Prediction of weed density: the increase of error with prediction interval, and the use of long-term prediction for weed management

Authors


Dr J. Wallinga, RIVM, PO Box 1, 3720 BA Bilthoven, The Netherlands. Tel: 31 30 2742553; Fax: 31 30 2744409; E-mail: jacco.wallinga@rivm.nl

Abstract

1. This paper addresses the errors that are associated with the long-term prediction of weed densities, and the effect of these errors on the performance of weed management decisions based on those long-term predictions.

2. A model of weed population dynamics was constructed and its parameters were estimated from experimental observations of population dynamics of the weed species Stellaria media in a crop rotation.

3. The observations showed that estimates of weed population growth rate differed between two locations.

4. The model was used to analyse error propagation for predicted weed densities in an enlarged prediction interval. It is concluded that errors due to an uncertain population growth rate increase linearly with the length of the prediction interval, and thus pose an upper limit to the horizon for long-term predictions.

5. It is shown that a limited ability to predict weed densities does not necessarily impair the practical use of weed population dynamic models in planning for long-term weed control programmes.

Introduction

Weed ecologists have put much effort into quantifying the life-history characteristics of weeds and incorporating this knowledge into models of population dynamics (e.g. Doyle, Cousens & Moss 1986; Cousens et al. 1987; Mortimer, Sutton & Gould 1989). One of the main objectives of this effort was to predict weed densities over a long period, and to use the predictions for the selection of weed control measures that have some desired effect. Only a few studies have compared model predictions of weed densities with field observations, and their outcome showed that predictions were not very accurate (Firbank et al. 1984; Firbank 1991; Cousens & Mortimer 1995). This led Cousens & Mortimer (1995) to the conclusion that it is not at all evident that reliable long-term predictions can be made and that these predictions can be used to plan for long-term weed control programmes.

Various models of annual weed population dynamics have been proposed (e.g. Watkinson 1980; Cousens & Mortimer 1995; Wallinga 1998). Previous studies on the reliability of model predictions of weed density often assumed that throughout the prediction interval the weed population is subjected to a previously determined kill rate (Firbank & Watkinson 1986; Freckleton & Watkinson 1998). Alternatively, one may assume that each year a farmer will continue to make the weed control decision that gives the best expected economic results (Pandey & Medd 1991; Wallinga 1998). For instance, a farmer may continue to control weeds only when the weed density exceeds a threshold level (cf. Doyle, Cousens & Moss 1986). It has been shown recently that such a weed control strategy gives rise to complex dynamics (Hughes & González-Andújar 1997; Wallinga & Van Oijen 1997). But as yet it is unknown how reliable long-term predictions of weed density are in this case. Sensitivity analysis is an approach to gain some idea of the confidence that should be put into the predictions. In a sensitivity analysis the absolute or relative effect of a change in the value of a model variable on the predicted weed density is quantified (Cousens & Mortimer 1995). Sensitive dependence of predicted density on uncertain variables may indicate to what extent prediction is fundamentally limited by the accuracy of measurement of demographic variables.

The objective of this paper is to examine the change of error in predicted weed density when the prediction interval is enlarged, and to examine how errors in long-term prediction affect the performance of weed control decisions that are based on those predictions. The error in predicted weed density as studied here applies to the deviation between a prediction based on the actual value (‘correct’ value) and a prediction based on the nominal value (‘incorrect’ estimate) of an uncertain variable. First, an experiment is described that examines the population dynamics of the weed Stellaria media L. in a crop rotation of winter wheat and sugar beet. The experimental results are used to identify uncertain demographic variables in a model of population dynamics of S. media. The effect of small changes in uncertain demographic variables on predicted density is determined. Subsequently, the effects of small changes in uncertain demographic variables on the control decisions are analysed. Finally, the implications for the practical use of weed population dynamic models in planning for long-term weed management are indicated.

Material and methods

Experiments

Two similar experiments were conducted at two research stations in The Netherlands, one near Wageningen, henceforth indicated as experiment 1, and another near Nagele, henceforth indicated as experiment 2. Both experiments were carried out over the period 1990–93, during which time the experimental plots were cropped in successive years with winter wheat, sugar beet and winter wheat. The most dominant weed at both sites was S. media. Two weed management regimes were imposed in both experiments: (i) weeds were left uncontrolled in winter wheat, but weeds were controlled in sugar beet; (ii) weeds were controlled in both winter wheat and sugar beet. Each management regime was carried out on 10 plots at each site. Full experimental details are given in Table 1. The number of S. media plants in each plot was assessed in the winter wheat crop in March 1991 and March 1993. Crop yield of winter wheat was measured in each plot in 1993. Weed control in the sugar beet crop sufficed to eliminate weed plants.

Table 1.  Experimental setting in which the population density of Stellaria media L. was observed over 3 years. Half of the plots were treated with herbicides when winter wheat was cropped, in the other half of the plots weed control was omitted when winter wheat was cropped
 Experiment 1Experiment 2
  • *

    One half of the replicates received slightly more fertilizer than the other half.

  • Fenmedifam is used as common name for 3-methoxycarbonylaminofenyl 3-methylfenylcarbamate.

LocationWageningen, The NetherlandsNagele, The Netherlands
Soil typeLoamy sand soilSandy loam soil
Plot size18 m × 10 m21 m × 10 m
1991
CropWinter wheatWinter wheat
Observation20 March 1991, for each plot the21 March 1991, for each plot the
 number of weeds was counted innumber of weeds was counted in
 three quadrats of 0.25 m2three quadrats of 1 m2
Weed controlFluroxypyr/noneMecoprop, 2,4D, fluroxypyr/none
Fertilizer189 kg N or 148·5 kg N*122 kg N or 92 kg N*
1992
CropSugar beetSugar beet
Weed controlEthofumesate, fenmedifam, metamitron Ethofumesate, fenmedifam, metamitron
 with additional hoeingwith additional hoeing
1993
CropWinter wheatWinter wheat
Observation25 March 1993, for each plot the22 March 1993, for each plot the
 number of weeds was counted innumber of weeds was counted in
 8 quadrats of 0·25 m28 quadrats of 0·25 m2
Weed controlFluroxypyr/noneFluroxypyr, MCPA/none
Fertilizer171 kg N or 141 kg N*180 kg N or 150 kg N*
Measurement6 August 1993, for each plot the kernel14 August 1993, for each plot the kernel
 dry weight was measureddry weight was measured

Model

Various models of annual weed population dynamics have been proposed (Watkinson 1980; Cousens & Mortimer 1995; Wallinga 1998). These models differ in their description of density-dependent reduction of weed seed production at higher weed densities. However, these higher weed densities are of relatively little interest in weed management, as they would cause excessive crop yield losses. At lower weed densities the above-mentioned models can be approximated by a simple linear model. This model is used to describe the dynamics of S. media in a 2-year crop rotation of winter wheat and sugar beet, where weed control is optional in winter wheat:

image(eqn 1a)

where N is the S. media density in winter wheat, t denotes time (with 2-year time steps), a is the relative growth rate of the weed population when weeds are left uncontrolled, b is the relative growth rate of the weed population when weeds are controlled, and r is a binary control variable that indicates whether weeds are controlled in winter wheat (r = 1) or whether weeds are left uncontrolled in winter wheat (r = 0). The resulting winter wheat yields are described by:

image(eqn 1b)

where Y is winter wheat yield (kg m−2 kernel dry wt), Ymax represents the crop yield when weeds are controlled (kg m−2 kernel dry wt), and the parameter c indicates the yield loss per weed plant when weeds are left uncontrolled (kg−1 kernel dry wt).

The annual revenue of cropping winter wheat is described by the following equation:

image(eqn 1c)

where R is annual revenue (Dfl year−1), p is the wheat price (Dfl kg−1 kernel dry wt), and h is the costs of weed control (Dfl m−2).

Parameter estimation

The relative growth rate of the weed population when weeds are left uncontrolled, a, was estimated for both experiments by linear regression of weed density observed in 1993 against weed density observed in 1991 in plots where weeds were left uncontrolled. The relative growth rate of the weed population when weeds are controlled, b, was estimated for both experiments by linear regression of weed density observed in 1993 against weed density observed in 1991 in plots where weeds were controlled. The average yield loss per weed plant, c, was estimated for both experiments by linear regression of winter wheat yield against weed density observed in 1993 in plots where weeds were left uncontrolled. The economic parameters were estimated as P = 0·40 Dfl kg−1 kernel dry wt, and h = 0·01 Dfl m−2.

Prediction of weed densities

It is assumed that each year a weed control decision is made that gives the best expected economic results. For a specific S. media density this decision is derived by a computer program that applies the following rules:

• consider all possible combinations of weed control decisions (either control or no control) in winter wheat crops in the following years;

• for each combination of decisions calculate with the model the effects of decisions on the annual weed densities, crop yields and economic returns;

• calculate for each combination of decisions the sum of discounted revenues;

• select the weed control decision that leads to the highest sum of discounted revenues.

A planning horizon of 6 years and a discount factor of 0·8 were used. This means that the sum of discounted revenues is calculated by adding the expected revenue of the current winter wheat crop weighted with a factor 0·80, the subsequent winter wheat crop weighted with a factor 0·82, and the following winter wheat crop weighted with a factor 0·84.

Prediction of density of a weed population that is controlled according to optimum control decisions proceeds as follows. For an initial weed density N(0) the corresponding weed control decision r(0) is derived using the above-mentioned rules. Both N(0) and r(0) are substituted into equation 1a to calculate the expected weed density in the following winter wheat crop N(1). For this weed density, the corresponding weed control decision r(1) is derived. Both N(1) and r(1) are substituted into equation 1a to derive the expected weed density in the subsequent winter wheat crop N(2), and so on.

Results

Experiments

The average density of S. media plants was 26·8 m−2 in experiment 1 in 1991. For each management regime the densities observed in 1991 were approximately linearly related to the densities as observed in 1993 (Fig. 1a); densities increased on average by a factor of 2·0 when weeds were not controlled, and densities decreased on average by a factor of 0·8 when weeds were controlled (Table 2). The kernel dry weight of winter wheat in 1993 declined approximately linearly with increasing density of S. media (Fig. 1b).

Figure 1.

Results of experiment 1 at Wageningen. (a) The density of Stellaria media per plot as observed in spring 1991 and in spring 1993 in winter wheat crops. (b) The relationship between the density of Stellaria media observed in spring 1993 and kernel dry weight of winter wheat in that year. The plots were either sprayed with a herbicide after observation of density in 1991 and 1993 (filled symbols, linear regression on this data is indicated by the drawn line) or weed control was omitted in 1991 and 1993 (open symbols, linear regression on this data is indicated by the broken line). Some plots received slightly more fertilizer (squares) than others (triangles). In 1992, all plots were cropped with sugar beet and weeds were controlled on all plots.

Table 2.  Results of experiment 1, conducted at Wageningen, and experiment 2, conducted at Nagele. Data are presented as mean (±standard error of the mean). Estimates of initial density are based on data from 20 plots, other estimates are based on data from 10 plots. Winter wheat yield is given as kernel dry weight
  Experiment 1Experiment 2
Initial density of S. media plants (m−2)N(0)26·8(±2·8)1·1 (±0·2)
Rate of increase without weed controla2·0 (±0·1)43·5 (±4·6)
Rate of increase with weed controlb0·8 (±0·1)0·7(±0·4)
Crop yield in absence of weeds (kg m−2)Ymax0·95 (±0·02)0·85 (±0·01)
Reduction of crop yield per weed plant (kg)c1·7 (±0·5) 10−32·4 (±0·3) 10−3

The average density of S. media plants was 1·1 m−2 in experiment 2 in 1991. For each management regime the densities as observed in 1991 were approximately linearly related to the densities as observed in 1993 (Fig. 2a). Density increased on average by a factor of 43·5 when weeds were not controlled and densities decreased on average by a factor of 0·7 when weeds were controlled (Table 2). The kernel dry weight of winter wheat in 1993 declined approximately linearly with increasing density of S. media (Fig. 2b).

Figure 2.

Results of experiment 2 at Nagele. (a) The density of Stellaria media per plot as observed in spring 1991 and in spring 1993 in winter wheat crops. (b) The relationship between the density of Stellaria media observed in spring 1993 and kernel dry weight of winter wheat in that year. The plots were sprayed with a herbicide after observation of density in 1991 and 1993 (filled symbols, linear regression on this data is indicated by the drawn line) or weed control was omitted in 1991 and 1993 (open symbols, linear regression on this data is indicated by the broken line). Some plots received slightly more fertilizer (squares) than others (triangles). In 1992, sugar beet was grown on all plots and weeds were controlled.

Analysis of the model

Population dynamics when weeds are controlled according to optimum decisions

For a wide range of S. media densities the weed control decision that leads to the highest sum of discounted future revenues is calculated. When model parameters are estimated from experiment 1 (that is, a = 2·0 and b = 0·8) the best decision is to control in the current year when the weed density exceeds 10·85 plants m−2, otherwise the best decision is to skip control in the current year. Figure 3(a) shows the resulting relationship between weed density in the current winter wheat crop and the best weed control option and the expected weed density in the next winter wheat crop. When model parameters are estimated from experiment 2 (that is, a = 43·5 and b = 0·7), the best decision is to control in the current year when the weed density exceeds 6·80 plants m–2, otherwise the best decision is not to control. Figure 3(b) shows the resulting relationship between weed density in the current winter wheat crop and the best weed control option and the expected weed density in the next winter wheat crop. The dynamics of a S. media population, subjected to optimum control decisions, are thus effectively described by:

Figure 3.

Population dynamics of Stellaria media in a crop rotation of winter wheat and sugar beet, subjected to economic optimum weed control in winter wheat. (a) The relationship between the density of S. media in a winter wheat crop and the density in the next winter wheat crop, where parameters are estimated from experiment 1, at Wageningen. (b) The relationship between the density of S. media in a winter wheat crop and the density in the next winter wheat crop, where parameters are estimated from experiment 2, at Nagele. The arrow indicates the threshold weed density; the optimum economic weed control decision is ‘no control’ if weed density is below this threshold density and ‘control’ if weed density exceeds this threshold density. The broken line indicates the line where the weed density remains equal over time, that is, N(t) = N(t–1).

image(eqn 2)

where N is the density of S. media in winter wheat, t denotes time (with 2-year time steps), a is the relative growth rate of the weed population when weeds are left uncontrolled, b is the relative growth rate of the weed population when weeds are controlled, and K is the density threshold for applying weed control.

Effect of error in estimates of initial density on predicted density

In Appendix 1 it is derived that, in the long term, the relationship between a small error in the estimate of initial density N(0) and the resulting error in predicted density N(t) is:

image(eqn 3)

This means that an error in the estimate of initial density is preserved over the long term, and that the magnitude of the resulting error in predicted density does not depend on the length of the prediction interval.

Effect of error in estimates of population growth rate on predicted density

In Appendix 2 it is derived that, in the long term, the relationship between a small error in the estimate of population growth rate a and the resulting error on predicted density N(t) is:

image(eqn 4)

where α is the relative frequency of years in which weeds are left uncontrolled and NE is the expected weed density when weeds are not controlled. This means that the absolute error in predicted densities increases linearly with the length of the prediction interval if there is an error in the estimate of growth rate. When model parameters are estimated from experiment 1, the value of the term αNE is estimated as 2·4 plants m−2 per time step. When model parameters are derived from experiment 2, the value of this term is estimated as 0·5 plants m−2 per time step (see Appendix 2).

Effect of an error in control decisions on crop yield

The relationship between an error in the estimated density threshold for weed control K and the expected annual crop yield, denoted by YA, is (see Appendix 3):

image(eqn 5)

When model parameters are estimated from experiment 1, an overestimate of 1 plant m−2 in the density threshold would cause a decrease in average crop yield of 0·37 10−3 kg m−2. When model parameters are estimated from experiment 2, an overestimate of 1 plant m−2 in the density threshold would cause a decrease in average crop yield of 0·17 10−3 kg m−2.

Discussion

The purpose of the present paper was to examine the change of error in predicted weed density when the prediction interval is enlarged, and to examine how errors in long-term prediction affect the performance of control decisions based on those predictions. Weed population dynamics were described by a simple model that approximates previously published models of annual weeds. It was assumed that over the prediction interval a weed control decision was made that gave the best expected economic returns. When predicted dynamics of weed density were analysed, it was found that an error in measurement of initial density will result in a prediction error that is independent of the length of the prediction interval, and that an error in measurement of growth rate will result in a prediction error that increases linearly with length of the prediction interval.

The model of annual weed population dynamics used in this paper (equation 1a) rests on the assumption that the weed population growth rates, a and b, do not depend on weed density. This means, for instance, that the weed density should stay high enough such that import of weed seeds from other field is negligible, and that the weed density should remain low enough to avoid limitation of seed production due to crowding effects. Figures 1(a) and 2(a) demonstrate to what extent this assumption of constant growth rates is realistic for the case of S. media in a winter wheat–sugar beet crop. The experimental data conform approximately to the linear fits, indicating that growth rates are about constant over the range of densities that is attained in this experiment.

The calculations of expected economic returns show that for the range of densities attained in the experiment the best decision is not to control if weed density is below a density threshold, and to control if weed density is above a density threshold. In these calculations two assumptions were made. First, the farmer can only decide to either control or leave weeds uncontrolled. In reality, a farmer will have more options for weed control, but this does not alter the density-dependent basis of the weed control decisions. Second, the farmer is supposed to maximize the expected economic returns in the long term (6 years in this case). If the farmer has a different objective, for instance maximizing economic returns over the current year, the threshold density for weed control will be higher, but weed control decisions would still be density-dependent. However, in the special case where the farmer has the objective of maximizing economic returns over a very long period, the resulting threshold density may approach zero. In this case the best weed control decision is to control always, and the weed control decision is no longer density-dependent. But, such an eradication programme will only be effective in the rare situations where import of weeds from other field is impossible; this option is not explored further in this paper.

The combination of weed population dynamics with optimum economic decisions results in a non-linear prediction model of the dynamics of a weed population that is subjected to optimum decisions (equation 2, Fig. 3a,b). Perhaps this model may seem to be a rather crude representation of the underlying population dynamic processes and the decision making in weed management. But this model provides a practical representation of the facts that weed control is the most important factor determining future weed densities, and that weed control is density-dependent. As a result of this density-dependent regulation, the density will show complex dynamics (Felsenstein 1979; Bélair & Milton 1988; Hughes & González-Andújar 1997; Wallinga & Van Oijen 1997). No study has yet addressed the question of how measurement errors of demographic variables, given these complex dynamics, will affect the predicted weed densities.

The experimental outcome shows that it is very likely that there will be errors in initial density. Table 2 shows a more than 20-fold difference in observed values for the initial weed density N(0) between experiment 1 and experiment 2. In principle, the weed density can be measured exactly for any location, but in practical situations such an exact measurement will be too labour-intensive and, as a result, the estimated weed density will have some measurement error. Equation 3 shows that an error in estimate of initial weed density results in an error in predicted weed density, and that the absolute error is preserved and does not, on average, increase or decrease.

Furthermore, the experimental outcome shows that it is also very likely that there will be an error in estimated value of growth rate. Table 2 also shows a more than 20-fold difference in relative population growth rate a between experiment 1 and 2. In practical situations it is impossible for farmers to estimate weed population growth rate since this requires leaving weeds uncontrolled. Hence, the uncertainty associated with the growth rate a is very large. Equation 4 shows that an error in the estimate of weed population growth rate results in an error in predicted density, and that the absolute error increases with increasing prediction interval. And, since density stays constant on average, the relative error will also increase with increasing prediction interval. Eventually the magnitude of the relative error will exceed unity, rendering predictions meaningless.

Equation 4 makes explicit the rate at which error in prediction increases, and on which variables the increase in error depends:

• error increases linearly with length of the prediction interval;

• the growth of error is positively related to the density that weeds achieve, on average, in the years when weeds are left uncontrolled (that is, if higher weed densities are tolerated, the prediction errors will increase more rapidly);

• the growth of error is positively related to the frequency of years in which the weeds are left uncontrolled (that is, if the weed population requires less intensive control, the prediction errors will increase more rapidly).

It follows that the prediction error increases more slowly for the most noxious weeds because these weeds can be tolerated only at very low densities, rarely allowing non-use of control measures.

Measurement errors for population growth rate a are unavoidable, and will cause errors in the predicted weed densities that will in turn cause an error in the estimated threshold density for weed control. What is the consequence of such an error in threshold density for the expected crop yields? It may be tempting to speculate that errors in predicted weed density will, by necessity, result in poor model-based weed control decisions and hence in large crop yield losses. But this is not the case. Consider, for example, a prediction made with estimates for a and b that differ as much from the actual values as the observed parameter values for experiment 1 differ from those for experiment 2; the threshold density will then be overestimated by 4·05 plants m−2. The long-term consequence of this overestimation for average crop yield can be calculated by equation 5: overestimation of the threshold as estimated for experiment 2 by 4·05 plants m−2 causes a reduction in average crop yield by 0·69 10−3 kg m−2. In relative numbers, a 95% underestimation of the population growth rate a and a 14% overestimation of the population growth rate b will result in a 60% overestimation of the threshold density for weed control, and this will cause an average annual crop yield loss that amounts to 0·08% of the maximum crop yield. This single example shows that poor model-based predictions of weed density do not necessarily lead to poor long-term control decisions.

Throughout this paper three simplifying assumptions were made. It was assumed that: (i) weed population growth rates are constant over the range of densities of interest; (ii) the weed control decision is restricted to either control or no control at all; (iii) the farmer has long-term economic objectives. The first assumption is shown to be in agreement with the data presented here; the latter two assumptions are shown not to be essential limitations. The three simplifying assumptions made it possible to demonstrate that, even for a simple prediction model, the dynamics of weed density are complex and that a measurement error in growth rate poses a limit to the prediction interval beyond which predictions of weed density are meaningless. There is no obvious reason to suppose that for more complex deterministic models the prediction of dynamics will be any easier. That is not to say that the results, as presented here, are the final word on reliability of predictions of weed density. Instead, the variation in growth rate as observed in the experiments shows that there is a good deal more to understand about weed population dynamics. This observed variation in growth rates can be included in the model by considering the model parameters a and b as stochastic variables. The present study then provides a null-hypothesis that remains to be tested by such a stochastic model: the error in predicted weed density increases linearly with length of the prediction interval.

Conclusion

For a simple, but realistic, model of population dynamics of annual weeds, it is shown that relative error in predicted weed density increases linearly with the length of the prediction interval. This sets a limit to the period over which density can be predicted. With the same model it is shown that poor model-based predictions of weed density do not necessarily lead to poor long-term control decisions.

Acknowledgements

Part of this research was funded by The Netherlands Grain Centre. The authors thank Rommie van der Weide who started the experimental work and Gareth Hughes who provided valuable comments on an earlier version of the manuscript.

Received 31 December 1997; revision received 8 February 1999

Appendices

Appendix 1

Effect of error in initial density on predicted density

The weed population is controlled when the density of weeds exceeds a density threshold. The resulting dynamics are given by:

image(eqn A1.1)

where K is the density threshold for weed control, a is the relative growth rate of the uncontrolled population (such that a > 1), b is the relative growth rate of the controlled population (such that 0 < b < 1). Previous studies have shown that weed density, after an initial transient phase, will remain within a limited range of densities around the threshold K that is given by (bK, aK] (Wallinga & Van Oijen 1997). The lower bound of this attracting range, bK, is achieved when in the previous year the weed density was just above the threshold K and weeds were controlled. The upper bound of the attracting range, aK, is achieved when in the previous year the weed density equalled the threshold K and weeds were not controlled. Felsenstein (1979) found that on a logarithmic scale the itinerary fills this range uniformly, which means in the present context that the logarithm of weed densities, ln N, is uniformly distributed over (ln bK, ln aK].

The relative frequency of years in which density does not exceed the threshold value, denoted by α, is then:

image(eqn A1.2a)

The relative frequency of years in which density exceeds the threshold value, denoted by β, is then:

image(eqn A1.2b)

The error in predicted population density N(t) due to an error in the initial density N(0) is obtained by taking the derivative of prediction density with respect to initial density. On the long-term, this derivative is:

image(eqn A1.3)

So, for large t,

image

Appendix 2

Effect of error in population growth rate on predicted density

The error in predicted population density N(t) due to error in the relative growth rate a is obtained by taking the derivative of predicted density with respect to relative population growth rate a. This derivative is:

image(eqn A2.1)

For convenience, the following notations are introduced:

image(eqn A2.2a)
image(eqn A2.2b)
image(eqn A2.2c)

In this notation, equation A2.1 becomes:

u(t) = p(t - 1) u(t - 1) + q(t - 1)

which can be expanded as

u(t) = p(t - 1) [p(t - 2) u(t - 2) + q(t - 2)] + q(t - 1)

and so on, until it finally yields

image

Since u(0) =

image

the above equation reduces to:

image(eqn A2.3)

Equation A2.3 can be decomposed into three terms:

image(eqn A2.4)

and for each of these terms the behaviour with increasing t is studied.

First, it is determined how x(t) increases with increasing t. Note that equation A1.3 rewritten in the notation of equation A2.2b means that

image

p(τ) approaches unity in the long term. Therefore, in the long term,

image

Recalling that q(t) is defined by equation A2.2c as weed density if weeds are left uncontrolled and zero if weeds are controlled, the resulting sum can be written as the product of prediction interval t, the frequency that weeds are left uncontrolled α, and the expected weed density when weeds are left uncontrolled NE:

image

Second, it is determined whether y(t) increases with increasing t, and again use is made of the fact that

image

p(τ) approaches unity on the long-term:

image(eqn A2.6)

Third, it is determined how z(t) increases with increasing t. For this purpose, z(t) is split up into two parts:

z(t) = z1(t) + z2(t)
image

where r = t − 1 − ρ(t), and ρ(t) increases with t such that when t → ∞, ρ(t) → ∞, and ρ(t)/t → 0, (these requirements are met when, for instance, ρ(t), is taken as (a log t), truncated to the nearest integer) z1(t) approaches zero because both

image

p(τ) and

image

p(τ) approach unity on the long-term. For the increase of z2(t) with t, an upper limit is derived by using the facts that q(s − 1) ≤ K,

image

p(τ) ≤ aρ(t), and

image

p(τ) Ð 0 (these restrictions follow in a straightforward manner from the definitions of q and p in equation A2.2b and equation A2.2c):

image

Writing out the last term then gives:

image(eqn A2.7)

This increase of the upper limit to z(t) with t is in the long term negligible as compared to the increase of x(t) with t. For instance, take ρ(t) = (a log t):

image

In summary, u(t) is decomposed into three terms x(t), y(t) and z(t) of which y(t) is zero and z(t) is negligible compared to x(t) when t is large:

u(t) ≈x(t).

Substituting equation A2.2a and equation A2.5 into the above equation gives:

image(eqn A2.8)

The term αNE can be expressed in terms of a, b and K, the variables that are used to describe the weed population dynamics. If the weed population follows the dynamics of equation 2, the logarithm of weed density is distributed uniformly over the interval (ln bK, ln K] (see Felsenstein 1979 and Bélair & Milton 1988 for a derivation of this result in a different context). The probability density function for weed densities that do not exceed the threshold value K is then

image

. The expected weed density, given that the density does not exceed density threshold K, is

image

. The frequency of years in which density does not exceed the threshold value K is given by equation A1.2:

image

Combining the expressions for NE and α gives:

image(eqn A2.9)

Appendix 3

Effect of an error in control decisions on crop yield

The expected weed density, given that density does not exceed the threshold, is denoted by NE. The expected crop yield, given that density does not exceed the threshold is denoted by YE:

YE = Ymax - cNE.(eqn A3.1)

The expected yield, given that density exceeds the threshold, is the maximum yield Ymax. Furthermore, α is the frequency of years in which density does not exceed the threshold. It follows then that the average yield over all years, denoted by YA, is given by:

YA = Ymax - cαNE.(eqn A3.2)

Substitution of equation A2.9 into equation A3.2 gives:

image(eqn A3.3)

Taking the derivative with respect to K gives:

image(eqn A3.4)

Ancillary