Pine Cones and Beetles
Tutorial
Use P1a to create data files for the red pine cone and
cone beetle populations per hectare studied by Mattson (1980 -
Table 1, Columns 2 and 3): Red Pine Cone Beetles - 1218, 7200,
1525, 2718, 1770, 1614, 1055, 3695, 500, 1000). Red Pine Cones
- omit first observation, 23632, 14636,13006, 28615, 20039, 18755,
35419, 7189, 14131, 21544. Divide each observation by 10 before
entry them in P1a and building two single-species models.
Remember to save the models to disk.
- Enter the directory in which your PAS programs reside
and load the PAS shell. Press [3] to access the
TWO-SPECIES MENU, then [1] to load the Time Series Analysis
- P2a. Place the disk containing the pine cone and cone
beetle data and models in the appropriate drive. Use [F1]
for a brief description of P2a. Press [F10] to proceed.
- When asked, enter the path on which your data files reside,
then enter the name you gave to the first data file; e.g., PCONES.
Press [P] for plant when asked to decide on the trophic
level. NOTE that the lower trophic level must be entered first
in trophic (feeding) interactions. Enter the name of the cone
beetle file; e.g., CNBEETLE. Press [H] to label it as a
herbivore.
- Press [B] when asked if pine cones benefit, harm or
have no effect on cone beetles, then [H] to indicate that
cone beetles harm pine cones. These answers allow PAS to
identify the interaction as EXPLOITATION (see figure).
Press [F10] to continue.
- A time series plot of the pine cone
and cone beetle data will appear in the graphics window. Use [F2]
and [F3] to replot the graph on arithmetic and logarithmic
scales and [F4] and [F5] to adjust its speed so
that you can clearly observe the motion of the trajectory.
- Press [F10] to continue and carefully observe the two-species
phase portrait (see figure). Use [F1]
key for information on phase plots.
- Use [F2] and [F3] to observe the motion of the
trajectory, noting that it moves about in a rather haphazard manner.
Press [F10] to proceed.
- The results of a regression analysis of the LV exploitation
model are now displayed (see figure).
The second column of the REGRESSION ANALYSIS table shows the statistics
for the simple regression of pine cone rate of increase (R)
on cone density (N). This is the intraspecific effect,
or the effect of pine cone density on its own rate of increase.
The third column shows the values of the statistics for the simple
regression of R on the density of cone beetles, Z,
or the interspecific effect. NOTE that the slope of the interspecific
effect is positive, which does not seem reasonable because we
would expect beetle feeding to have a negative effect on the rate
of increase of pine cones. The last column shows the statistics
for the multiple regression of R on N and Z.
NOTE that the significance of the interspecific effect improves
from 0.58 to 0.91 in the multiple regression and that the relationship
is now negative, as would be expected in a trophic interaction.
NOTE that the coefficient of determination has been increased
by the addition of the interspecific effect so that the multiple
regression equation explains about 83 per cent of the variation
in the data (% variation explained = rm * 100). NOTE that
the multiple regression statistics are also displayed in the top
right window and are labeled as follows: the intercept (A),
the slope of the intraspecific effect (C1), the slope of
the interspecific effect (C2), the coefficient of determination
(r2), and the coefficients of partial determination (pn)
and (pz).
- Press [F10] to continue and examine the regression
analysis of the RD model (see figure).
This analysis is identical to the previous one with the exception
that the interspecific interaction is defined in terms of the
consumer/resource ratio; i.e., Z is now the density of
cone beetles divided by the density of pine cones. NOTE that the
RD model explains 86% of the variation in the data, and that this
is slightly better than the LV model.
- Press [R] to accept the RD model and compare the LV
and RD models for the cone beetle, noting
that the latter gives a slightly better fit. NOTE that the intraspecific
effect is quite strong in the simple regression (r2 = 0.58)
but is much weaker in the multiple regression (pn = 0.009).
This implies that adding an intraspecific effect contributes little
to the explanation of cone beetle dynamics. On the other hand,
the interspecific effect in the RD model explains almost all of
the variation in the cone beetle rate of change, suggesting that
they are largely determined by the abundance of pine cones.
- Press [R] to accept the RD model and examine the preliminary
interaction structure (see figure).
- Press [F10] to continue, then [R] three times
to retain the first three interactions. Press [D] to delete
the intraspecific effect on the cone beetle rate of change.
- Press [S] to simulate with the model and then default
the run length and initial conditions. Then press [D] to
simulate deterministically (see figure);
i.e., in the absence of environmental variability. NOTE that the
cone population quickly damps to a stable equilibrium and that
the beetle population stabilizes in a similar way.
- Press [F10] to continue and observe the simulated trajectory
(grey line) on the phase plane. Notice
that the simulated trajectory is contained within the data (magenta
points) because random variation was excluded from this simulation.
- Press [S] to simulate again and default the run length
and initial conditions. Then press [S] again to perform
a stochastic simulation and default to the estimated standard
deviations. Observe the time series plot on arithmetic
and logarithmic scales; i.e., use the
[F2] and [F3] keys.
- Press [F10] and notice how the simulated trajectory
exceeds the bounds of the data, suggesting that the variation
is too great (this is not too surprising as some of the measured
variation probably includes sampling error).
- Press [S] to simulate again, defaulting the run length
and initial conditions, then press [S] again for another
stochastic simulation. Reduce the standard deviation of both species
to 0.3 and press [S] to use the same series of random numbers
on both species. Run several simulations under the same conditions
and notice how some simulated trajectories approximately cover
the data on the phase plot.
- Press [F6], [D] and [Y] to save the model
on your data disk; i.e., model parameters will be written to the
files PCONES.MO2 and CNBEETLE.MO2. You can also save the time
series data from your last simulation. If you do so make sure
that you give the files a different name from the original data.
For fun you can run the simulated data through P1a and
P2a.
- Press [R] to reconstruct the model, then [F10]
and [R] four times to retain all the interactions, including
the intraspecific effect on the cone beetle rate of change. Repeat
the simulation experiments with this model, noting that the intraspecific
interaction of cone beetles has little affect on the behavior
of the model. When you have finished playing with the model, terminate
the analysis by pressing [Q] or [Alt]+[X]
Reference:
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