Mountain Pine Beetle Tutorial

Adults of the mountain pine beetle bore into the bark of living pine trees and introduce pathogenic fungi into the phloem and sapwood. At the same time, they produce powerful aggregating pheromones which attract other beetles to the attacked tree. The combined action of insect boring and pathogen infection usually causes the death of the tree.

The population dynamics of mountain pine beetles are characterized by long periods of relative stability, when the insects are restricted to trees weakened by lightning strikes, root diseases, and other stress factors, followed by occasional outbreaks during which a large numbers of trees may be killed. During the low-density, stable phase, beetles cannot colonize normal healthy trees because the defensive secretions of the tree (resins)

1. mask or otherwise interfere with pheromone emission,
2. pitch out the attacking beetles with resin flow,
3. confine the beetles and fungal pathogens in an inhospitable resinous lesion (Berryman 1972, Raffa and Berryman 1983, 1986).

When mountain pine beetle populations increase to some critical density, however, they are capable of overwhelming trees of intermediate or even high resistance because

1. high initial attack by pioneer beetles increases the probability of pheromone emission,
2. mass attack diminishes the resin flow experienced by each beetle, and
3. rapid colonization makes it more difficult for the tree to resist the insect and pathogen infection.

For any stand of pines, there is a threshold beetle density per hectare above which the beetle population escapes from the constraints imposed by host resistance, and can then spread through the normal healthy forest. The escape threshold and maximum tree mortality rate are determined by the average resistance of the stand and its variance, escape thresholds being lower in stands of low mean resistance, and mortality rates higher in stands of low resistance variability.

Let us now proceed to deduce a model for the mountain pine beetle using the single-species Modeling & Simulation program P1b.

• Enter the directory where your PAS programs reside, then type > PAS and [Enter] to load the PAS Shell. Press [2] to enter the single-species environment, then [2] again to access P1b.
• Press [F10] then [N] to build a new model, then enter a name for the organism (Pine Beetle), the sampling unit (hectare), and number of units per sample (1).
• We must now estimate the maximum per capita rate of increase of the mountain pine beetle (note that the parameters are entered in the upper right window). We obtained this from figure 4.9 in Berryman (1986) which shows beetle-killed trees increasing at a log_e rate of 1.5 during an outbreak. Thus, enter 1.5 for the maximum per-capita rate of increase.
• We now need to estimate the interaction coefficient, or the slope of the R-function. We can do this by determining the beetle carrying capacity, or sustained long-term maximum equilibrium density. We did this by consulting yield tables for thinned 80-year old lodgepole pine stands (Cole and Edminster 1985, Table 6) to obtain estimates of stand density (900 trees/hectare), total volume (200 cubic meters/ha), and mean annual incremental growth (2.7 cubic meters/hectare). From this we calculated that each tree has an average volume of 200/900 = 0.22 cubic meters and thus 2.7/0.22 = 12.3 tree equivalents are produced per hectare per year by tree growth. These 12.3 trees can be utilized each year without depleting the stand and, assuming that about 600 beetles can breed in each tree equivalent, the beetle equilibrium density is estimated to be 12.3 * 600 = 7380 beetles/ha/yr. The interaction coefficient is obtained by dividing the maximum per-capita rate of increase by the equilibrium density 1.5 / 7380 = 0.0002. Enter 0.0002 for the coefficient of interaction.
• We now need to identify the correct time delay d in the negative feedback at the upper basin of attraction. Because large beetle populations can deplete their food supplies, by killing trees faster than they are produced, we probably need a time lag of at least two. Enter 2 for d.
• Next we must provide a coefficient of curvature Q. As bark beetle productivity has been found to vary with the square root of attack density (see Berryman 1974), enter 0.5 for Q.
• P1b now informs you that your selection of parameters creates a carrying capacity which is too large. This is because the smaller value of Q increases K according to the formula K = exp[(ln A - ln C) / Q]. Hence we can find an appropriate value for C, which gives us the correct K = 7380, from the equation C = exp(ln A - ln K * Q). In this case C = exp(ln 1.5 - ln 7380 * 0.5) = 0.0175.
• Press [F7] to change parameters, [Y] to alter the R-function, and change C to 0.0175.
• Now press [F8] to simulate, defaulting all conditions and omitting environmental forcing. Notice that the trajectory cycles around the equilibrium line with damped oscillations, and that the first outbreak lasts about 7 years.
• Press [F10] when you have finished simulating and use [F1] to refresh your memory about low-density basins of attraction. Remember that bark beetle population growth is restricted at low densities by host resistance.
• Press [Y] to build a low-density R-function. We now need an estimate of the low density interaction coefficient. Assuming that one tree per hectare is infested per year by 600 beetles in the endemic state, then L = 600 * 0.0025. If we wish to keep the coefficient of curvature at 0.5, then Cl = exp(ln1.5 - ln 600 * 0.5) = 0.0612. Enter this value for Cl.
• Because bark beetles can only infest dead or dying trees when their density is low, they are not likely to impact the food supply for future generations and, therefore, the time lag should be one. Enter [1] for dl.
• Enter 0.5 for Ql.
• Use [F1] to refresh your memory about escape thresholds then provide an estimate for U, say 2000 beetles per hectare. The two equilibrium R-function is now shown on the screen (see figure). This is our model for the mountain pine beetle. It has two basins of attraction separated by an unstable separatrix.
• Press [F8] to simulate and default all operating conditions. Notice the very stable low-density equilibrium in a constant environment.
• Press [F8] to simulate again, default the time horizon, enter 2001 and 2010 for the initial densities, and default the standard deviation. Notice the outbreak cycle which settles to a stable equilibrium.
• Press [F8] to simulate again and set the time horizon at 50 years. Set both initial densities at 10 and the standard deviation at 0.6. Repeat this simulation a number of times until you have observed one or two outbreaks. Notice that the population fluctuates around its low-density equilibrium most of the time but exhibits occasional outbreaks of 5 to 7 years duration and peak densities of 20000 to 60000 beetles/ha/yr (between 30 and 100 trees killed/ha/yr).
• Use [F6] to save some of the more interesting trajectories on disk. You should also print them out if your computer is hooked up to a printer. You can use the data you save to perform a data analysis with P1a.
• Continue simulating with different run-times, standard deviations, and initial densities until you have a good feel for the dynamics of the beetle population.

References:

• Berryman, A. A. 1972. Resistance of conifers to invasion by bark beetle-fungus associations. BioScience 22: 598-602.
• Berryman, A. A. 1974. Dynamics of bark beetle population: Towards a general productivity model. Environmental Entomology 4: 579-585.
• Berryman, A. A. 1986. Forest Insects: Principles and Practice of Population Management. Plenum Press, New York.
• Cole, D. M. and C. B. Edminster. 1985. Growth and yield of lodgepole pine. Pages 263-290 in D. M. Baumgartner, R. G. Krebill, J. T. Arnott and G. F. Weetman (Eds.) Lodgepole pine: The species and its management. Cooperative Extension Service, Washington State University, Pullman.
• Raffa, K. F. and A. A. Berryman. 1983. The role of host plant resistance in the colonization behavior and ecology of bark beetles. Ecological Monographs 53: 27-49.
• Raffa, K. F. and A. A. Berryman 1986. A mechanistic computer model of mountain pine beetle populations interacting with lodgepole pine stands and its implications for forest managers. Forest Science 32: 789-805.