Persistence of species obeying difference equations

The question of persistence of interacting species is one of the most important in theoretical ecology; when the system is governed by difference equations this question is particularly difficult to resolve because of the complicated dynamics of the model. The problem has usually been tackled via the concepts of asymptotic stability and global asymptotic stability, however the first is not a strong enough restriction since only orbits starting near a rest point are guaranteed to converge to the rest point, while the second as well as usually being extremely difficult to establish is surely too strong a condition, since it rules out for example a stable limit cycle. It is proposed here that a more biologically realistic criterion, and incidentally one which turns out to be more tractable, is that of cooperativeness, where all orbits are required eventually to enter and remain in a region at a non-zero distance from the boundary (corresponding to a zero value of at least one population). A theorem is proved giving conditions for cooperativeness, and is applied to some examples of predator-prey interactions, simple conditions on the parameters being obtained even for some ranges of the parameters where the dynamics are chaotic.     

The role of herbivorous fishes and urchins in coral reef communities

Synopsis Herbivorous fishes and invertebrates are conspicious elements of coral reef communities where they predominate both in numbers and biomass. Herbivores and the coral reef algae on which they feed represent a co-evolved system of defense and counter-defense. Algal species have developed toxic, structural, spatial and temporal defense or escape mechanisms, while the herbivores employ strategies that involve anatomical, physiological and behavioral adaptations. Current research demonstrates that many reef fishes are highly selective in the algae they consume. Food selection in these fishes may be correlated with their morphological and digestive capabilities to rupture algal cell walls. Sea urchins select more in accordance with relative abundance, although certain algal species are clearly avoided.The determinants of community structure on coral reefs have yet to be established but evidence indicates a strong influence by herbivores. Reef herbivores may reduce the abundance of certain competitively superior algae, thus allowing corals and cementing coralline algae to survive. We discuss how the foraging activities of tropical marine herbivores affect the distribution and abundance of algae and how these activities contribute to the development of coral reef structure and the fish assemblages which are intimately associated with reef structure.This paper forms a part of the proceedings of a mini-symposium convened at Cornell University, Ithaca, N.Y., 18–19 May 1976, entitled Patterns of Community Structure in Fishes (G. S. Helfman, ed.).