The Soay sheep population is unusual in that it fluctuates dramatically in size with time (see history and figure, right). In some years the population can shrink by up to 70% as deaths massively outnumber births. A key objective of our research on the population dynamics of the Soay sheep has been to understand why we see such fluctuations in population size.
The broad approach we have taken is to start identifying the key processes that we suspect generate the fluctuations in population size. Having done this, we then derive models that incorporate these processes. Next, we parameterize these models using the field data we collect. We then compare predictions from the models with what we observe in the field. We also use mathematical and computational tools to analyze our models and allow us to understand why they make the predictions they do.
Our first model was remarkably simple. Grenfell et al. 1992 derived a model that only incorporated density-dependence. This model proposed that sheep compete with one another for food. When their numbers are low, birth rates are high and the population rapidly grows, quickly overshooting carrying capacity – a number describing the maximum number of sheep the island can support during winter. Insufficient food to go round means many sheep die and the population crashes.
Although Grenfell’s model did well in capturing the ups and downs in population size that we observed early in the study, it soon became clear that it was a rather crude approximation of reality. So our next step was to include weather as well as density-dependence. We found that for population crashes to occur we need both high density and bad weather (Grenfell et al. 1998).
But the sheep surprised us. Although this second model was an improvement upon our first, it still didn’t particularly accurately describe the way the population fluctuated. Our next step was to write down a model that explicitly included birth and death rates that varied between both males and females and sheep of different ages. This model showed that these different groups of animals were influenced by weather and population density in contrasting ways. A consequence of this was we discovered that population density, weather and the age- and sex-structure of the population interacted to influence the population dynamics (Coulson et al. 2001).
Still we weren’t satisfied with the performance of the model. One issue was with the way we had characterized the weather. We had found that using local weather was a challenge because there are so many different ways in which it can be measured. In contrast, large-scale weather indices, like the North Atlantic Oscillation, provided a convenient summary of local weather. However, it proved a rather crude predictor of conditions on St. Kilda. We refined our understanding of the role that local weather plays by modeling the consequences of the timing of individual storms on over-winter survival (Hallett et al. 2004). We found that the timing of specific storms had the potential to substantially impact the number of sheep that survived each winter.
Figure (From Ozgul et al. 2009): change in the average body size (z bar) of female (a) lambs, (b) yearlings, (c) adults aged between two and six, and (d) senescent sheep aged seven and over. The solid lines show observed average weights in each age-class; the dashed lines show the mean weights we predicted.
A second issue was that the sheep were getting smaller (see figure), so we next needed to think about incorporating body weight into our models. First, we needed to statistically identify the extent to which the population dynamics were likely influenced by the size of sheep on the island (Pelletier et al. 2007). We found that in years when the weather was bad, the distribution of sheep body weights contributed substantially to whether the population increased or not, while in good years body weight contributed little.
Armed with this insight we were now in a position to start modeling why the population size of sheep had been showing a tendency to (noisily) increase while the sheep had been getting smaller (Ozgul et al. 2009). The models we developed here suggested that climate change had altered the way sheep competed for food. A consequence of this was that growth rates of the average sheep were reduced, and more smaller sheep were surviving through the winter than used to be the case.
The approach developed in Ozgul et al. 2009 led to us to appreciate that we could write down some very general theory from which we could not only track the dynamics of population size, but also of sheep life history and characters like body size and coat colour (Coulson et al. 2010). Our aim now is to use this approach to gain further understanding of why the sheep population fluctuates in the way it does.
Of course, there were many statistical and theoretical challenges associated with constructing population models. We have spared the reader the gory details of the various challenges we met along the way. However, the interested reader can find the literature cited, and other papers that describe these challenges, and the solutions we found, on the publications page.