory the “complexity vs stability” problem in model ecosystems.

Abstract

This work represents an attempt to formalize and partially resolve in the framework of Liapunov's stability theory the "complexity vs stability" problem in model ecosystems. Sufficient (and sometimes necessary and sufficient) conditions are derived for the stability of models under structural perturbations caused by time-varying nonlinear interactions among species in the community. The conditions guarantee that the community is stable despite arbitrary bounded variations in the interactions, which include the reduction of the number of species in the community, and the changing of species from predator-prey to competitive interaction over the time interval, as well as interaction effects such as cutting trophic links between species, predator switching, and saturation of predator attack capacity. Instability of ecomodels is formulated in the context of variable interactions among species. Sufficient conditions for instability are derived, which can help in producing the necessary condition for stability of the ecomodels. Stability regions are also studied, in view of the fact that most of the useful models of population dynamics are not globally stable. Hierarchic models of ecosystems are considered which describe communities decomposed into "blocks" of species representing various trophic levels or subcommunities. Stability of the total trophic web is determined on the basis of the aggregate model where each subcommunity is represented by a single variable, and which has order equal to the number of subcommunities. The aggregate model not only represents a reduction of the dimensionality of the stability problems, but also provides an insight into essential structural properties of the interacting subcommunities.