Textured Psyllid Tutorial

The textured psyllid is an Australian insect that sucks sap from the leaves of eucalyptus trees. The population dynamics of this insect have been studied in detail by Clark (1964a,b) who found that the psyllid exhibited occasional eruptive outbreaks but was normally maintained at sparse densities by predators. Clark's studies, and a more recent experiment by Loyne et al. (1983), came to the conclusion that birds were the most significant factor in maintaining sparse populations. However, if the psyllid populations exceeded a certain threshold density, birds were no longer able to control them and they exploded to very high densities. During outbreaks, psyllid populations only declined when they severely impacted their host trees, causing dieback and sometimes tree death. In addition, the leaves of heavily infested trees contain increased levels of phenolics which impacted the survival of young psyllid nymphs.

The detailed work of Clark can be used to construct a model of textured psyllid populations dynamics:

• Start the analysis as before and, when prompted, enter a name for the insect (Psyllid), a unit of measure (shoot), and number of units per sample (1).
• Clark's data showed maximum rates of increase of around 2 during outbreaks. Enter 2 for A.
• The psyllid carrying capacity can also be estimated from Clark's data, which showed that the average psyllid density during outbreaks was about 150 eggs per eucalyptus shoot. As C = 2 / 150, enter 0.0133 for the interaction coefficient.
• Because large psyllid populations severely impact the quantity and quality of food for the next generation, the time delay should probably be 2. Enter 2 for d.
• As there is no evidence for non-linearity in the R-function, enter 1 for Q.
• Press [F8] to simulate and default all conditions with no trend or step. Notice the cyclic dynamics with period of 10-12 generations and maximum amplitude of around 1200 eggs per shoot. This is a much higher density than is normally recorded, Clark's data showing maximum densities around 500. We can stabilize the R-function by decreasing the curvature parameter Q. Let's make Q = 0.5. Remember, however, that if we do this we must also alter C to obtain the correct value for K; i.e., C = exp(ln 2 - ln 150 * 0.5) = 0.1633.
• Press [F7] then [Y] to alter the R-function, and enter 0.1633 for C and 0.5 for Q. Default the other parameters.
• Press [F8] to simulate, defaulting all operating conditions. Notice that the amplitude of the outbreaks (around 500 eggs/shoot) is much more in line with the data.
• Press [F10] to indicate satisfaction with the upper R-function, then [Y] to build a low-density attractor.
• We estimated from Clark's data that the average density of psyllids during the sparse phase was about 5 eggs per shoot, and that the escape threshold was about 15 eggs per shoot. Enter Cl = 2 / 5 = 0.4.
• Birds are thought to regulate psyllids at low densities by aggregating in areas where psyllids are abundant. As these responses are rapid, the time delay should be short. Enter 1 for dl.
• Enter 1 for Ql.
• Enter 15 for the escape threshold U and observe the two-equilibrium R-function (see figure).
• Press [F8] to simulate, then default the run length, set the initial density to 2, and the standard deviation to 0. Note the stable oscillations (2-point limit cycles) in a constant environment.
• Use [F8] to repeat the simulation in variable environments, with run lengths of 100 standard deviations of 0.3 - 0.8 (default the other options). Examine the simulations on the logarithmic plot by pressing [F3].
• Examine the outbreak behavior (see figure) and, if possible, compare it to Clark's interpretation [see Fig. 4 in Clark (1964b)]. Save one or two of the more interesting data sets for later analysis with P1a.

References:

• Clark, L. R. 1964a. Predation by birds in relation to the population density of Cardiaspina albitexture (Psyllidae). Australian Journal of Zoology 12: 349-361.
• Clark, L. R. 1964b. The population dynamics of Cardiaspina albitexture. Australian Journal of Zoology 12: 362-380.
• Loyne, R. H., R. G. Runnalls, G. Y. Forward and J. Tyers. 1983. Territorial bell miners and other birds affecting populations of insect prey. Science. 221: 1411-1413.