Nonlinear Dynamics and Chaos

Abstract

Nonlinear dynamics deals with more‐or‐less regular fluctuations in system variables caused by feedback intrinsic to the system (as opposed to external forces). Chaos is the most exotic form of nonlinear dynamics, in which deterministic interactions produce apparently irregular fluctuations, and small changes in the initial state of the system are magnified through time.7

Keywords: chaos; population cycles; population dynamics; nonlinear time‐series analysis

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 Biology22: 153–173; Luckinbill LS (1973) Ecology54: 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 Ecology42: 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) Nature364: 232–235; Middleton AD (1934) Journal of Animal Ecology3: 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) Nature225: 58–60; Rinaldi S, Muratori S and Kuznetsov Y (1993) Bulletin of Mathematical Biology55: 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 (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.

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References

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Ellner S and Turchin P (1995) Chaos in a noisy world: new methods and evidence from time‐series analysis. American Naturalist 145: 343–375.

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Further Reading

Ferrière R and Fox GA (1995) Chaos and evolution. Trends in Ecology and Evolution 10: 480–485.

Gleick J (1987) Chaos: Making a New Science. New York: Viking.

Hastings A, Hom CL, Ellner S, Turchin P and Godfrey HCJ (1993) Chaos in ecology: is Mother Nature a strange attractor? Annual Review of Ecology and Systematics 24: 1–33.

May RM (1973) Stability and Complexity in Model Ecosystems. Princeton: Princeton University Press.

May RM (1986) When two and two do not make four: nonlinear phenomena in ecology. Proceedings of the Royal Society of London B 228: 241–266.

Pahl‐Wostl C (1995) The Dynamic Nature of Ecosystems: Chaos and Order Intertwined. Chichester: Wiley.

Royama T (1992) Analytical Population Dynamics. London: Chapman and Hall.

Schaffer WM and Kot M (1986) Chaos in ecological systems: the coals that Newcastle forgot. Trends in Ecology and Evolution 1: 58–63.

Stone L and Ezrati S (1996) Chaos, cycles and spatiotemporal dynamics in plant ecology. Journal of Ecology 84: 279–291.

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Kendall, Bruce E(Apr 2001) Nonlinear Dynamics and Chaos. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1038/npg.els.0003314]