In time series analysis, the partial autocorrelation function or PACF is often used to indicate the lag structure of the generating process. The PACF shows the correlations between n(t) and n(t-d) with the effect of earlier lags removed; i.e.,

where

and

j = 1, 2, 3, ..... d-1

However, PACF is not a good indicator of negative feedback at lag one in biological populations. This is because positive autocorrelation due to reproduction (+feedback) may mask the negative feedback effect. A better indicator is the correlation between the differenced series R(t-1)=n(t)-n(t-1) and density at lag one n(t-1) which, because R is the same as the per-capita rate of change, we will call the rate correlation at lag one

where is the mean of the per-capita rates of change. P1a modifies the traditional PACF to a partial rate correlation function, PRCF, as follows

Note that PACF at lag two or greater is the same as PRCF at the same lag. Hence, the PRCF provides us with the partial correlations between the per-capita rate of change R and lagged population density n(t-d). PAS plots the PRCF as a histogram.

The PRCF provides information on the lag structure of the negative feedback mechanisms acting on the population. For example, the PRCF for the cone beetle indicates dominant -feedback with lag one. This may lead us to suspect that intra-specific competition is the main regulatory mechanism acting on this population.