Chaos and evolution

There is growing interest in applying nonlinear methods to evolutionary biology. With good reason: the living world is full of nonlinearities, responsible for steady states, regular oscillations, and chaos in biological systems. Evolutionists may find nonlinear dynamics important in studying short-term dynamics of changes in genotype frequency, and in understanding selection and its constraints. More speculatively, dynamical systems theory may be important because nonlinear fluctuations in some traits may sometimes be favored by selection, and because some long-run patterns of evolutionary change could be described using these methods.     

Chaos, fractals and nonlinear dynamics in evolution and phylogeny

Biological evolution is a dynamic system that can be modelled using physical-time-evolution equations, Even simple iterative models can have complex dynamics, and replication, the fundamental evolutionary property of living things, is an iterative process. All living things can be conceived in abstract geometric terms as elements comprising an infinite fractal set in n-dimensional euclidian space. Phylogeny, the ancestral-descendant time series connecting individuals through successive generations, is also fractal. This article shows how dynamic models and fractal geometry can be applied to phylogeny and evolutionary theory, providing a basis for refuting linnaean categorical ranks in taxonomy, for recognizing limits to the naturalness of any classification and for understanding the physics of the evolutionary process.     

Chaos, oscillation and the evolution of indirect reciprocity in n-person games

Evolution of cooperation among genetically unrelated individuals has been of considerable concern in various fields such as biology, economics, and psychology. The evolution of cooperation is often explained by reciprocity. Under reciprocity, cooperation can prevail in a society because a donor of cooperation receives reciprocation from the recipient of the cooperation, called direct reciprocity, or from someone else in the community, called indirect reciprocity. Nowak and Sigmund [1993. Chaos and the evolution of cooperation. Proc. Natl. Acad. Sci. USA 90, 5091-5094] have demonstrated that directly reciprocal cooperation in two-person prisoner's dilemma games with mutation of strategies can be maintained dynamically as periodic or chaotic oscillation. Furthermore, Eriksson and Lindgren [2005. Cooperation driven by mutations in multi-person Prisoner's Dilemma. J. Theor. Biol. 232, 399-409] have reported that directly reciprocal cooperation in n-person prisoner's dilemma games (n>2) can be maintained as periodic oscillation. Is dynamic cooperation observed only in direct reciprocity? Results of this study show that indirectly reciprocal cooperation in n-person prisoner's dilemma games can be maintained dynamically as periodic or chaotic oscillation. This is, to our knowledge, the first demonstration of chaos in indirect reciprocity. Furthermore, the results show that oscillatory dynamics are observed in common in the evolution of reciprocal cooperation whether for direct or indirect.     

Chaos and the (un)predictability of evolution in a changing environment

Among the factors that may reduce the predictability of evolution, chaos, characterized by a strong dependence on initial conditions, has received much less attention than randomness due to genetic drift or environmental stochasticity. It was recently shown that chaos in phenotypic evolution arises commonly under frequency‐dependent selection caused by competitive interactions mediated by many traits. This result has been used to argue that chaos should often make evolutionary dynamics unpredictable. However, populations also evolve largely in response to external changing environments, and such environmental forcing is likely to influence the outcome of evolution in systems prone to chaos. We investigate how a changing environment causing oscillations of an optimal phenotype interacts with the internal dynamics of an eco‐evolutionary system that would be chaotic in a constant environment. We show that strong environmental forcing can improve the predictability of evolution by reducing the probability of chaos arising, and by dampening the magnitude of chaotic oscillations. In contrast, weak forcing can increase the probability of chaos, but it also causes evolutionary trajectories to track the environment more closely. Overall, our results indicate that, although chaos may occur in evolution, it does not necessarily undermine its predictability.     

Chaos and language

Human language is a complex communication system with unlimited expressibility. Children spontaneously develop a native language by exposure to linguistic data from their speech community. Over historical time, languages change dramatically and unpredictably by accumulation of small changes and by interaction with other languages. We have previously developed a mathematical model for the acquisition and evolution of language in heterogeneous populations of speakers. This model is based on game dynamical equations with learning. Here, we show that simple examples of such equations can display complex limit cycles and chaos. Hence, language dynamical equations mimic complicated and unpredictable changes of languages over time. In terms of evolutionary game theory, we note that imperfect learning can induce chaotic switching among strict Nash equilibria.     

Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales ( e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales (e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

Chaos in percepts?

Multistability in perceptual tasks has suggested that the mechanisms underlying our percepts might be modeled as nonlinear, deterministic systems that exhibit chaotic behavior. We present evidence supporting this view, obtaining an estimate of 3.5 for the dimensionality of such a system. A surprising result is that this estimate applies for a rather diverse range of perceptual tasks. Received: 22 April 1993/Accepted in revised form: 6 August 1993     

Order out of Chaos

The current crisis in the world's fisheries indicates the need for a different management method than that now used by Western scientists, which regulates the quantity of fish taken. The authors propose a method called parametric management, which takes into account the complex, chaotic nature offish stocks and emphasizes preserving regular biological processes in the life cycle of fish by controlling how people fish. Supporting data come from 28 folk societies, the Maine lobster industry, and the authors' mathematical model of fish stocks.     

Chaos in biology.

This paper presents a brief and pedagogical account of the relevance of chaos theory in biology. A few caveats to avoid misleading interpretations are underlined, for instance the required determinism and consistency of the experimental time series. The selective advantage offered by a properly controlled chaotic dynamics is discussed on the examples of cardiac rhythm and brain dynamics.     

Chaos and population control of insect outbreaks

We used small perturbations in adult numbers to control large fluctuations in the chaotic demographic dynamics of laboratory populations of the flour beetle Tribolium castaneum . A nonlinear mathematical model was used to identify a sensitive region of phase space where the addition of a few adult insects would result in a dampening of the life stage fluctuations. Three experimental treatments were applied: one in which perturbations were made whenever the populations were inside the sensitive region ("in-box treatment"), another where perturbations were made whenever the populations were outside the sensitive region ("out-box treatment"), and an unperturbed control. The in-box treatment caused a stabilization of insect densities at numbers well below the peak values exhibited by the out-box and control populations. This study demonstrates how small perturbations can be used to influence the chaotic dynamics of an ecological system.