Sandhill Crane Tutorial

• Load P1a and enter the numbers of sandhill cranes counted during the spring surveys along the Platte River in Nebraska from 1959 to 1978 (Buller 1979). You can enter total for the unit of measure and .001 for the number of units because the data should be divided by 1000. Notice how the trajectory spends most of the time either above or below its mean value, and that the return time variance is much larger than the mean on both arithmetic and logarithmic scales. Plot the autocorrelation function and note that it does not fluctuate evenly above and below the zero correlation line. This, and the fact that VRT is much larger than MRT, indicates a non-stationary series.
• Press [F10] and then [Y] to sequence the data. Use the arrow keys to move the sequencing line around (if the arrow keys do not work, you may need to turn off your [Num Lock] or [Caps Lock]). Place the sequencing line in the center of the series and press [A] to divide the series into two 10-year sequences (see figure).
• Examine the first sequence and plot a regression line through it. Examine the ACF, MRT and VRT. What does this tell you?
• Press [F10] then [N] to see the phase portrait (see figure). Use [F8] to see the PRCF and note the dominant negative correlation at lag 1 (see figure). Press [F7] to plot a regression line through the data.
• Press [F10] then [N] to leave the lag at 1, then use [F7] to fit a curvilinear model to the data. Remember that sandhill cranes have low reproductive rates so start the convergence process at around A = 0.7; i.e., an R-value of 0.7 is roughly equivalent to a maximum reproductive rate of about 2 offspring per pair per year. Notice how the convex R-function explains a high percentage of the variation in the data (the coefficient of determination r² = 0.89).
• Press [F8] to simulate with the model and set the run length at 40, default the initial density, and set the standard deviation at zero. Note the persistent "saw-toothed" oscillations which repeat themselves every 4 years. In technical terms this is called a "4-point limit cycle" and is quite unusual in models built from real data. Run some simulations with environmental variability.
• When you are ready, move on to build a 2-lag model. Note that the additional lag does not improve the model of this sequence.
• Continue and examine the time-series of sequence number two. Notice that the sequence is stationary and has similar ACF and PRCF to the first sequence.
• Build a curvilinear 1-lag model by starting with the same A-value as previously. Note the very high coefficient of determination, and that the model is quite similar to the first one. However, simulations demonstrate that this model is damped-stable in a constant environment (see figure).
• Move on to examine the 2-lag model, noting a slight improvement in the coefficient of determination. Run some simulations in constant and variable environments, noting any differences to the 1-lag models.
• Continue and examine the complete series, noting the different equilibrium lines for each series. Use [F1] to obtain information about adjusting the series to a common mean, then press [A] to adjust the series.
• Examine the adjusted series and its ACF then continue to build one and two lag models of the adjusted series. Use simulation experiments to show that both models are damped-stable. Notice that the 1-lag curvilinear model has the highest coefficient of determination.

CONCLUSIONS

Although the sandhill crane series exhibits a shift in its mean density, both series have similar kinds of "saw-toothed" fluctuations. This suggests that some external variable changed in the 10^th year causing a shift in the equilibrium density. Another possible explanation is that the method for censusing the crane population was changed at this time. The convex curvilinear relationship between the per-capita rate of increase and population density at lag 1 indicates strong negative feedback of length one, probably due to intentional competition. A possible mechanism that could create this kind of competition is territorial behavior. This suggests that the dynamics of sandhill crane populations are regulated almost entirely by territoriality.

Refernces:

• Buller, R. J. 1979. Lesser and Canadian sandhill crane populations, age structure, and harvest. US Department of the Interior Special Scientific Report -- Wildlife N. 221, Washington, DC.