Symmetry option Competitive regime
3.10. Application of the Coupled Map Lattice to Crop Data.
3.10.1. Introduction.
In this section the CML model is adjusted to model a eld experiment on carrot plants (Daucus carota) carried out at Horticulture Research International at Wellesbourne in Warwickshire in the summer of 1985. Seedlings were planted on three sowing dates in a grid and harvested over the subsequent 100 days. Three of the sowing patterns were examined:single,doubleandtriple
sowing dates. In the rst case all plants were sown on day 1. In the second case alternate plants were sown on day 1 and the remainder on day 14. The nal experiment planted every third plant on day 1, half of the gaps were lled on day 14 and the rest on day 21. Plants of each cohort were harvested on days 41, 50, 62, 70, 83 and 97. Four dierent inter-plant spacings were used: 5, 71
2, 10 and 15cm there were three replicates of each sowing treatment.
The sowing and harvesting patterns are shown in gures 13 - 15. The letters
a
,b
andc
refer respectively to the sowings on days 1, 14 and 21. Upper case letters indicate harvested plants and lower case letters give `guard plants'. The blocks of harvested plants were removed and weighed, starting from one end of the grid. Further experimental details are given in Benjamin (1988) as experiment 2.3.10.2. Extension of the Basic Model.
The carrot experiment is ideal for simulation by a CML because of the rectilinear pattern of the plants. The width of the CML model was changed to 12, as in the experiments. The length was changed to 29, 42 and 55 for the single, double and triple sowings, to allow 12 harvestings to be simulated. This was twice as many as the experiment, to allow further development of the stand to be studied.
Seedlings were `sown' on the CML at the three sowing dates in the patterns given by gures 13 - 15, with mean initial mass 10;5 (relative to the maximum mass described in section 3.2
and equation (17)) and with a standard deviation of . At the appropriate iterations, 6 plants were removed of each type (
a
,b
orc
) by resetting their masses to zero and the mean massesattained were recorded. The model was run for the single, double and triple sowings and for all four competitive symmetries (table 3). Standard deviations of = 0%, 1% and 10% of the initial plant size were used. A range of intrinsic growth rates were used: g = 5, 10 and 20 (gm;2day;1). The results were averaged over 25 runs to produce smooth curves.
3.10.3. Results and Discussion.
Figure 16 shows the mean mass for the single, double and triple sowings for the three growth rates. These curves are for the 0% standard deviation, but the other cases resulted in values diering by less than 0.1%. Variation of symmetry type has signicantly dierent eects only for the multiple sowings and for suciently higher growth rates (comparing rstly gures 16a, d, g with c, f, i and secondly a, b, c, with g, h, i). Since there is a plant in every CML cell (density 1.0) in the single-sowing even-aged stand, there is no signicant dierence between the plants to be enhanced by greater asymmetry of competition. In the multiple-aged stands, the rst-sown plants have an advantage over later cohorts. This is magnied by asymmetric interference as is particularly clear in gures 16h - i.
The intrinsic growth rate g (equation (15)) also signicantly aects the resulting population structure. If the growth rate is low, the rst cohort does not gain such an advantage over later plants and there is less eect of asymmetric competition. If the growth rate is high, the rst plants gain such a size advantage that even slight asymmetry leads to great size dierences between the cohorts. Hence there is the most dierence between the symmetry types at intermediate growth rates (gures 16d - f).
Figure 17 shows the experimental data for the 12 combinations of plant spacings and sow- ing treatments, each with three replicates. The double and triple sowing experiments show signicant dierences between the rst and subsequent cohorts, which indicates considerable asymmetry of interaction. These dierences arise at a fairly early stage, indicating a relatively high intrinsic growth rate. As the spacing between plants rises, the second and third sowings produce relatively larger plants. The larger growing space delays interactions, so there is less opportunity for size dierences to arise. Wider spacing is thus equivalent to a lower growth
rate at a xed spacing, which is a constraint of the CML model.
Although this model is intended only to study the type of interactions, data not being available for parameter estimation, a basic comparison can be made by tting a constant multiple of the model output to the empirical data by a least squares technique. Figure 18 shows ttings for sample single, double and triple sowings. Figure 18a shows the results for a single sowing at the maximum and minimum spacings, with a good t to the data by the absolutely asymmetry model with g = 20. Figure 18b shows the 5cm spacing for the double sowing, with a reasonable t for absolutely asymmetry and g = 20. The largest spacing, gure 18c, is apparently mid-way between the g = 10 and g = 20 model results. Thus as expected, the wider spacing is equivalent to a lower growth rate. The triple sowing data provide a less good t, as the later cohorts of carrots carried on growing larger for slightly longer than the model predicted (gure 18d). In conclusion, the plant CML model was able to show that carrot plants grown on a rect- angular lattice experienced predominantly asymmetric competition, resulting in high levels of suppression in later cohorts of plants.