Published in Science: a collaboration with Alan Hastings and leading scientists from Canada, the US and UK, looks at the importance of transient dynamics in ecological systems. The work comes out of an ongoing NIMBioS (National Institute for Mathematical and Biological Synthesis) working group.
Transient phenomena in ecology
Hastings, Abbott, Cuddington, Francis, Gellner, Lai, Morozov, Petrovskii, Scranton and Zeeman
The importance of transient dynamics in ecological systems and in the models that describe them has become increasingly recognized. However, previous work has typically treated each instance of these dynamics separately. We review both empirical examples and model systems, and outline a classification of transient dynamics based on ideas and concepts from dynamical systems theory. This classification provides ways to understand the likelihood of transients for particular systems, and to guide investigations to determine the timing of sudden switches in dynamics and other characteristics of transients. Implications for both management and underlying ecological theories emerge.
Much of ecological theory and the understanding of ecological systems has been based on the idea that the observed states and dynamics of ecological systems can be represented by stable asymptotic behavior of models describing these systems. Beginning with early work by Lotka and Volterra through the seminal work of May in the 1970s, this view has dominated much of ecological thinking, although concepts such as the idea of tipping points in ecological systems have played an increasingly important role. In contrast to the implied long time scales of asymptotic behavior in mathematical models, both observations of ecological systems and questions related to the management of ecological systems are typically focused on relatively short time scales.
A number of models and observations demonstrate possible transient behavior that may persist over very long time periods, followed by rapid changes in dynamics. In these examples, focusing solely on the long-term behavior of systems would be misleading. A long transient is a persistent dynamical regime—including near-constant dynamics, cyclic dynamics, or even apparently chaotic dynamics—that persists for more than a few and as many as tens of generations, but which is not the stable long-term dynamic that would eventually occur. These examples have demonstrated the potential importance of transients but have often appeared to be a set of idiosyncratic cases. What is needed is an organized approach that describes when a transient behavior is likely to appear, predicts what factors enhance long transients, and describes the characteristics of this transient behavior. A theory of long ecological transients is a counterpart to the related question of tipping points, where previous work based on an analysis of simple bifurcations has provided broad insights.
Just as ideas based on the saddle-node bifurcation provide a basis for understanding tipping points, a suite of ideas from dynamical systems provides a way to organize a systematic study of transient dynamics in ecological systems. As illustrated in the figure, a relatively small number of ideas from dynamical systems are used to categorize the different ways that transients can arise. Translating these abstract results from dynamical systems into observations about both ecological models and ecological system dynamics, it is possible to understand when transients are likely to occur and the various properties of these transients, with implications for ecosystem management and basic ecological theory. Transients can provide an explanation for observed regime shifts that does not depend on underlying environmental changes. Systems that continually change rapidly between different long-lasting dynamics, such as insect outbreaks, may most usefully be viewed using the framework of long transients.
An initial focus on conceptual systems, such as two-species systems, establishes the ubiquity of transients and an understanding of what ecological aspects can lead to transients, including the presence of multiple time scales and particular nonlinear interactions. The influences of stochasticity and more realistic higher-dimensional dynamics are shown to increase the likelihood, and possibly the temporal extent, of transient dynamics.
The development of such a framework for organizing the study of transients in ecological systems opens up a number of avenues for future research and application. The approach we describe also raises important questions for further development in dynamical systems. We have not, for example, emphasized nonautonomous systems, which may be required to understand the implications of a changing environment for transients. Systems with explicit time dependence as well as stochastic nonlinear systems still present great mathematical challenges.
Implications for management and basic ecological understanding depend on both the results we describe and future developments. A recognition of the difficulty of prediction caused by long transients, and of the corresponding need to match dynamics to transient behaviors of models, shows that basing either management or interpretation of ecological observations only on long-term dynamics can be seriously flawed.
The full article can be accessed here.
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