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Abstract

It is 10 years since Mortimer, Sutton & Gould (1989) reviewed the use of models to describe the population dynamics of weeds, and the utility of such modelling in the design of weed control programmes in annual crops. Their motivation was clearly stated: ‘A primary and strategic aim of weed ecology is to be able to explain and ultimately predict which weed species may become abundant and moreover the levels of abundance they may achieve under particular management practices.’ In the decade since this was written, have we moved closer to being able to explain and predict the population dynamics of weeds? Mortimer et al. (1989) approached weed population dynamics via ‘vertical’ studies of weed population dynamics, in which mathematical models of density-dependent responses are used to predict long-term trends in population density. Thus, as a starting point, Mortimer et al. (1989) adopted the underlying relationship to describe the future weed population Nt+1 in terms of the population in the previous generation Nt and a non-linear, density-dependent function F(Nt). Typically, F(Nt) is characterized from short-term experimentation, as there are few long-term ‘horizontal’ data sets (time series) from which information about weed population dynamics can be extracted. Simple deterministic models like equation 1 may give rise to complex (chaotic and cyclical) dynamics (May & Oster 1976). Thus, we are immediately faced with a problem. 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? In weed population biology, the way round this difficulty has often been to adopt a particular form of F(Nt), and estimate its parameters from experimental data – typically in the form of a graphical plot of Nt+1 against Nt. If these estimates fall clearly in the zone of parameter space for which the model dynamics are known to be non-chaotic, the dynamics of the population are regarded as non-chaotic. On the basis of this type of analysis, conducted for a large number of data sets, it may appear that the qualitative dynamics of weed populations are, essentially, such that complex dynamics will not occur in real (as opposed to model) systems. This is one way to answer the question of the utility of simple models with (possibly) complex dynamics. Thus, Cousens & Mortimer (1995) state: ‘It would therefore appear that complex behaviour such as chaos is likely to be more of a mathematical property of our models than a behaviour to be expected of real populations of annual plants.’Cousens's (1995) review of vertical studies of weed population dynamics is in emphatic agreement, noting also that incorporation of additional model parameters describing seed mortality and seed dormancy virtually rules out the possibility of oscillatory dynamics. Under the scheme of such vertical studies, an understanding of the dynamics of weed populations ‘in the real world’ depends on the characteristics of intrinsically regulatory factors and a knowledge of their interaction with various exogenous factors (Cousens & Mortimer 1995). F(Nt) describes an intrinsic regulatory signal, and the first task of the modeller is, essentially, to extract this signal from the obscuring ‘noise’. Cousens's (1995) review tells us that this is easier said than done. Population data from ‘the real world’ rarely – if ever – lie exactly on some smooth density-dependent function of N. Among the difficulties then faced (see e.g. Nisbet, Blyth & Gurney 1989; Morris 1990) are that one particular model has been chosen when different models may fit the same data equally well yet make different predictions about dynamics, and that simple models may fail to capture observed population dynamics. A recent example is provided by an elegant experimental study of the dynamics of the annual greenhouse weed Cardamine pensylvanica in controlled-environment growth chambers (Crone & Taylor 1996). The experiment produced a long (by the standards of plant population ecology) time-series data set, extending over 15 generations. All the experimental populations exhibited complex dynamics over the period of the experiment. A number of population models were fitted to the time-series data. In addition to models of the form of equation 1, models of the form were fitted. All the models explained significant amounts of variation, but only models that incorporated two-generation lagged density dependence (equation 2) reproduced the qualitative (cyclical) dynamics of the experimental populations. Models incorporating only the one-generation lag predicted stable dynamics. In further work on the same theme, Crone (1997a,b) went on to investigate the role of parental density effects (i.e. density dependence relating Nt+1 to Nt−1) in single-species populations and in interacting populations. Modelling studies indicated (as might be expected) that the inclusion of parental density effects decreased the range of parameter values for which a stable equilibrium population was the predicted outcome. Experimental studies (again with Cardamine pensylvanica) showed that, although weaker than offspring density effects (i.e. density dependence relating Nt+1 to Nt), parental density effects were large enough to change the predicted dynamics of populations. Another recent discussion of complex population dynamics has an agricultural context (Wallinga & van Oijen 1997). For their general model, Mortimer et al. (1989) adopted the widely studied form F(N) = R(1 + aN)−b, in which R is the asymptotic per capita rate of increase at low population density, and a and b are parameters describing the form and intensity of density dependence. However, in agriculture, weed populations may be controlled, and so not be left to follow dynamics depending only on their endogenous demographic parameters and Mother Nature's array of environmental effects. To model the effects of (density-independent) weed control, Mortimer et al. (1989) modified their version of equation 1 to include a control parameter Λ = ρR, in which ρ represents the proportional reduction of the weed population, to give: Now, suppose that the requirements of agricultural production meant that the weed population was never allowed to build up to a level where density dependence [as described here by the (1 + aNt)−b term] seriously came into play and, further, that weed control was deemed unnecessary at low weed population densities. If K is a threshold density (K > 0) above which – on economic grounds – it has been decided that weed control is worthwhile, then, when N > K, and, when N ≤ K, This model was investigated by Wallinga & van Oijen (1997). They showed that the discontinuity imposed by the adoption of a discrete choice threshold for weed control gave rise to complex dynamics. Such a finding gives credence to the suggestion (Berryman & Millstein 1989) that chaotic dynamics may arise from human interventions (sometimes in pursuit of ‘control’) in agricultural systems. Weed population dynamics is not the only area of crop protection where such concerns arise. For example, Shaw (1994) discussed the implications of complex dynamics (induced by seasonal forcing) in models of plant disease, and Cavalieri & Kocak (1995) and Gonzalez-Andujar & Perry (1995) discussed the implications of complex dynamics for the biocontrol of the insect pest Ostrinia nubialis. Overall, the problem for population ecology was accurately summarized by Renshaw (1994). Failure to accept the properties of non-linear models of the form of equation 1– particularly where there is significant lagged density dependence, or where human intervention or seasonality causes discontinuities – may be misleading in a forecasting scenario. As mentioned earlier, the paucity of horizontal studies of weed population dynamics is attributable largely to the lack of appropriate time-series data. This lack, in turn, probably reflects a view of methodological difficulties voiced (doubtless on behalf of many others) by Cousens (1995), who asked: ‘How do we … distinguish in field data between environmentally driven fluctuations of an otherwise asymptotic behaviour and pure chaos driven by density dependence? We are back to the problem of interpreting horizontal density studies and away from the supposed advantages of vertical studies.’ However, Ellner & Turchin (1995) have argued that the separation of chaotic and stochastic dynamics in ecological systems, apart from being methodologically intractable, is unnecessary. The methods outlined by Ellner & Turchin (1995), based on statistical theory for parameter estimation in non-linear time-series models, characterize both the dynamic feedbacks regulating population behaviour and the ‘dynamic noise’ affecting how state variables change over time. Under this scheme, a combination of vertical and horizontal studies is required. For an entomological example, Cushing et al. (1998) combined models, experiments and statistical analysis of data to build up a predictive model. They developed a mathematical model that was impressively successful in describing and predicting the population dynamics of Tribolium castaneaum populations. Their results show convincingly that certain non-linear phenomena (complex dynamics) may occur. Their study demonstrates that mathematical models – even ‘simple’ mathematical models – are capable of providing accurate descriptions, explanations and predictions for the dynamics of biological populations. Weed management depends on a knowledge of weed biology (Bhowmik 1997), and models provide a basis for integrating this knowledge and studying qualitative scenarios (Gonzalez-Andujar & Fernandez-Quintanilla 1991). If we are to ‘demonstrate what dynamics actually occurs in nature’ (Cousens 1995) and apply the results in predictive weed management, we believe that the integration of vertical and horizontal approaches, as demonstrated by Cushing et al. (1998), represents a useful way forward.