Figure 1. Examples of nonlinear dynamics in laboratory populations. (a) Blowflies (Lucilia cuprina). (b) The protozoa Didinium nasutum (predator) and Paramecium aurelia (prey). (Data from Nicholson AJ (1957) Cold Spring Harbor Symposia on Quantitative Biology 22: 153–173; Luckinbill LS (1973) Ecology 54: 1320–1327.)

Figure 2. Examples of nonlinear dynamics in field populations. (a) Coffee leaf-miners (Leucoptera spp.) in Tanzania. (b) Larch budworm (Zeiraphera diniana) in Switzerland. (c) Voles (Microtus and Clethrionomys) in Finland. (d) Red grouse (Lagopus lagopus scotius) in Scotland. (Data from Bigger M (1973) Journal of Animal Ecology 42: 417–434; Baltensweiler W and Fischlin A (1988) In: Berryman AA (ed.) Dynamics of Forest Insect Populations, pp. 331–351. New York: Plenum Press; Hanski I et al. (1993) Nature 364: 232–235; Middleton AD (1934) Journal of Animal Ecology 3: 231–249.)

Figure 3. Time-series and phase space representations of a discrete-time host–parasitoid model (left panels) and a continuous-time predator–prey model with seasonal variation in the prey carrying capacity (right panels). Top rows: limit cycles. Middle rows: quasiperiodic cycles. Bottom rows: chaos. For the host–parasitoid model the parasitoid attack rate increases from top to bottom. For the predator–prey model the strength of seasonality increases from top to bottom (there is no seasonality in the top pane); note that the period of the predominant oscillation remains longer than one year. (Models from Beddington JR, Free CA and Lawton JH (1976) Nature 225: 58–60; Rinaldi S, Muratori S and Kuznetsov Y (1993) Bulletin of Mathematical Biology 55: 15–35.)

Figure 4. Bifurcation diagrams of (a) the logistic model (illustrating the period-doubling route to chaos) and (b) the host–parasitoid model from Figure 3 (illustrating the torus route to chaos). In each case a model parameter varies along the horizontal axis. Above that parameter value, successive values of one of the state variables are plotted on the vertical axis. A single point represents an equilibrium, two points represent periodic dynamics with period 2, and so on. Chaos and quasiperiodic dynamics result in a dense infilling of points. Notice the ‘windows’ of periodic dynamics within the chaotic and quasiperiodic regimes.