## Analysis dissecting density

A number of related approaches have sought to dissect the density-dependent 'structure' of species' population dynamics by a statistical analysis of time series of abundance. Abundance at a given point in time may be seen as reflecting abundances at various times in the past. It reflects abundance in the immediate past in the obvious sense that the past abundance gave rise directly to the present abundance. It may also reflect abundance in the more distant past if, for example, that past abundance gave rise to an increased abundance of a predator, which in due course affected the present abundance (i.e. a delayed density dependence). In particular, and without going into technical details, the log of the abundance of a population at time t, Xt, can be expressed, at least approximately, as:

Xt = m + (1 + P,)XH + P2Xt_2 + ... + fidX^ + u, (14.1)

an equation that captures, in a particular functional form, the idea of present abundance being determined by past abundances (Royama, 1992; Bjornstad et al., 1995; see also Turchin & Berryman, 2000). Thus, m reflects the mean abundance around which there are fluctuations over time, Pj reflects the strength of direct density dependence, and other Ps reflect the strengths of delayed density dependences with various time lags up to a maximum d. Finally, uc represents fluctuations from time-point to time-point imposed from outside the population, independent of density. It is easiest to understand this approach when the Xts represent deviations from the long-term average abundance such that m disappears (the long-term average deviation from the mean is obviously zero). Then, in the absence of any density dependence (all Ps zero) the abundance at time t will reflect simply the abundance at time t — 1 plus any 'outside' fluctuations ut; while any regulatory tendencies will be reflected in P values of less than zero.

Applying this approach to a time series of abundance (i.e. a sequence of Xt values) the usual first step is to determine the statistical model (Xt as the dependent variable) with the optimal number of time lags: the one that strikes the best balance between accounting for the variations in Xt and not including too many lags. Essentially, additional lags are included as long as they account for a significant additional element of the variation. The P values in the optimal model may then shed light on the manner in which abundance in the population is regulated and determined. An example is illustrated in Figure 14.9, which summarizes analyses of 19 time series of microtine rodents (lemmings and voles) from various latitudes in Fennoscandia (Finland, Sweden and Norway) sampled once per year (Bjornstad et al., 1995). In almost all cases, the optimum number of lags was two, and so the analysis proceeded on the basis of these two lags: (i) direct density dependence; and (ii) density dependence with a delay of 1 year.

Figure 14.9a sets out the predicted dynamics, in general, of populations governed by these two density dependences (Royama, 1992). Remember that delayed density dependence is reflected in a value of P2 less than 0, while direct density dependence is reflected treating red grouse for nematodes

14.5.1 Time series dependence abundance determination expressed as a time-lag equation

Fennoscandian microtines Figure 14.9 (a) The type of population dynamics generated by an autoregressive model (see Equation 14.1) incorporating direct density dependence, P1, and delayed density dependence, P2. Parameter values outside the triangle lead to population extinction. Within the triangle, the dynamics are either stable or cyclic and are always cyclic within the semicircle, with a period (length of cycle) as shown by the contour lines. Hence, as indicated by the arrows, the cycle period may increase as P2 decreases (more intense delayed density dependence) and especially as P1 increases (less intense direct density dependence). (b) The locations of the pairs of P1 and P2 values, estimated from 19 microtine rodent time series from Fennoscandia. The arrow indicates the trend of increasing latitude in the geographic origin of the time series, suggesting that a trend in cycle period with latitude, from around 3 to around 5 years, is the result of a decreased intensity of direct density dependence. (After Bj0rnstad et al., 1995.)

Figure 14.9 (a) The type of population dynamics generated by an autoregressive model (see Equation 14.1) incorporating direct density dependence, P1, and delayed density dependence, P2. Parameter values outside the triangle lead to population extinction. Within the triangle, the dynamics are either stable or cyclic and are always cyclic within the semicircle, with a period (length of cycle) as shown by the contour lines. Hence, as indicated by the arrows, the cycle period may increase as P2 decreases (more intense delayed density dependence) and especially as P1 increases (less intense direct density dependence). (b) The locations of the pairs of P1 and P2 values, estimated from 19 microtine rodent time series from Fennoscandia. The arrow indicates the trend of increasing latitude in the geographic origin of the time series, suggesting that a trend in cycle period with latitude, from around 3 to around 5 years, is the result of a decreased intensity of direct density dependence. (After Bj0rnstad et al., 1995.)

in a value of (1 + P1) less than 1. Thus, populations not subject to delayed density dependence tend not to exhibit cycles (Figure 14.9a), but P2 values less than 0 generate cycles, the period (length) of which tends to increase both as delayed density dependence becomes more intense (down the vertical axis) and especially as direct density dependence becomes less intense (left to right on the horizontal axis).

The results of Bj0rnstad et al.'s analysis are set out in Figure 14.9b. The estimated values of P2 for the 19 time series showed no trend as latitude increased, but the P1 values increased significantly. The points combining these pairs of Ps, then, are shown in the figure, and the trend with increasing latitude is denoted by the arrow. It was known prior to the analysis, from the data themselves, that the rodents exhibited cycles in Fennoscandia and that the cycle length increased with latitude. The data in Figure 14.9b point to precisely the same patterns. But in addition, they suggest that the reasons lie in the structure of the density dependences: on the one hand, a strong delayed density dependence throughout the region, such as would result from the actions of specialist predators; and on the other hand, a significant decline with latitude in the intensity of direct density dependence, such as may result from an immediate shortage of food or the actions of generalist predators (see Figure 10.11b). As we shall see in Section 14.6.4 (see also Section 10.4.4), this in turn is supportive of the 'specialist predation' hypothesis for microtine cycles. The important point here, though, is the illustration this example provides of the utility of such analyses, focusing on the abundances themselves, but suggesting underlying mechanisms. 