Target plants with neighbours that were sown later, i.e. with neighbours that were smaller throughout the experiment, grew to a larger size than targets with neighbours that were sown earlier (Fig. 2a, one-way anova, *F* = 642.9, degrees freedom (error) = 164, *P* < 0.001). In the treatment with the neighbours that were sown latest, targets still grew to a much smaller size than control plants (Fig. 2a), even though the neighbours were considerably smaller than the targets throughout the experiment.

#### EFFECT OF NEIGHBOUR SIZE

With the parameters α and β as estimated above on isolated (control) plants from the size experiment, the mean value of the ‘observed’ competitive effect *x*_{ij} was significantly greater than zero in eqn 7–10 (respectively, mean: 0.028, 0.490, 0.08, 0.056; standard error: 0.002, 0.05, 0.005, 0.004; *P* < 0.001 for each assuming normality). Furthermore, *x*_{ij} was positively correlated with neighbour size, indicating that larger neighbours had a greater competitive effect (for eqn 7–10, respectively: *r*_{s} = 0.150, 0.167, 0.150, 0.150, *n* = 1014, *P*≤ 0.001).

This non-independence of residuals left after eliminating effects of intrinsic growth justifies a more detailed search for functions describing how neighbours affect targets. In the size experiment, because the distance between targets and neighbours was fixed, the non-ZOI kernels could be simplified to

*F*_{2}(*w*_{i}, *w*_{j}, *d*_{ij}) = γ*f*(*w*_{i}, *w*_{j}),(eqn 13)

where γ is a constant, confining attention to functions describing size effects alone. For the ZOI model the effects of size and distance cannot be separated, because they interact non-linearly to determine the region of overlap between the two ZOIs, *z*(*w*_{i}, *w*_{j}, *d*_{ij}) (see below).

Four explicit functions were used for *f*(*w*_{i}, *w*_{j}) as listed (A–D) in Table 2. Each function was tested in eqn 3–5, giving 12 functions in all. In addition, a ZOI kernel was used in eqn 6; this function was

Table 2. Tests of the effect of plant size on growth of target and neighbour plants. The tests use non-linear regression on calculated values of competitive effect *x*_{ij} (eqn 7–10) in the size experiment. A–D are functions defined at the end of the Table; D is an APA (area potentially available) function of Soares & Tomé (1999); ρ is a parameter used in the ZOI function, not estimated in the regression. The term γ is a parameter estimated by regression; its standard error is also given. *R*^{2} measures the proportion of variation in *x*_{ij} explained by the kernel. The percentage MSE measures how close the predicted and observed final plant sizes are, relative to the sizes predicted from growth of an isolated plant (eqn 2); lower percentage MSE values indicate a better predictive ability Function | | γ | Standard error | *R*^{2} | Percentage MSE (× 10^{−6}) |
---|

From eqn 3: α − β*w*_{i} *−* γ*f*(*w*_{i}, *w*_{j}) | A | 0.00438 | 0.00030 | 0.022 | 9.07 |

B | 0.0308 | 0.0020 | 0.035 | 6.38 |

C | 0.000580 | 0.000043 | 0.001 | 10.1 |

D | 0.0624 | 0.0043 | 0.024 | 9.28 |

From eqn 4: | A | 0.0827 | 0.0078 | 0.026 | 23.0 |

B | 0.558 | 0.053 | 0.023 | 29.8 |

C | 0.0114 | 0.0011 | 0.023 | 18.6 |

D | 1.12 | 0.11 | 0.016 | 31.9 |

From eqn 5: | A | 0.0127 | 0.00075 | 0.035 | 5.64 |

B | 0.0895 | 0.0050 | 0.053 | 4.56 |

C | 0.00168 | 0.00011 | 0.001 | 6.67 |

D | 0.180 | 0.011 | 0.036 | 6.07 |

ZOI from eqn 6: α(1 − *F*_{2}) *–* β*w*_{j} | ρ | | | | |

0.5 | 0.109 | 0.0074 | 0.028 | 27.7 |

1.0 | 0.106 | 0.0072 | 0.027 | 27.2 |

1.5 | 0.106 | 0.0071 | 0.028 | 27.2 |

2.0 | 0.106 | 0.0072 | 0.028 | 27.3 |

Definitions of *f*(*w*_{i}, *w*_{j}): | A *w*_{i} | | C *w*_{i}, *w*_{j} | | |

B *w*_{i}, *w*_{j} | | D *w*_{j}/(*w*_{i}, *w*_{j}) | | |

- (eqn 14)

where the area of the ZOI of individual *i*, *Z*(*w*_{i}), is proportional to the area of the largest circle that could be placed inside a square of area equal to *i*'s box area:

- (eqn 15)

The constant ρ scales the relation between box area and ZOI and four values (0.5, 1, 1.5, 2; Table 2) were used in fitting the kernel. The function *z*(*w*_{i}, *w*_{j}, *d*_{ij}) is the area of overlap of the ZOIs of *i* and *j*, determined by the rules of trigonometry (details not shown), and *L*(*w*_{i}, *w*_{j}) describes the fractional allocation of resources to *j* in this region of overlap

- (eqn 16)

Initially, we set φ = 1 corresponding to perfect size symmetry, where contested resources are divided between competitors in proportion to their sizes (Schwinning & Weiner 1998); effects of asymmetric competition are considered below.

Regression analysis thus involved fitting a single parameter γ for each function. Because the analysis was based on residuals after removing effects of intrinsic growth of targets, *R*^{2} values were relatively small (Table 2). The analysis shows the highest *R*^{2} values came from models of the form of eqn 5, suggesting that eqn 5 is the most appropriate way to modify the Gompertz equation to include the competitive effect of a neighbour.

To evaluate the predictive power of the functions in the long term, we compared the plant sizes observed at the end of the experiment with plant sizes predicted by numerical integration using the functions (see for example eqn 11 and 12). Regardless of the choice of function, incorporating the effect of the neighbour improved the prediction of final size by several orders of magnitude as measured by the small percentage MSE values (Table 2, Fig. 4). Fits to the observed data were good, bearing in mind that the equations contain only three parameters (only one more than the equation for isolated growth), and that over the time period in question plant size increased by a factor of 2000–4000. Equation 5 gave percentage MSE values somewhat lower than eqn 3, 4 and 6, as can be seen from visual inspection of the graphs (Fig. 4a). Equation 5 also gave better predictions for the neighbour sizes than did eqn 3 and 4 (Table 2, Fig. 4b). Equation 6 was insensitive to the scaling from box area to ZOI (parameter ρ), and was relatively unsuccessful in predicting target size unless the target and neighbour plants were similar in size (Fig. 4a). None of the equations gave good predictions for neighbours sown earliest, probably because these plants were senescing. The results suggest a ranking Equation 5, 3, 6, 4 in terms of their success incorporating competition into the Gompertz equation.

To test for asymmetric competition between plants of different sizes, we introduced a second parameter φ, replacing *w* with *w*^{φ} in the functions (the ZOI model already included the asymmetry parameter φ, see eqn 16). Effectively a value of φ > 1 increases the differential in size between competing plants, making the interaction more asymmetric, whereas φ < 1 reduces the size differential and makes the interaction less asymmetric. The regression analysis was repeated with two parameters: γ and φ (Table 3).

Table 3. Tests of asymmetric competition between target and neighbour plants. Regression analysis is the same as in Table 2 except that an asymmetry parameter φ is estimated as well as γ. The percentage MSE in this case measures how close the predicted and observed final plant sizes are in the two-parameter models, relative to the one-parameter models in Table 2; values < 100% (shown in boldface) indicate that the asymmetry parameter improves the prediction | γ | Standard error γ | φ | Standard error φ | *R*^{2} | Percentage MSE γ only |
---|

From eqn 3: |

| A | 0.00303 | 0.0019 | 1.18 | 0.30 | 0.023 | **97.0** |

| B | 0.0290 | 0.0024 | 1.56 | 0.31 | 0.038 | 338 |

| C | 0.00512 | 0.0039 | 0.452 | 0.19 | 0.008 | 118 |

| D | 0.0620 | 0.0044 | 3.23 | 0.69 | 0.037 | **95.0** |

From eqn 4: |

| A | 0.00335 | 0.0046 | 2.57 | 0.64 | 0.036 | **60.9** |

| B | 0.527 | 0.061 | 1.54 | 0.43 | 0.026 | 125 |

| C | 0.00173 | 0.0028 | 1.47 | 0.39 | 0.025 | **80.1** |

| D | 0.842 | 0.081 | 37.3 | 49.3 | 0.037 | 147 |

From eqn 5: |

| A | 0.00703 | 0.0039 | 1.29 | 0.27 | 0.036 | **91.6** |

| B | 0.0829 | 0.0059 | 0.071 | 0.094 | 0.060 | 187 |

| C | 0.0123 | 0.0081 | 3.42 | 0.50 | 0.013 | 114 |

| D | 0.177 | 0.011 | 3.74 | 0.69 | 0.059 | 106 |

From eqn 6: | ρ | | | | | | |

| 0.5 | 0.184 | 0.027 | −0.0516 | 0.17 | 0.033 | **61.2** |

| 1.0 | 0.169 | 0.023 | 0.00353 | 0.18 | 0.033 | 64.5 |

| 1.5 | 0.167 | 0.023 | 0.019 | 0.18 | 0.033 | 65.2 |

| 2.0 | 0.166 | 0.023 | 0.0234 | 0.18 | 0.033 | 65.0 |

The extra parameter φ did little to improve the fit. In eqn 3–5, the estimated value of φ was most often greater than one, implying that competition was more asymmetric than was implied by the functions without φ. In the ZOI model eqn 6, it is striking how close to zero estimates of φ were, implying competition close to complete symmetry, where contested resources are divided equally between competitors regardless of size (Schwinning & Weiner 1998). The percentage MSE values show that prediction of final size of the plants was not improved by allowing for asymmetry in eqn 3–5: often asymmetry actually made the prediction worse, and in view of this we assume φ = 1 when using these equations below. However, the asymmetry parameter did provide some improvement in the fit of the ZOI model (Table 3), so we retain the extra parameter φ in this case. Nevertheless, even with the inclusion of the asymmetry parameter φ, the predictions of final size from the ZOI are still inferior to the non-ZOI functions with symmetric competition (Fig. 4).

On the basis of these results, we carry forward the following five equations for analysis of the distance experiment:

- (eqn 17)

- (eqn 18)

*RGR*_{ij} = α−β*w*_{i}−γ(*w*_{j}/*w*_{i})(eqn 19)

*RGR*_{ij} = α−β*w*_{i}−γ*w*_{j}(eqn 20)

- (eqn 21)

Equations 17 and 18 are from eqn 5 with competition functions B and A, respectively (Table 2); eqn 19 and 20 are from eqn 3 with functions B and A, respectively (Table 2); eqn 21 is the ZOI model as described by eqn 14–16.

The *R*^{2} and percentage MSE values indicate that the best equation overall was eqn 17. Equation 18 is also kept, as it contains a particularly simple form of the function *f*(*w*_{i}, *w*_{j}) and, at the same time, gave quite accurate predictions. We retain eqn 19 and 20 because these gave predictions not greatly inferior to eqn 17 and 18 and include the effect of neighbours in an especially simple way. Lastly we keep the ZOI model eqn 21, in view of its interest in plant ecology.

#### EFFECT OF NEIGHBOUR DISTANCE

We now use growth of pairs of plants in the distance experiment to find functions that describe the effect of spatial separation of the target and neighbour on growth of the target. Once again, we computed the residual *x*_{ij} left after removing the effect of intrinsic growth of the plants: Equation 9 (corresponding to eqn 17 and 18), eqn 7 (corresponding to eqn 19 and 20), and eqn 10 (corresponding to eqn 21). Values of *x*_{ij} were negatively correlated with *d*_{ij}, irrespective of which equation was used (*r*_{s} = −0.092 and *P* = 0.006 for each). Because the distance between the target and its neighbour was the main remaining variable, there is a justification for a more detailed search for functions that can best describe the effect of this distance.

To deal with distance in eqn 17–20, the parameter γ was replaced by functions of distance *g*(*d*)_{ij}, giving kernels of the form

*F*_{2}(*w*_{i}, *w*_{j}, *d*_{ij}) = *g*(*d*)_{ij}*f*(*w*_{i}, *w*_{j});(eqn 22)

four simple decay functions for *g*(*d*)_{ij} were used to describe this dependence (Table 4). The *g*(*d*)_{ij} functions had two parameters: the maximum deleterious effect *κ* of a neighbour on the flux in size of the target when the plants are at the same location (*d*_{ij} = 0), and the distance σ at which the competitive effect is half of the maximum value. These functions were applied to each of eqn 17–20 giving 16 functions as listed in Table 4. This separation of effects of distance and size was not possible in the ZOI function, eqn 21; here the two parameters γ and φ were retained from the size experiment and, as before, we examined four values of the constant ρ in the ZOI function.

Table 4. Tests of some functions to describe the effect of neighbour distance on growth of target plants. The tests use non-linear regression on calculated values of the competitive effect of a neighbour, *x*_{ij}. Functions describing size dependence from the size experiment are given by the right-hand sides of eqn 17–20. Functions describing distance dependence are given by A–D at the end of the Table; these replace γ in eqn 17–20. Two parameters *κ* and s are estimated; SE_{κ} and SE_{σ} are standard errors of the estimates. Equation 21 is the ZOI model, with parameters γ and φ estimated as for the size experiment. *R*^{2} measures the proportion of the variation in *x*_{ij} explained by the kernel. The percentage MSE measures the deviation between the observed final plant size and the size predicted from the kernel, as described in the text; lower percentage MSE values indicate a better predictive ability *g*(*d*_{ij}): | | *κ* | SE_{κ} | σ (mm) | SE_{σ} | *R*^{2} | Percentage MSE |
---|

From eqn 17: |

| A | 0.0764 | 0.010 | 28.9 | 12.1 | 0.013 | 11.23 |

| B | 0.0644 | 0.0064 | 38.4 | 9.5 | 0.011 | 11.41 |

| C | 0.0921 | 0.021 | 17.0 | 11.0 | 0.014 | 11.05 |

| D | 0.0660 | 0.0072 | 36.4 | 12.0 | 0.011 | 11.05 |

From eqn 18: |

| A | 0.0130 | 0.0018 | 31.3 | 13.1 | 0.012 | 11.82 |

| B | 0.0110 | 0.0039 | 40.2 | 10.0 | 0.011 | 11.95 |

| C | 0.0157 | 0.0034 | 18.3 | 11.5 | 0.013 | 11.62 |

| D | 0.0112 | 0.0012 | 39.5 | 13.0 | 0.011 | 11.94 |

From eqn 19: |

| A | 0.0307 | 0.0061 | 33.8 | 22.9 | 0.004 | 8.95 |

| B | 0.0269 | 0.0039 | 39.1 | 14.5 | 0.004 | 9.00 |

| C | 0.0329 | 0.0090 | 29.4 | 29.7 | 0.004 | 8.94 |

| D | 0.0272 | 0.0043 | 39.6 | 20.0 | 0.004 | 9.01 |

From eqn 20: |

| A | 0.00580 | 0.0011 | 33.3 | 19.8 | 0.015 | 9.18 |

| B | 0.00498 | 0.00068 | 39.9 | 13.4 | 0.015 | 9.26 |

| C | 0.00647 | 0.0017 | 24.6 | 21.6 | 0.015 | 9.12 |

| D | 0.00504 | 0.00074 | 40.9 | 18.6 | 0.015 | 9.30 |

| ZOI | ρ | γ | SE_{γ} | φ | SE_{φ} | |

From eqn 21: |

| 0.5 | 0.206 | 0.034 | 0.509 | 0.77 | −0.002 | 10.33 |

| 1.0 | 0.149 | 0.021 | 0.576 | 0.68 | 0.010 | 9.17 |

| 1.5 | 0.131 | 0.018 | 0.452 | 0.59 | 0.017 | 8.86 |

| 2.0 | 0.120 | 0.016 | 0.361 | 0.56 | 0.017 | 8.86 |

Definitions of *g*(*d*_{ij}): |

A: Exponential | | | C: Hyperbolic I | *κ*[1 + (*d*_{ij}/σ)]^{−1} | | | |

B: Gaussian | | | D: Hyperbolic II | *κ*[1 + (*d*_{ij}/σ^{2})]^{−1} | | | |

We carried out a non-linear regression analysis to fit the two parameters of the functions (Table 4). As before, the *R*^{2} values were rather small, because we were dealing with the residuals after removing the effect of intrinsic growth of targets. For the predictions of growth in the long term, each function provided a substantial improvement over the isolated growth model (eqn 2), although the percentage MSEs remained much larger than those from the size experiment (Table 2). These larger percentage MSE values are not surprising, because there were more sources of variation in the distance experiment (*w*_{i}, *w*_{j},* d*_{ij} varied with treatment and time), and the observed mean target sizes did not increase monotonically with *d*_{ij} (Fig. 5). The values for *R*^{2} and percentage MSE were largely independent of the function chosen for *g*(*d*)_{ij} (Table 4).

The results show that the largest *R*^{2} values, and the greatest improvement over the isolated growth model (lowest percentage MSEs) was given by the ZOI model (eqn 21) with scaling parameter ρ = 1.5 (Table 4). The fitted value for the asymmetry parameter φ was close to 0.5, in contrast to the φ-values close to zero for the ZOI model fitted to the size experiment (Table 3): apparently, differences in the conditions in the glasshouse at different times of year induced differences in the asymmetry of competition. The best non-ZOI function was eqn 21, which gave *R*^{2} and percentage MSE values close to the ZOI model (Table 3). The predictions from the different functions for the final size of a target grown with a neighbour at distance zero ranged from 1724 to 2465 mm^{2} (results not shown). These values are reasonable, given that isolated plants grew to 3380 mm^{2} in the same period (Fig. 2). [A prediction for final target size with a neighbour at distance zero of much less than half the size of an isolated plant of the same age would correspond to overcompensating density dependence, which has rarely been observed in plant populations (Silvertown & Lovett-Doust 1993).]