Nonlinearities in mathematical ecology: Phenomena and models: Would we live in Volterra's world?

Presented is a critical survey of canonical nonlinear models in theoretical population ecology, namely single-species population, prey–predator, competition, migration within a metapopulation, and trophic chains. Various nonlinear effects, like hysteresis, structural instability, dissipative structures, dynamic chaos, etc., do exist in these models, but the problem how to detect these phenomena in real ecosystems is not yet solved. In the mathematics of nonlinear models, the central question is whether the simplest, i.e., Volterra-type, nonlinearity is sufficient to reproduce a variety of nonlinear phenomena in a given model or we need a more sophisticated formalism. Examples are considered where the Volterra models fail. Although fundamental physical principles, like, e.g., the mass conservation law, should work in ecology too, the ecological origin of the models often causes mathematical effects which are distinct from those in theoretical physics. For example, the trophic-chain model does reveal a kind of chaotic behaviour, but the “ecological strange attractor” occupies an intermediate position between Lorenz's and Feigenbaum's attractors; moreover, the phase volume of our system contracts, hence the system is dissipative (like a Lorenz's one) in spite of its matter conservation property. Nevertheless, when applied properly, physical concepts, like, e.g., the thermodynamic notion of exergy, give better insight both to the patterns of nonlinear ecosystem behaviour and to comparison of the patterns.     

Are all signals the same? Ontogenetic change in the response to conspecific and heterospecific chemical alarm signals by juvenile green sunfish (<Emphasis Type="Italic">Lepomis cyanellus</Emphasis>)

Within aquatic communities, individuals may gain survival benefits by responding to the chemical alarm signals of heterospecific prey guild members. Piscivorous individuals, however, should be selected to use such chemical signals as foraging cues. A variety of centrarchid species, such as largemouth bass (Micropterus salmoides), undergo an ontogenetic change in their response to the chemical alarm cues of heterospecific guild members, switching from antipredator to foraging responses. This ontogenetic shift should occur when potential foraging benefits outweigh any survival advantage gained from an antipredator response. To test this model, we exposed juvenile green sunfish (Lepomis cyanellus) to the skin extracts of conspecifics, a heterospecific prey guild member (finescale dace, Phoxinus neogeaus) or an allopatric heterospecific (green swordtails, Xiphophorus helleri). Juvenile sunfish exhibited a significant positive relationship between standard length and time spent moving and a significant negative relationship between length and time in a spine-erect posture, when exposed to dace skin extract, but not to either swordtail or conspecific skin extracts. Smaller individuals of less than 90 mm standard length (SL) decreased time moving and increased time with spines erect (indicating an antipredator response) while larger individuals (>90 mm SL) increased time moving and decreased time with spines erect (indicating a foraging response), when exposed to dace skin extract. Conversely, juvenile sunfish, regardless of size tested, always exhibited an antipredator response to conspecific skin extract. Sunfish exhibited no change in behaviour in response to swordtail skin extracts. These data further support our model of a threat sensitive trade-off in the response to chemical alarm signals by juvenile centrarchids.