Biological Chaos and Complex Dynamics

by

David A. Vasseur

• LAST REVIEWED: 28 April 2023
• LAST MODIFIED: 23 May 2012
• DOI: 10.1093/obo/9780199830060-0024

Introduction

Classically, chaos is defined as a lack of order; however, in a scientific context it refers to the lack of predictability of a process or sample. Chaos differs from randomness in that chaotic systems are purely deterministic; that is, they are entirely determined by a set of mathematical formulas and initial conditions, with no random elements involved. Particularly intriguing is that chaotic systems are especially sensitive to initial conditions; simulations of a chaotic system initiated at only slightly different states will quickly diverge so that predicting the state of one iteration of the chaotic system from a second is not possible. This lack of predictability is the origin of the term “butterfly effect,” which was made famous by the meteorologist Edward N. Lorenz in his talk “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” In addition to chaos, other forms of complex dynamics, such as regular oscillations and quasiperiodic oscillations, are preeminent features of many biological systems. Recognition of these types of dynamics came first from early observations on economically important species, such as fish and small mammals, and soon after from mathematical models that ecologists used to understand the basic processes of population demography. Early evidence for chaos and complex dynamics challenged the classical notion that ecological systems are dominated by equilibrium-based processes (a “natural” or reference state of the system to which it would return if left unperturbed by the external influence of humans, climate, and so on) and raised great skepticism about our ability to accurately predict the fate of populations for conservation and management. Although there is considerable challenge discerning chaos and complex dynamics from noisy equilibrium-based dynamics, the former has come to be accepted as a viable but (in most instances) rare population attribute. The intense study of these subjects during the 1980s and 1990s has given biologists and ecologists an increased awareness of when and where to expect chaos and complex dynamics. Research in the early 2000s has focused less on the identification of these patterns in real data and experiments and more on the reasons why natural systems seem to exhibit more simple and stable dynamics than many models predict.

General Overviews

A number of general overviews have provided a general description of the mathematical phenomena of chaos and complex dynamics and their applications in biological systems. Perhaps the most well-known overview and popular history of chaos is Gleick 2008; however, Strogatz 2000 provides an accessible mathematical introduction to chaos and complex dynamics that is often motivated with examples from the biological sciences. Turchin 2003 provides the most comprehensive overview of chaos and complex dynamics in ecology, combining a mix of models and empirical evidence. Cushing, et al. 2003 demonstrates that the complex dynamics of real populations can be derived by integrating the important biological mechanisms into simple mathematical models. The body of work discussed therein is arguably the best understood of any ecological examples of chaos. Many notable review papers emerged during the 1990s, when the search for chaos and complex dynamics was at the forefront in ecology. Of these, Hastings, et al. 1993 provides a comprehensive review of work describing approaches to detect chaos and complex dynamics in real data. Logan and Allen 1992 and Stone and Ezrati 1996 review mounting evidence for chaos and provide important perspectives on the role of chaos and complex dynamics in ecology. Ferrière and Fox 1995 reviews the role of nonlinearity in models of evolutionary change with specific reference to complex dynamics and chaos.

• Cushing, J. M., R. F. Costantino, Brian Dennis, Robert A. Desharnais, and Shandelle M. Henson. 2003. Chaos in ecology. Vol. 1, Experimental nonlinear dynamics. Boston: Academic Press.

  This work is a synthesis of a series of experiments and theory that used Tribolium as a model organism to demonstrate chaos. The authors describe experiments and models aimed at understanding chaos and set their model system into the landscape of other important examples.

• Ferrière, R., and G. A. Fox. 1995. Chaos and evolution. Trends in Ecology and Evolution 10:480–485.

  DOI: 10.1016/S0169-5347(00)89194-6

  The authors argue that the nonlinearities commonly describing ecological dynamics may have previously overlooked the importance of understanding changes in gene frequency and long-term fluctuations in selection. They argue that when chaotic processes are at work, evolution may not necessarily be seen as an adaptive process.

• Gleick, James. 2008. Chaos: Making a new science. New York: Penguin.

  The author gives a thought-provoking introduction to chaos theory and the people who dominated its history. This is perhaps the most popular book ever written on the topic and is easily accessible to all levels of interested readers.

• Hastings, Alan, Carole L. Hom, Stephen Ellner, Peter Turchin, and H. Charles J. Godfray. 1993. Chaos in ecology: Is Mother Nature a strange attractor? Annual Review of Ecology and Systematics 24:1–33.

  The authors review methods for detecting chaos in natural data with the intent of providing a more solid understanding of nonlinear dynamics for the field of ecology. Available online for purchase or by subscription.

• Logan, J. A., and J. C. Allen. 1992. Nonlinear dynamics and chaos in insect populations. Annual Review of Entomology 37:455–477.

  DOI: 10.1146/annurev.en.37.010192.002323

  The authors review the mounting evidence for complex dynamics and chaos in insect populations. Available online for purchase or by subscription.

• Stone, Lewi, and Smadar Ezrati. 1996. Chaos, cycles, and spatiotemporal dynamics in plant ecology. Journal of Ecology 84:279–291.

  DOI: 10.2307/2261363

  The authors revisit the idea that chaos and complex dynamics are not important in plant communities by amassing literature on the subject. Available online for purchase or by subscription.

• Strogatz, Steven H. 2000. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering. Studies in Nonlinearity. Cambridge, MA: Westview.

  The author provides an accessible mathematical introduction to chaos and complex dynamics with many biological examples discussed throughout. Elements of equilibrium theory and linearization techniques are well explained with the aid of many helpful graphs and figures.

• Turchin, Peter. 2003. Complex population dynamics: A theoretical/empirical synthesis. Princeton, NJ: Princeton Univ. Press.

  The author provides the most comprehensive and up-to-date review of the experimental and natural evidence for chaos and complex dynamics and describes the ecologically relevant models.