## Methods of Determining Patterns and Mechanisms

One of the problems inherent in studying population cycles is that cycles often take place over many years. For example, the snowshoe hare and lynx cycles have a period of roughly 10 years. Ecologists were only able to appreciate this phenomenon because of the careful records taken by Hudson Bay Company fur trappers. For most organisms, we do not have the long-term population data sets needed to observe cycles, so ecologists will likely discover more cycling populations as long-term data sets are established and evaluated in the future.

Cycles in population may, at first glance, seem easy to detect. For example, we can look at the snowshoe hare and lynx data and observe clear up and down portions of the graph in a repeating pattern (Figure 1). But how do researchers detect if there is a regular pattern to the fluctuations.? There must be enough detailed data from several years of observation to even begin the analysis. If the cycle has a period of 10 years, then at least 20 and perhaps 30 years of data are needed. After these data are collected, ecologists generally use time-series analysis to detect whether a true repeating pattern occurs. In practice, this usually takes the form of calculating the autocorrelation function (ACF), which measures the correlation in population density between pairs of years in the data. The ACF can tell you the most probable period of the population cycle, that is, whether a cycle takes 5, 7, or 9 years to complete.

More complex methods can also detect if the fluctuations are characteristic of first-, second-, or higher-order functions in the time series. A cycle with a period of t years does not necessarily mean that the only effect of this year's population will be felt t years in the future. Population densities in the intervening years will also have an impact on densities t years in the future.

Which years have the greatest impact on current populations are related to the order of the system

dynamics, as mentioned above. How do we determine these orders if they are not evident by looking at graphs. Ecologists use partial ACFs (PACFs) or their derivatives to determine the dominant process order of a density-dependent system. PACFs tell you where, in time, the greatest effects of population density occur. For example, systems with second-order dynamics usually have density-dependent lags of between 3 and 5 years, and systems with zero-order dynamics have almost instantaneous lags. It should be noted that although these analyses generally identify cycles of a certain period, there are many fluctuations around these periods in the actual data. Stochastic events are present in any ecological system and may perturb whatever periodic cycle we identify such that a population that has a general period of 3 years may have some periods of 4 years and others of 2.

These demanding forms of analysis have been used recently to help pinpoint actual mechanisms of population regulation in cycling populations. These methods are used to complement field manipulations, since field manipulations are rarely employed long-enough or at large-enough scales to replicate or change long-term dynamics.

Despite the difficulty of manipulations, such experiments have been executed, especially in the snowshoe hare-lynx systems in boreal Canada and the vole systems in Europe. Large-scale manipulations using exclosures or food additions have the twin advantages of being relatively realistic and having the ability to look in-depth at a single factor at a time over large areas.

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