Blackheaded Budworm Tutorial

Before you start this tutorial, you may want to locate and make a copy of the paper that contains the data (Morris 1959). Notice that the data vary from 3.1 to 533 budworm larvae per 100 square feet of foliage. We will rescale the data to larvae per 1000 sq. ft. by multiplying them all by 10.

Switch on your computer and enter the PAS MAIN MENU. Press [2] to access the SINGLE-SPECIES MENU, and then [1] to run a Time Series Analysis.

• Press [F10] to proceed until asked how you intend to enter data, then press [K]. Enter the data as described in the cone beetle tutorial using Budworm for the organism, sq. ft. for the sample unit, 1000 for the number of units, and then enter the population data 220, 1120, 5330, 2250, 120, 31, 33, 310, 1500, 2370, 3000, 1830 (Morris 1959).
• Examine the time-series plot of the budworm data (see figure).
• Adjust the speed with [F4] and [F5]. Press [F3] to replot on log scale. Notice how the trajectory goes through two relatively smooth cycles and that MRT > 2 and VRT < MRT. This implies that the delay in the density dependent feedback is greater than one and that the series is stationary.
• Use [F7] and [F8] to see the regression line plotted through the time series and the ACF. What does this tell you?
• Press [F10] then [N] twice to omit sequencing and detrending.
• Examine the phase portrait noting, in particular, the wide clockwise orbit (see figure). Use [F1] to refresh your memory about what to look for and to replot the trajectory on both arithmetic and logarithmic scales. How does this phase plot differ from the cone beetle's?
• Use [F7] to plot a regression line through the data and [F8] to see the PRCF. Note the large negative correlation at lag 2. What do you infer from this?
• Press [F3] to make sure the data are plotted on the logarithmic scale, then [F10] to continue.
• Change the time lag to 2 and then plot a regression through the data. Notice the high coefficient of determination, r².
• Press [N] to leave the lag at 2, then [F7] to curve fit. Enter [2] for the initial value of A, the maximum per-capita rate of increase. Press [Q] when the r² value no longer improves (see figure).
• Press [F8] to simulate and enter a run length of 50, default the initial densities, and set the standard deviation to 0. Note the stability properties of the deterministic trajectory. Press [F10] to see the correspondence between simulation and data (see figure). Press [F10] to continue.
• Press [F8] to simulate again. Enter a run length of 100 years, default the initial densities and standard deviation. Notice how environmental variability affects the amplitude of the cycles. Notice the correspondence between data and simulated trajectories on the phase plane. Repeat the simulation with different values for the standard deviation.
• When you have finished simulating, press [F10] to continue and [Y] to build a two-lag model. Enter lags of [1] and [2]. Compare the multiple r² to that for the 1-lag model (see table here). Run some simulations in constant and noisy environments. Try different combinations of 2-lag models. Which one has dynamic behavior most like the real data?
• When you have finished with the 2-lag model, move on and record the best logarithmic model to disk. Then press [B] to go back and build an arithmetic model; i.e., repeat the exercise but this time on the arithmetic scale. Save the arithmetic model if you prefer it.

CONCLUSIONS

The analysis indicates that blackheaded budworm cycles are the result of a lag of two in the negative feedback, possibly due to the delayed interaction between the budworm and its insect parasitoids (see Morris l959). In constant environments, the cycles usually damp to a stable equilibrium (e.g., after 100 generations or so in some of the models). Quasi-periodic cycles are generated by some models, such as the linear one-lag arithmetic model. Environmental variability sustains cyclic dynamics in models with stable deterministic solutions and greatly increases their amplitude (Berryman 1986). Stochastic disturbances have a lesser influence on cycle periodicity which is largely entrained by the intrinsic feedback delays. These conclusions seem to be supported by observations in the field (Miller 1966).

References:

• Berryman, A. A. 1986. On the dynamics of blackheaded budworm populations. Canadian Entomologist 118: 775-779
• Miller, C. A. 1966. The blackheaded budworm in Eastern Canada. Canadian Entomologist 98: 592-613
• Morris, R. F. 1959. Single-factor analysis in population dynamics. Ecology 40:580-588