Red Pine Cone Beetle Tutorial

Before you start this tutorial, you may want to locate and make a copy of the paper on red pine cone beetles (Mattson 1980.) Notice that the data vary from 500 to 7200 beetles per acre. We will rescale the data to beetles per 0.1 acre by dividing them all by 10 As P1a uses real data you must decide where it is to be stored. We suggest you store your data on a 5.25 or 3.5 inch floppy disk. Let us suppose that you decide to store data in a directory called INSECTS on a floppy disk in drive A: You can create the directory by using the DOS make-directory command; e.g., type >MD A:INSECTS and then press [Enter]. We can now start the tutorial. Switch on your computer. Enter the directory on your hard drive in which the PAS programs reside by typing >CD\PAS. Now enter PAS and, when the PAS MAIN MENU appears, press [2] to access the SINGLE-SPECIES MENU then [1] to run the Time Series Analysis program P1a.

• If this is your first time using P1a, press [F1] to obtain a brief description of the program. You can print out the text using the [Print Scrn] or [Shift] + [Print Scrn] key if you wish.
• Press [F10] to move on and when you are asked how you intend to enter data, press [K] for keyboard entry.
• When prompted, enter a name for the organism; in this case, Cone Beetle.
• At the next question, enter the unit of observation. In this case Acre.
• Next enter the number of units per sample; in this case you should enter the data rescaling constant 0.1.
• Enter 121.8 when asked for the cone beetle numbers at time 1. If you make a mistake in data entry, you can correct it immediately by entering the word change, or you can wait until the end of the data input to make changes.
• Enter 720 when asked for the cone beetle numbers at time 2.
• Continue entering data 152.5, 271.8, 177, 161.4, 105.5, 369.5, 50, 100.When they are all entered type the word end and press [Enter] to terminate data input.
• Check over your data and, if there are any mistakes, press [A] to alter data. When all data are correct, press [C] to continue.
• The Cone Beetle Time-Series Plot now appears on the screen. Before you do anything else, press [F6] to save your data then press [S] to save them to disk. Default to the current data path (NOTE: default means to press [Enter] without typing anything. When asked, enter a name for your data file; e.g., CNBEETLE.
• Press [F2] and [F3] to replot the data on arithmetic and logarithmic scales and [F4] and [F5] to speed up and slow down the trace (see the *Function Keys* to the lower right of your computer screen.) Try to adjust the speed so that you can clearly see the motion of the trajectory. Notice how the trajectory moves from above to below its mean value every year or so. Look in the * Statistics * window and note that the mean return time is less than two; i.e., MRT < 2. Also note that the return time variance is less than the mean; i.e., VRT < MRT. These statistics indicate that the series does not exhibit trends or discontinuities (it is stationary), and that the population is being regulated by fast acting negative feedback. NOTE: Numerical data are reported in engineering notation; i.e., VRT = 6.67916E-01 means that the decimal point must be moved one place to the left while MEAN=2.22960E+02 means that the decimal must be moved two places to the right.
• Use [F1] to obtain information on what to look for in the time series and associated statistics.
• Use [F7] to plot a regression line through the time-series to check for trend (see figure here.) Notice that the regression line has a fairly steep slope suggesting that the population may be undergoing a declining trend (this observation is at odds with the return time statistics). Check for trend on both arithmetic [F2] and logarithmic [F3] scales.
• Use [F8] to plot the Autocorrelation Function or ACF (see cone beetle ACF here.) Use [F1] for information about the ACF. Notice that all the correlations tend to alternate from positive to negative at successive lags. This suggests that the time series has a periodicity of two time steps and that it is stationary, or has no trends or discontinuities.
• When you have finished looking at the time-series plot, press [F10] to move on, then press [N] to omit sequencing the data (you can see how sequencing is done in Tutorial 3.)
• You must now make a decision about detrending the data. As the diagnosis is ambivalent, let us try detrending, by pressing [Y].
• Press [P] to plot a regression line through the data, then default to detrend the data around its mean value (see detrending screen here and detrended time series here.)
• Press [F8] to see the ACF for the detrended data and notice that the correlations have improved somewhat (see ACF for detrended data here.) This indicates that detrending may have removed some external influence which caused the mean cone beetle density to decline over the years.
• Press [F10] to continue and display the cone beetle Phase Portrait or Phase Plot -- a plot of the cone beetle's per-capita rate of change on population density over the ten years. Use [F1] to find out about phase plots. Notice that the trajectory bounces rapidly from above to below the equilibrium (R=0) line (NOTE: capital R is used for the per-capita rate of increase of the organism while lower case r is reserved for the statistical correlation coefficient). Use [F2] and [F3] to observe the motion of the trajectory several times (see phase portrait here.)
• Press [F7] to plot a regression line through the data (see figure here.) Notice the high coefficient of determination in both arithmetic and logarithmic regressions (the coefficient of determination, r² in the *Statistics* window, measures the proportion of the variation in the data explained by the regression line).
• Press [F8] to see the Partial Rate Correlation Function, or PRCF. The PRCF shows the correlation between the per-capita rate of increase R and the logarithm of population density at increasing lag. Note that the correlation at lag one has a very high negative value (the value is, in fact, the square root of the coefficient of determination in the *Statistics* window) (see PRCF here.)
• Press [F10] to move on, and then [Y] to plot the rate of increase against a different lag. Enter [2] to plot R on N(t-2).
• Press [F7] to plot a regression line through the data and notice that it has a positive slope and a small positive coefficient of determination (see figure.)
• Press [Y] to change the time lag and then enter [1] to plot R on N(t-1) again.
• Press [F2] to make sure that you are in arithmetic mode, then press [N] to leave the lag at one.
• A linear regression line is now drawn through the data (NOTE: the lines joining data points are omitted in this graph.) This is the first estimate of the one-lag logistic R-function. Use [F1] to get a brief description of the R-function. Notice the R-function parameters in the *Statistics* window .
• Press [F8] to simulate the dynamics of cone beetle populations with this model. Enter [20] for the run length, default the initial density, and set the standard deviation at zero. Notice how the population grows to equilibrium with a few minor oscillations (see figure.) Press [F10] and note that the deterministic trajectory appears as a single line in phase space because it falls exactly on the R-function.
• Press [F10] then [F8] to simulate again, but this time default to the standard deviation calculated when the model was fit to the data. Notice the sharp "sawtoothed" oscillations similar to the original data (see it here.)
• Press [F10] to see how the simulated trajectory superimposes itself on the original data in phase space. We verify the "reasonableness" of the model by evaluating the correspondence between simulated trajectory and the original data (see here.)
• Press [F10] to return to the R-function and notice that the data seem to have a slightly concave form.
• Press [F7] to do a curve fit, then enter [4] for the initial value for A, the y-intercept; i.e., estimate by eye where you think the curve will intercept the y-axis. Watch how the r² value improves with each iteration. When r² no longer improves much, press [Q] to quit. Note the high value of r² and the small value of the parameter Q, the coefficient of curvature (use [F1] for details of the curving parameter) (see curvilinear cone beetle R-function here.)
• Press [F8] to simulate, set the run length at twenty, default the initial density, and set the standard deviation at zero. Notice that this model is less stable than the linear one as it exhibits oscillations of larger amplitude before settling to equilibrium (see figure.) Perform some simulations in variable environments (s>0) and notice how some of them resemble the data (figure.) Compare the simulated trajectory with the real data on the phase plane. This comparison provides the best verification of the model because the time dimension is ignored. Simulation tests provide information on how well the modeled R-function and random external effects describe the actual cone beetle dynamics (go here for the comparison of simulated trajectory with cone beetle data on the phase plane.)
• Press [F10] to move on then [Y] to build a two-lag model. Enter [1] for the first lag and [2] for the second. Notice from the regression table that a linear model with two lags has a lower r² value than a non-linear model with one lag (still visible in the *Statistics* window.)
• Press [F8] to simulate with the two-lag model and observe the stability properties and dynamics in a variable environment (see pictures of deterministic and stochastic simulations with the two-lag model.)
• Press [F10] to move on and then [N] to omit different lags. A summary of your one- and two-lag models is presented. Use the [Print Scrn] key to print out this summary, then press [R] to record the model to disk. Press [1] to record the better one-lag logistic model.
• Press [C] to continue. This is the end of your first P1a data analysis. We suggest that you now press [S] to restart the analysis but this time do not detrend the data. This will enable you to determine if detrending significantly improved the model and altered its dynamic behavior. Save the undetrended model if you wish, either as a separate file or overwrite the first model. You can also try building logarithmic models by being in logarithmic mode when you do the analysis.

CONCLUSIONS

The analysis shows that the cone beetle population is stable in a constant environment (the deterministic solution when s = 0). The time lag of 1 suggests that the population was regulated by fast acting negative feedback, as might be caused by intraspecific competition for limited resources, possibly the density of pine cones available from year to year. This implies that most of the variation in cone beetle abundance is due to variation in the supply of cones.

Reference:

• Mattson, W. J., Jr. 1980. Cone resources and the ecology of the red pine cone beetle, Conophthorus resinosae (Coleoptera: Scolytidae). Annals of the Entomological Society of America 73: 390-396