Gypsy Moth Tutorial

• Access P1a and enter the numbers of gypsy moth egg clusters per acre obtained from Burgess (1938) [5214, 5407, 2635, 3658, 4400, 3751, 3702, 2273, 4032, 2134, 2387, 402, 110, 50, 61, 127, 303, 722, 407, 66, 42, 112, 344, 181]. Notice how the data spend most of the time either above or below the mean and that the return time variance is much larger than the mean, even on the logarithmic scale (see figure). Plot the ACF and note the monotonic decline with increasing lag. Is this time-series stationary?
• Press [F10] and then [Y] to sequence the data. Use the arrow keys to move the line to the point where you want to divide the data; e.g., move one or two positions to the left (see figure). Press [A] to analyze the first sequence.
• Examine the arithmetic plot and draw a regression line through the data (see figure). Examine the ACF. What do you infer from this?
• Proceed to detrend the first sequence and then examine its phase portrait (see figure). Continue to build 1-lag and 2-lag models for this sequence, performing simulations to check the stability properties of the models and their similarity to the real data. Which model provides the best description of the data?
• Continue and examine the time-series of sequence number two. Plot the ACF and note VRT, MRT. Is this series stationary?
• Move on to examine the phase plot (see here), noting the wide clockwise orbit reminiscent of the blackheaded budworm phase portrait. Examine the PRCF and note the dominant lag.
• Move on and plot the per-capita rate of change against the dominant lag (see figure).
• Continue to construct a non-linear R-function, then build a 2-lag model. Which model provides the best description of the data?
• Move on to examine the complete series and the two estimated equilibria (see figure). Use [F1] to find out that you cannot adjust these two sequences to a common mean because the lags are different, implying that different feedback mechanisms were operating.
• Press [C] to continue then [M] to build a two-equilibrium model. Examine the separatrix between the basins of attraction of the two equilibria (use [F1] for information about the separatrix). Notice that the current separatrix does not clearly identify two basins of attraction.
• Press [Y] to change the separatrix and enter [7] for the new value. Notice that the separatrix now clearly defines the two basins of attraction.
• Press [N] to accept this separatrix, then review the models for the two sequences, noting that the 2-lag models are generally better than the 1-lag models. If you have a printer use the [Print Scrn] key to save the model parameters for both sequences.

CONCLUSIONS

The gypsy moth data exhibit divergent behavior, the population being attracted to two different equilibrium points. It seems likely that the upper basin of attraction was determined by competition for food, the foliage of its deciduous host trees. The declining trend in the upper equilibrium could be due to trees being killed or their vigor reduced by the continuing gypsy moth outbreak. Eventually the gypsy moth population was reduced to a lower equilibrium by delayed negative feedback, possibly by insect parasitoids introduced from Europe to control the moth. Interactions with insect parasitoids frequently produce cycles of abundance of their prey (see Tutorial 2), a phenomenon which is observed in gypsy moth populations in their native home (see Sisojevic 1979 as well as Berryman 1991a,b and Liebhold and Elkinton 1991 for a discourse on this issue).

References:

• Berryman, A. A. 1991a. The gypsy moth in North America: a case of successful biological control? Trends in Ecology and Evolution 6: 110-111.
• Berryman, A. A. 1991b. Gypsy moth dynamics: a reply from Alan Berryman. Trends in Ecology and Evolution 6: 264-265.
• Burgess, A. F. and W. L. Baker. 1938. The gypsy and brown-tail moths and their control. United States Department of Agriculture, Circular No. 464.
• Liebhold, A. M. and J. S. Elkinton. 1991. Gypsy moth dynamics. Trends in Ecology and Evolution 6: 263-264.
• Sisojevic, P. 1979. Interactions in the host-parasite system, with special reference to the gypsy moth-tachinids (Lymantria dispar L. - Tachinidae). In Papers of the 6th Interbalcanic Plant Protection Conference, Izmir, Turkey. Ministry of Food, Agriculture and Animal Husbandry, Turkish Republic, Research Section, No. 13.