Chaos and Ecology.

`The very simplest nonlinear dierence equations can possess an extraordinarily rich spectrum of dynamical behaviour...' - May 1976

`Chaos gives us a very dierent picture of the world in which we live'- Yorke 1989

As stated in section 1.2, there are various situations which promote chaos and these are fre- quently applicable to ecological systems. High dimensionality is one such criterion, so that ecosystems with many species and interactions may easily be chaotic (May, 1976 Fielding, 1991 Ferri#ere & Gatto, 1994). Similarly, size, age or spatial structuring of populations have been shown to promote chaos (Hastings, 1992 Sole & Valls, 1992a Lloyd, 1995 Ruxton, 1995).

Chaos is also more likely in the presence offeedback, either positive (raised growth rates) or negative (overcompensation, inducing time lags or delays), both of which are common in ecolog- ical community interactions34. There has recently been a demonstration that noise can cause

transient chaos to become permanent (Rand & Wilson, 1991a) the ubiquity of noise in ecology thus further promotes chaos.

In spite of these points, there is limited direct evidence of chaos in the natural world. Ecologists have oered various explanations for this. Some consider that chaos is maladaptive and that natural selection will act against chaotic systems ($Lomnicki, 1989 Mani, 1989). It is suggested that the large uctuations seen in chaotic (and oscillatory) systems will result in susceptibility to stochastic extinctions (Thomas et al., 1980 Mueller & Ayala, 1981 Fielding, 1991). Dispersal has been cited as a mechanism to promote non-chaotic dynamics (Hastings, 1993). A more controversial idea is that natural selection has excluded chaos in nature, but that anthropogenic intervention has increased the tendency of systems to be chaotic (Berryman & Millstein, 1989a Pool, 1989b).

The revolutionary articles of May (1974 1975 1976) introduced the concept of chaos in in- vertebrate population ecology using the logistic map and simple two-dimensional competition models. There has since been a deluge of ecological models which exhibit chaotic dynam- ics (Renshaw, 1994), including predator-prey models (Beddington et al., 1975 Hanski et al., 1993), resource-predator-prey models (Rand et al., 1994) and Leslie matrices (Guckenheimer et al., 1977). There has also been considerable interest about chaos in epidemiology (Bartlett, 1990), since the SEIR equations, which model the spread of diseases such as measles, were suggested to show chaotic dynamics35.

Spatially-extended models have provided several cases of chaos, both in forced continuum sys- tems (Tsonis et al., 1989) and discrete lattices of logistic and other one-dimensional maps

34Pimm & Hyman, 1987 Berryman & Millstein, 1989b Allen, 1990 Hunter & Price, 1992 Power, 1992

McGlade et al., 1994.

35Schaer & Kot, 1985 Pool, 1989a Rand & Wilson, 1991a Grenfell, 1992 Sidorowich, 1992 Bolker &

(Kaneko, 1985 1986 1989 Sole & Valls, 1992a Bascompte & Sole, 1994), the Nicholson-Bailey equation for host-parasite equations (Hassell et al., 1991 1994) and Lotka-Volterra systems (Sole & Valls, 1991 Sole et al., 1992b). Two-species CML have produced a new class of spatial dynamics, chaotic Turing structures, which consist of patches which have chaotic dynamics within them, but stable boundaries (Sole et al., 1992a Sole & Bascompte, 1993).

There has been less conclusive identication of chaos in empirical data, because of methodolog- ical problems, caused by the tendency of ecological time series to be short and noisy (Lloyd, 1994). Phase space reconstruction produced some evidence of chaos in lynx populations (Schaf- fer, 1984 1985a Godfray & Blythe, 1990) and measles epidemics (Schaer, 1985b). Forecasting techniques (Sugihara & May, 1991 Sugihara et al., 1990), tting to the SEIR equation (Olsen & Schaer, 1990 Grenfell, 1992) and spectral analysis (Olsen et al., 1988) have all claimed to prove that measles epidemics are chaotic. Some of these techniques have however been ques- tioned, in particular the forecasting techniques (Rand & Wilson, 1991b). The dynamics of voles (Hanski et al., 1993) and ant trails (Cole, 1991) are also potentially chaotic, while little clear evidence of cycles or chaos in plant population data has been found (Thrall et al., 1989 Rees & Crawley, 1991 Tilman & Wedin, 1991b).

Many ecologists have focused on detecting chaos by tting simple maps to experimental data. Frequently a one-dimensional map has its parameters estimated from a time series, then chaos is accepted or rejected for the natural or laboratory system depending on whether the map is chaotic for the estimated parameters. In the 1970s studies based on such techniques were used to suggest that insect populations rarely exhibited chaos or cycles. The key paper by Hassell et al. (1976) led most ecologists to reject any relevance of chaos to the natural world and to view it as a mathematical artefact of certain models (Pool, 1989b). More recent studies of two- and higher-dimensional maps have been much more suggestive of chaos (one-dimensional systems are inherently unlikely to be oscillatory or chaotic (Gatto, 1993)). This has produced various criticisms of the whole use of model tting in the detection of chaos, as the procedure is too sensitive both to the dimension and form of the selected map and the method of parameter estimation used (Nisbet et al., 1989 Morris, 1990 Bascompte & Sole, 1994).

The key to identifying chaos in time series depends on distinguishing it from genuinely random noise. A natural system certainly has some noise - environmental stochasticity - to which some measurement error is inevitably added. At rst it was thought that determinism and stochas- ticity were indistinguishable (May, 1976), but in recent years much eort has been put into their separation. In natural systems this is equivalent to identifying density-dependent (nonlinear) and density-independent (noise) eects (May, 1986 McGlade, 1994 Ellner & Turchin, 1995). In mathematical terms, the dierence between noise and chaos is dimensional: deterministic dynamics can be reduced to a nite-, often low-dimensional attractor, whereas uncorrelated noise is of innite dimension (chapter 7 Casdagli, 1991). In natural systems there will always be a balance between the two extremes of pure determinism and pure stochasticity.