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II.3 Biological Chaos and Complex Dynamics Alan Hastings OUTLINE 1. Fluctuations in populations 2. Brief guide to dynamic systems 3. Chaotic dynamics in models in ecology and population biology 4. Search for chaos in data 5. Resolution as noisy clockwork 6. Other complex dynamics The cause of fluctuations in ecological populations has long been the subject of study, with the goal of understanding the relative importance of exogenous versus endogenous forces in explaining observed dynamics. The discovery of the likelihood of chaotic dynamics in simple discrete-time models that could be used to describe singlespecies population dynamics spurred much research focused on understanding chaos and its importance and likelihood in ecological systems. To understand the importance of chaos, we consider the role of fluctuations in ecological systems, the generation of chaotic dynamics in models, and the determination of chaos from time series. This naturally leads to more general questions on the role of complex dynamics in ecology and to a more synthetic view of the causes of observed fluctuations. GLOSSARY asymptotically stable solution. A solution that is approached by all nearby solutions is asymptotically stable. This is also known as an attractor. chaos. Chaos is a property of an attractor in a dynamic system that can be roughly characterized as sensitive dependence on initial conditions and can be detected by the presence of a positive Lyapunov exponent. cycle. A cycle is a solution that repeats at regular intervals . equilibrium. An equilibrium of a model is a solution that does not change in time. Lyapunov exponent. A Lyapunov exponent represents the exponential rate of divergence (if positive) or convergence (if negative) of (two) solutions started on or near an attractor. 1. FLUCTUATIONS IN POPULATIONS A key observation that is central in ecology is that populations fluctuate in time. These fluctuations can exhibit some regularity or can be irregular. Periodical cicadas emerge with great regularity, whereas outbreaks of other insects such as locusts are both dramatic and irregular. The cause of fluctuations in populations in ecology has been a central question in ecology for many years. Early in the history of ecology, Volterra and Lotka focused on the regular oscillations produced by interactions between predator and prey in their models. Shortly thereafter, Gause attempted to reproduce these oscillations in laboratory systems using microorganisms and found that sustained oscillations were difficult to reproduce. In the simple laboratory systems, either the predator ate up all the prey and then starved or the predator could not find enough food and starved with only the prey surviving. This set up a problem that remains until today, namely, what allows predator and prey to coexist. Also, many of the mechanisms that might allow coexistence of species might lead to more complex dynamics, and more often coexisting species fluctuate. In any examination of natural populations, fluctuations in numbers have been found to be the almost universal outcome. These fluctuations could range from relatively regular cycles, such as those observed in small mammal populations, or more dramatic changes, such as outbreaks of insect populations. A classic debate in ecology has focused on the causes of these fluctuations. One potential source of fluctuations could be external influences, such as changes in weather or climate. These exogenous forces could be responsible for changes in the dynamics of populations, producing cycles that were either regular or irregular. Another cause of changes in the numbers of populations would be endogenous forces within the population that would lead to cycles or more complex dynamics. Enlarging the question to look at fluctuations not just in time but also over space was thought of as one way to decide. Population fluctuations that are synchronous over space would either have to be caused by exogenous forces that were synchronous over space (the Moran effect) or endogenous forces such as dispersal that would synchronize populations. One particular kind of fluctuating population that we will return to later that deserves special attention is the incidence of disease. In particular, the numbers of individuals with various childhood diseases (measles, mumps, rubella, and others) in the prevaccination era have been intensely studied. These particular fluctuations have played a special role for several reasons. The data are of much higher quality, with more, and more accurate, observations than for the numbers of many organisms. Another important aspect is that the underlying interactions producing the dynamics of diseases are relatively simple and relatively easily described . Finally, because data are known for multiple diseases in multiple locations, deeper understanding is possible...


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