### Introduction

- Top of page
- Summary
- Introduction
- Lack of overcompensating yield–density responses
- Stability in the long-term dynamics of weeds
- Threshold management
- Can we predict weed population dynamics?
- References

The most basic framework for understanding the dynamics of biological populations recognizes that changes in population numbers are the result of two types of process. On the one hand there is the deterministic component of population dynamics that results from interactions between individuals and other predictable ecological processes. On the other, there is the stochastic component of population dynamics that results from random variations in birth and death rates, for example owing to the direct effects of weather or disturbance. This latter component tends to make weed population dynamics more unpredictable. It is of course fundamental to understanding the dynamics of any population to determine the relative roles of these two forms of process in determining year-to-year variations in population numbers.

One of the most important developments in ecology during the 1970s was the recognition that, even in the absence of stochastic variations, entirely deterministic systems are capable of producing patterns of population change that are apparently indistinguishable from random noise (May & Oster 1976). This form of dynamics results when density dependence within populations is overcompensating and the growth of populations from low densities is high. At high densities, population growth is disproportionately reduced by increasing density, so that population numbers are reduced to low levels. From this low level, growth is then rapid, and high levels are soon reached. Consequently populations fluctuate around a long-term average, and in the extreme these patterns of fluctuation may be entirely unpredictable. The existence of such dynamics could thwart attempts to predict population dynamics from one year to the next.

Several articles recently have explored the potential role of chaotic dynamics in determining the numbers and dynamics of weed populations (Gonzalez-Andujar 1996; Wallinga & Van Oijen 1997; Wallinga *et al*. 1999; Gonzalez-Andujar & Hughes 2000). One recent review has concluded that we are in a poor position to predict weed population numbers (Gonzalez-Andujar & Hughes 2000). Specifically, they asked: ‘if the models we use to describe weed population dynamics can give rise to complex, possibly chaotic dynamic behaviour, will they ever be useful in predicting levels of weed abundance at the field scale?’ We believe that in emphasizing a potential role for chaotic dynamics, the review by Gonzalez-Andujar & Hughes (2000) is misleading. In this paper we wish to promote an alternative and more optimistic point of view. Specifically we make three points: (i) chaos is unlikely in weed populations, except resulting from some highly specialized forms of population management; (ii) temporal variability is more likely to compromise the numerical accuracy of models; (iii) the aim of weed modelling is generally not to predict weed numbers accurately at very local scales, but rather to aid in the development of strategic control programmes, an activity that is unlikely to be compromised by the kind of population behaviour that Gonzalez-Andujar & Hughes (2000) highlight as being important.

### Lack of overcompensating yield–density responses

- Top of page
- Summary
- Introduction
- Lack of overcompensating yield–density responses
- Stability in the long-term dynamics of weeds
- Threshold management
- Can we predict weed population dynamics?
- References

The occurrence of chaotic dynamics requires that some component of the life cycle shows a density-dependent response that is overcompensating, i.e. when the proportional decline in parameter value with increasing density exceeds the change in density. This is a necessary but not sufficient condition and has to be accompanied by a high finite rate of population growth from low densities in order for non-linear population dynamics to result. For instance, consider the familiar hyperbolic logistic model for population growth (Hassell 1975; Watkinson 1980):

*N*_{t +1 } = λ *N*_{t} (1 + *aN*_{t} ) ^{−b}(eqn 1)

In this equation the number of individuals, *N*, at time *t*+ 1 is related to those at time *t* by the finite rate of population increase (λ) and density-dependent feedback parameters *a* and *b*. In equation 1 the finite rate of increase and the parameter *b* contribute to model stability. For populations to exhibit unstable dynamics in this model the condition *b*(1 −λ^{−1/b}) > 2 must hold (Hassell 1975). That is, populations need to exhibit not only overcompensating competition-density responses (i.e. *b* > 1), but also large finite rates of population increase (i.e. λ >> 1).

The most common source of density dependence in annual plant populations that has been demonstrated to date appears to be through competition between plants for resources; this results in a decrease in mean performance with increasing density (Watkinson 1980, 1996). In populations where this is the case, chaos is unlikely for three reasons (for more extensive discussion of these issues see Rees & Crawley 1989, 1991; Silvertown 1991): (i) the law of constant final yield (Kira, Ogawa & Sakazaki 1953) implies that density dependence is perfectly compensating in many populations as a consequence of the modular construction of plants (Watkinson 1980); (ii) asymmetric competition results in only the smallest individuals in a population experiencing the most intense effects of competition (Weiner 1988) and, at high densities, results in self-thinning, further promoting stability (Watkinson 1980); (iii) the lack of size thresholds for reproduction in many plants implies that even the smallest individuals will produce some seed when the effects of competition are intense, and that ‘collapsing’ yield–density responses are consequently unlikely (Rees & Crawley 1989, 1991). It should also be noted that weed populations rarely approach the kinds of densities where such mechanisms operate because control (e.g. through herbicide application) maintains populations at low levels.

The most commonly cited example of an annual plant showing higher-order dynamics as a consequence of overcompensating density-dependent reductions in seed production is the population of *Erophila verna* studied by Symonides (Symonides, Silvertown & Andreasen 1986). However, such dynamics were found only under a restricted set of germination rates and in very small areas on a single dune. Other studies have shown the dynamics of this species to be stable (Rees, Grubb & Kelly 1996).

The most commonly cited data for an annual weed indicating that higher-order dynamics may be possible in fact implicate a size threshold for reproduction as the underlying mechanism (Thrall, Pacala & Silander 1989). Based on the observation by Pacala (1986) that individuals of *Abutilon theophrasti* smaller than 0·41 g were incapable of setting seed, Thrall, Pacala & Silander (1989) tested whether this was likely to lead to oscillatory dynamics under field conditions. Bazzaz *et al*. (1992) also studied the potential impacts of varying CO_{2} levels on the stability of population dynamics in this species using glasshouse experiments. Although models and yield–density relationships derived from glasshouse experiments predicted that oscillatory dynamics were possible, field populations did not show evidence for such dynamics (Thrall, Pacala & Silander 1989). This was because the species possesses a long-lived pool of dormant seeds (Thrall, Pacala & Silander 1989) that tends to stabilize dynamics, confirming the earlier predictions of MacDonald & Watkinson (1981). Because many weeds possess a seed bank (Thompson, Bakker & Bekker 1997), this component of the life history alone would appear to preclude chaos in many weed species. More recently, Buckley *et al*. (2001) have shown in the annual/biennial weed *Tripleurospermum perforatum* that increasing frequency of delayed flowering at high densities may lead to overcompensating yield–density responses, but that under field conditions overcompensating population growth-density responses are unlikely owing to the stabilizing effects of density-dependent mortality and the presence of a seed bank.

It is also worth noting that all the yield–density relationships in the studies described above were derived from sown rather than naturally regenerating populations. Under such conditions the role of asymmetric competition will be minimized and it has been noted that competition–density functions may be rather different in sown compared with naturally regenerating stands (Freckleton *et al*. 2000; Buckley *et al*. 2001).

If chaos resulting from intrinsic sources of population instability is to occur in plant populations, then this must result from overcompensating processes at stages of the life cycle other than the period of growth and competition. Tilman & Wedin (1991) found that increasing nutrient supply could lead to high rates of litter deposition, which, it was argued, might lead to chaotic population dynamics. Similarly, Thompson (1994) and Crone & Taylor (1996) have suggested that increasing litter deposition could result in complex population dynamics of the annuals *Arabidopsis thaliana* and *Cardamine pensylvanica*, respectively. The critical factor generating complex dynamics in these cases is the effects of the presence of parents on seedling recruitment, i.e. the density dependence is lagged (Crone 1997). In arable weeds, however, all litter is removed between generations. Only in pasture systems, where the component species are allowed to regenerate from year to year, would such a mechanism be likely (for a contrast of weed dynamics in arable and pasture systems see Watkinson, Freckleton & Dowling 2000a). However, high levels of litter cover would prevent the emergence of any species and hence such a situation would be economically non-viable. Furthermore, the presence of large amounts of litter is unlikely in grazed systems.

Other stages of the life cycle have also provided evidence for density dependence within the course of a single growing season. In the arable weed *Anisantha sterilis*, for example, it was found that the proportion of seeds emerging from the seed bank declined as a function of density (Lintell Smith *et al*. 1999). In this example, however, the response was exactly compensating and hence incapable of generating chaotic dynamics. The reason for this may be general: after a large enough number of plants have recruited, ground cover is complete and light levels (or red : far-red ratios) are reduced to the point where further seed germination is not possible (Fenner 1985). This form of recruitment thus sets a fixed upper limit to population size and has an extremely stabilizing effect on population dynamics. Notably, the maximal finite rate of increase of the population of *Anisatha sterilis* described by Lintell Smith *et al*. (1999) was greater than 100 individuals individual^{−1} year^{−1} and hence should have been capable of revealing any source of intrinsic instability. Numbers of mature plants were, however, remarkably stable from one year to the next.

In summary, the asymmetric nature of interactions between individual plants, the general lack of reproductive thresholds and the possession of a long-lived seed bank by many species would seem to make it unlikely that weeds, in common with most plant species, would exhibit non-linear dynamics as a consequence of intrinsic sources of overcompensating density dependence (Watkinson 1980; Rees & Crawley 1989, 1991). Moreover, many weed populations (particularly arable weeds, although this is not always the case for pasture weeds) are subject to continuous control. Hence the finite rate of increase is reduced to low levels, perhaps not even greater than unity in most years, and higher-order dynamics are unlikely irrespective of the form of the yield–density response at high densities.

### Stability in the long-term dynamics of weeds

- Top of page
- Summary
- Introduction
- Lack of overcompensating yield–density responses
- Stability in the long-term dynamics of weeds
- Threshold management
- Can we predict weed population dynamics?
- References

In approaching the study of the long-term dynamics of weed populations it is clearly necessary to distinguish between stability and variability. Stability relates to the tendency of populations to return to an equilibrium following a perturbation. Variability may result from the impacts of extrinsic factors such as weather and control.

The potential long-term stability of weed populations has been evidenced by a number of studies. Long-term studies, for example, have shown evidence for the stability of numbers of plants in a spatial context, with small high-density patches persisting over a number of years (Wilson & Brain 1991; El Titi 1991; Wilson & Lawson 1992; Clark, Perry & Marshall 1996). There is therefore strong evidence that weed populations show spatiotemporal stability. Apart from the potential for oscillatory dynamics in *Abutilon theophrasti* monocultures, the only potential evidence for chaotic dynamics in a weed is for the annual weed *Cardamine pensylvanica*, resulting from the impacts of litter on recruitment in glasshouse-grown populations (Crone & Taylor 1996; Crone 1997). There are no field data that show chaotic or other higher-order dynamics in annual weeds.

To be able to demonstrate the existence of unstable population dynamics, long-term data are required. A mid-term (4-year) study on uncontrolled populations of *Anisantha sterilis*, *Galium aparine*, and *Papaver rhoeas*, in which populations were allowed to grow to very high densities, failed to yield evidence for anything other than a steady approach to a constant equilibrium (Watkinson *et al*. 1993). Longer term data are surprisingly rare, however.

#### ANALYSIS OF PUBLISHED DATA ON THE BROADBALK EXPERIMENT

One such source of information that has been used to explore the variability in numbers of weed populations is the Broadbalk experiment (Firbank 1991, 1993). Figure 1 shows an example of the dynamics of 12 species of common arable weeds for a 12-year run of published estimates of population sizes (Thurston 1968). The dynamics of these species show a wide range of behaviours, including apparent increases (e.g. *Poa trivialis*, *Equisetum arvense*), decreases (e.g. *Ranunculus arvensis*, *Cirsium arvense*) and quasi-cyclic behaviour (*Medicago lupulina*).

In a tentative attempt to determine whether any of the variations in numbers in Fig. 1 could be attributed to non-linear dynamics, we fitted equation 1 to these data. The model was fitted using a non-linear fitting procedure. The data were transformed to rates of population change, and the best-fit parameter values were estimated assuming an exponential distribution of errors, and a Rosenbrock pattern search procedure (Rosenbrock 1960). Because the number of observations was small, we were unable to obtain directly standard errors for the model parameters. However, as *N* in equation 1 becomes large, equation 1 can be logarithmically transformed and approximated by:

- log(
*N*_{t+1}/*N*_{t}) ≈ log(λ) − log(*a*) −*b* log(*N*_{t})(eqn 2)

The slope of the linear regression of log population growth rate on population density (we term this β to distinguish it from *b*) can thus be used to provide an approximate estimate of the standard error of *b*.

When analysing short time-series of this sort, the estimates of *b* and β are biased for small values and tend to overestimate the effects of density dependence. We therefore conducted a further test on these data in order to determine whether population dynamics showed evidence for any form of density dependence. We used the randomization test of Pollard, Lakhani & Rothery (1987) in order to determine whether the data deviated from a Brownian motion random walk. This test randomizes the data and uses the distribution of the correlation between log population change and log population size for the randomized data to estimate the probability that the observed data show a correlation stronger than expected under Brownian motion. This test rejects Brownian motion if the probability of obtaining the observed correlation in the randomized data sets is smaller than the nominal rejection level. It should be noted that this test is powerful even for small data sets, such as the ones analysed here.

The results of this analysis are summarized in Table 1, as well as in Fig. 2a, where the values of λ (the maximal mean population growth rate in the absence of density dependence) and *b* (the density-dependent parameter) are plotted together. Superimposed are the stability boundaries for equation 1 (Hassell 1975). Only three species (*Tripleurospermum maritimum*, *Papaver* spp. and *Medicago lupulina*) showed evidence for a form of population dynamics other than a straightforward damped approach to an equilibrium level. In each of these three cases, dynamics were modelled as an oscillatory approach to equilibrium. Of these three species, however, two (*Tripleurospermum maritimum* and *Papaver* spp.) were very close to the boundary for exponential damping. No species yielded parameter estimates consistent with limit cycles or chaotic behaviour. The values of β estimated from fitting equation 2, the linear regression of change in log population size on population size, were very similar to the estimates of *b* from the non-linear model (Table 1). The approximate standard errors for *b* therefore strongly suggest that the observed values are considerably lower than would be required to generate chaotic dynamics. Finally, the tests for density dependence indicated that density dependence was strong in only three of the 12 populations. Therefore nine out of the 12 exhibited a pattern of dynamics more consistent with a random walk, which argues strongly against the general existence of chaotic dynamics in these data.

Table 1. Analysis of data from the Broadbalk experiment ( Fig. 1 ) for evidence of chaotic or regulatory behaviour. Equation 1 was fitted to the data from Fig. 1 using a non-linear modelling procedure (see legend to Fig. 2 for details). λ and *b* are the best-fit parameters of the model that determine stability, and the overall *R*^{2} for the model is given. Owing to the small number of observations it was not possible to get standard errors directly for the parameters of the model and we used a simplification of the model to estimate the confidence intervals for β, the slope of the relationship between log rate of population change and log density (see text for details). The standard error for this slope is presented, and the superscript is the statistical significance. Finally, because such tests may be biased for small samples, we used a randomization test for density dependence (Pollard's test; Pollard, Lakhani & Rothery 1987 ) in order to determine whether data exhibited significant evidence of density dependence. The test statistic is the probability of obtaining a correlation between the rate of population change and population density in 10 000 randomizations of the observed data. Small values of these probabilities indicate significant density dependence, while values greater than 0·05 indicate that time-series are not statistically distinguishable from a Brownian process with no density dependence Species | λ | *b* | β | ± SE | *R*^{2} | Pollard's test |
---|

*Ranunculus arvensis* | 3·91 | 0·83 | 0·69 | 0·31^{0·05} | 0·33 | 0·21 |

*Vicia sativa* | 4·67 | 0·78 | 0·46 | 0·27^{0·13} | 0·27 | 0·21 |

*Medicago lupulina* | 65·77 | 1·62 | 1·56 | 0·27^{< 0·001} | 0·79 | 0·00 |

*Papaver* spp *.* | 10·32 | 1·36 | 1·05 | 0·32^{0·01} | 0·52 | 0·05 |

*Tripleurospermum maritimum* | 12·00 | 1·34 | 1·06 | 0·24^{< 0·001} | 0·65 | 0·04 |

*Alopecurus myosuroides* | 1·66 | 0·18 | 0·40 | 0·25^{0·15} | 0·20 | 0·63 |

*Poa annua* | 0·99 | 0·54 | 0·22 | 0·31^{0·50} | 0·09 | 0·10 |

*Cirsium arvense* | 2·46 | 0·24 | 0·19 | 0·14^{0·20} | 0·16 | 0·59 |

*Tussilago farfara* | 12·98 | 0·90 | 0·97 | 0·32^{0·01} | 0·49 | 0·78 |

*Equisetum arvense* | 1·41 | 0·26 | 0·13 | 0·21^{0·55} | 0·04 | 0·94 |

*Poa trivialis* | 4·56 | 0·52 | 0·76 | 0·37^{0·08} | 0·34 | 0·24 |

*Agrostis stolonifera* | 3·91 | 0·83 | 0·66 | 0·21^{0·13} | 0·52 | 0·28 |

We emphasize that this modelling exercise has to be considered as tentative, largely because the census error in the estimates of population size cannot be accounted for and may impact on estimates of the strength of density dependence (Shenk, White & Burnham 1998). Also we take no account here of the fact that this group of weeds includes a range of perennials (these in many ways behave as annuals in this system, although regeneration can occur through either seeds or rhizome fragments) and that a number of the annuals possess a persistent seed bank (the presence of the seed bank will tend to stabilize dynamics further, although equation 1 may provide a good approximation to population dynamics in many cases, cf. the bottleneck model of MacDonald & Watkinson 1981). Moreover the time-series are too short to look for evidence of lagged density dependence. Despite these caveats, the parameter estimates are so far from the range required for higher-order dynamics, we suspect that the current estimates are robust.

Published estimates of λ and *b* from a number of short-term studies also indicate that the dynamics of weeds along with other annual plants are likely to be relatively stable (Fig. 2b). We compared the estimates of λ and *b* shown in Fig. 2a with estimates taken from a range of sources in the literature (see figure legend for details). These included data on both weedy and non-weedy species. It is clear in Fig. 2b that the range of values fitted in Fig. 2a are entirely in line with previously published estimates, indicating that there is little evidence for unstable dynamics in these species.

To summarize this section, we do not find it surprising that some weed species show great variability in numbers across a number of years. Examples of such variability are clear in Fig. 1. If such variations were driven by chaos or non-linear dynamics, then this would be an important and significant factor in trying to explain year-to-year variations in population sizes. On the other hand, for most species it seems more likely that such variations are exogenously driven, and we should therefore seek extrinsic determinants of variability in weed population dynamics.

### Threshold management

- Top of page
- Summary
- Introduction
- Lack of overcompensating yield–density responses
- Stability in the long-term dynamics of weeds
- Threshold management
- Can we predict weed population dynamics?
- References

More recently, it has been proposed that low-density populations may exhibit a form of unstable dynamics under certain forms of threshold management (Wallinga & van Oijen 1997). Although described as ‘chaos’, the dynamics of populations in these models are rather more mundane than they may appear and do not represent chaos in the same way that the mechanisms discussed above yield chaos as a result of constant intrinsic functions relating population growth to density. Quite simply what happens is that a threshold is set, below which control is not applied, otherwise populations are controlled every year. Starting from a high density, control is applied each year until numbers are driven below the threshold. At this point control ceases and the population expands rapidly, exceeds the threshold and the cycle is completed, with control again reimposed.

Although the dynamics of this model may be shown mathematically to be chaotic (Hofbauer & Sigmund 1998), the dynamics of the system may more profitably be argued to be exogenously driven with variability resulting mainly from shifts between two different phases of population growth. Importantly, dynamics are highly predictable in this model. If, for example, control ceases and the current density of weeds is known, then given the finite rate of population increase it is trivial to calculate the time until control has to be reimposed. More complex predictions are also possible, but the statistical properties of these systems (e.g. the uncertainty in model predictions) can be accurately characterized (Wallinga *et al*. 1999) and the average effects of broad-scale changes in control practices may be predicted (Watkinson, Freckleton & Dowling 2000a). The chaotic dynamics of the threshold models is thus very different in nature from that of models that are continuous functions of density.

### Can we predict weed population dynamics?

- Top of page
- Summary
- Introduction
- Lack of overcompensating yield–density responses
- Stability in the long-term dynamics of weeds
- Threshold management
- Can we predict weed population dynamics?
- References

There is considerable interest in being able to predict the dynamics of weed populations (Grundy, Mead & Burston 1999; Collingham *et al*. 2000; Kleijn & Verbeek 2000; Watkinson *et al*. 2000c; Grigulis *et al*. 2001). For many species long-term variability may be extreme without being chaotic (Fig. 1). Evidence for the role and nature of such variability may be hard to obtain, however, and there is a clear need for more long-term time-series on weed systems (Cousens 1995). In the absence of such data, however, it is possible, by combining the statistical analysis of field data with detailed analysis of existing literature within a population modelling framework, to try to explore the potential factors impacting on the variability of annual weed populations (Jones & Medd 1997; Freckleton & Watkinson 1998a,b; Lintell Smith *et al*. 1999; Watkinson, Freckleton & Dowling 2000a;Watkinson *et al*. 2000c). Using this approach it has been possible to address two kinds of problem. First, sensitivity analysis of fitted models can be used to determine which parts of the life cycle are most important in determining patterns of population change. For example, in this way it has been shown that temporal variability in populations of *Chenpodium album* is likely to be driven primarily by variation in seed survival and emergence between years (Freckleton & Watkinson 1998b). Similarly, this approach has been used to explore differential sensitivities of the various components of the life cycle to control during the invasion and equilibrium stages of population growth in *Avena fatua* (Watkinson, Freckleton & Dowling 2000a). Secondly, the approach may be used to analyse the impacts of broad-scale changes in management, such as changed cropping patterns (Jones & Medd 1997), cultivation (Lintell Smith *et al*. 1999) and the introduction of genetically modified crops (Watkinson *et al*. 2000c). For many species, understanding the responses to broad-scale patterns of population management holds the key to predicting population dynamics, while the role of year-to-year variations may be minimal. For example, in *Anisantha sterilis* changes in the form of cultivation determine whether the species is able to persist or not and year-to-year variations are of less importance (Lintell Smith *et al*. 1999).

In these kinds of application it is important to note that models may be used to predict the impacts of changing management, even when the dynamics are of the ‘chaotic’ form highlighted by Gonzalez-Andujar & Hughes (2000). Watkinson *et al*. (2000c), for example, used a threshold form of model to predict the effects of the introduction of genetically modified crops on the long-term abundance of *Chenopodium album* in which dynamics are effectively chaotic, as described above. Because the model considered average abundance within fields over a long time period, the details of year-to-year fluctuations did not affect model predictions.

Importantly, this latter example highlights the likely irrelevance of this form of chaos for many of the common applications of weed population models. Models are not used on a field-by-field basis to predict weed infestations. Rather the strength of weed population models is in a more strategic implementation. Examples include explaining a nation-wide decline in a previously common weed (Firbank & Watkinson 1986); predicting strategy sets that provide economic control (Cousens *et al*. 1986; Doyle, Cousens & Moss 1986; Watkinson, Freckleton & Dowling 2000a); contrasting the impacts of broad-scale changes in farm practice for the dynamics of taxonomically similar weeds (Lintell Smith *et al*. 1999); predicting the impacts of new technology on the species that feed on weed seeds (Watkinson *et al*. 2000c). Weed population models can thus be used to address a range of questions, and the issue of stability is rendered unimportant or averaged over. We disagree with the conclusion of Gonzalez-Andujar & Hughes (2000) that the state of the art is poor and compromised by questions of population stability. The range of applications outlined here indicates that weed population modelling is in an extremely buoyant state and the issue of stability is at best of marginal interest.