The (complete) modelling process can be divided in several phases: Formulation, Implementation, Verification, Calibration, Analysis and Evaluation (Haefner, 2005). This section deals with the most fundamental tenet of model formulation, i.e. to what kind of detail the system should be represented in ECOSEDYN in order to provide a reasonable tool by which the effect of various weed management scenarios on the long-term weed seedbank can be compared.
2.4.1
Traditional modelling approaches
The first efforts to project weed population development over time were made by Sagar and Mortimer (1976) in the form of demographic models. The basic structure of the model consists of the lifecycle of an annual plant, usually divided into four states or stages; seeds in the soil (seed bank), seedlings, mature plants, seeds on parent plant. Demographic processes (germination, reproduction, survival, death) are expressed as transition and mortality rates (usually % per year). Cousens and Mortimer (1995) classify these models as multi-stage single cohort models and since 1976, many studies have followed this approach (Vidotto et al., 2001; Gonzalez Andujar and Fernandez Quintanilla, 2004; Puricelli et al., 2007). The values for transition rates (e.g. germination) and mortality rates in these models are obtained through conducting field experiments.
Similarly, projection matrices (van Groenendael et al., 1988; Caswell, 2001) have been used in identifying points in the life cycle that are of particular interest for designing intervention strategies (Parker, 2000; Davis et al., 2003; Westerman et al., 2007). Alternatively, these matrix models were used to forecast the development of (mostly perennial) weed populations over time given different cropping systems (Pino
et al., 1998; Davis et al., 2004b) or weed management options. The transition rates between the distinguished life stages consist of the multiplication of so-called lower- level parameters.
It is essential to appreciate that in the case of difference equations and projection matrix models, the ‘rates’ constitute the combined effect of all agronomic and biological factors with their interactions. Large inter-annual differences of demographic rates have been reported (Reader, 1985; Bierzychudek, 1999; Fernandez
patterns. The current increase in extreme weather events only increases that probability. Another factor that compromises the use of ‘rate-based’ models is that certain ‘vital rates’ are not independent from each other (van Tienderen, 1995; Ramula and Lehtila, 2005). Hence, correlated rates ought to be assessed simultaneously and over a number of years.
By replacing the ‘rates’ with components that represent the mechanisms, including the effect of the weather if relevant, these two disadvantages can be avoided (Colbach and Debaeke, 1998). The construction of population dynamic models by integrating a range of component models, each representing the best available empirical or mechanistic models is a recent trend (Rasmussen and Holst, 2003; Colbach et al., 2006) that looks very promising, both from a research as well as an agronomic point of view.
2.4.2
Defining appropriate level of complexity
Models never attain the complexity of the ‘real’ system and one of the most important choices to be made regards the way and the detail in which various aspects of the system are mathematically represented in ECOSEDYN. For example, should processes be represented by empirical or mechanistic models, should ECOSEDYN include spatial aspects and if so in which processes (component models) should it be incorporated and at what level of detail? These characteristics affect the way the component models operate and interact with each other.
To answer these questions, the objectives of a study, the understanding of the system and the data availability have to be considered. To come to a sound decision regarding the complexity warranted in ECOSEDYN, the relevant literature was thoroughly scrutinised and is summarised in the following sections.
2.4.2.1 Process abstraction
The hierarchical level at which the processes of the ‘system’ are represented mathematically is one of the key factors to distinguish between models. On the one hand a group of models exist that include little or none of the mechanisms responsible for a particular behaviour but merely consist of statistical models fitted to experimentally derived data.These are usually referred to as descriptive (Penning de Vries et al., 1989), empirical or phenomenological models (France and Thornley, 1984; Haefner, 2005). In the second group of models – referred to as explanatory,
mechanistic or process-oriented models - phenomena are separated into individual processes that are then represented quantitatively. Similarly to empirical models, the parameters of the equations are derived from experiments. For the sake of clarity, this thesis will use the terminology of empirical vs. mechanistic models. In practice a large group of models are somewhere in between these two ends (Russell, 1996). Azam-Ali (1994) referred to this group as semi-empirical or ‘index’ models. In these models, an index value with a clear biological meaning is defined whose value is related to parameters that will mechanistically influence the output. The index value is then experimentally derived for a number of different situations. For example, some crop growth models use an index parameter called the harvest index (seed biomass as a proportion of total crop biomass), to simulate crop seed growth over time (e.g. Bindi
et al., 1999).
The value of an empirical model is determined by the quality of the data input. Despite data quality, an empirical model may provide a poor description in conditions other than it was developed for. Substituting an empirical model for a mechanistic model generally implies disentangling the various processes that are then each represented by mechanistic or empirical models themselves (France and Thornley, 1984). The net result is a model with increased process representation and usually more parameters. Mechanistic models that represent processes that are insufficiently understood or for which no data was available to test and calibrate have to include assumptions to cover this void. Hence, a mechanistic model is not just constrained by the quality of the data but
also by the inherent assumptions present in the model (see Figure 2-8). Recent reviews of both weed population dynamics and individual component models have argued against ‘black box’ approaches and for the distinction between
different processes (Forcella et al., 2000;
Colbach et al., 2005). To an extent this is being realized as, for example, considerable
Figure 2-8 Model features along the ‘empiricism –
emergence growth’ pathway (Vleeshouwers and Kropff, 2000). On the other hand, Colbach et al. (2005) argued for realism in deciding upon the level of complexity in models:
“Biological aspects and environmental effects should only be specified if they interact with cropping systems”
“Model structures should not be overburdened with processes and complexities that have no immediate bearing on their use”
In general, a narrow-focussed research project on modelling seed dormancy justifies the construction of a more mechanistic model than a project seeking to evaluate the effect of weed management strategies on long-term weed population dynamics. The three guiding principles in choosing the structure of the overall framework and the mathematical representation of components were therefore: the objective of ECOSEDYN (weed management scenarios), the system to be modelled and the data, knowledge and time available. For example, given that crop sowing time was one of the components of the weed management scenarios to be tested, a model structure was needed that could deal with flexible sowing times.
2.4.2.2 Time
Models without an explicit representation of future states are usually referred to as static models whereas their counterparts are called dynamic models (Haefner, 2005). Given that the objective of this study is to build a model that is able to project the population development given certain management strategies, the factor time has to be included. Static models have to be replaced by dynamic ones, for example, the relationship between plant biomass and seed production is normally determined at crop harvest and is commonly modelled by a linear regression of plant biomass against seed weight. To incorporate time in this model requires experiments that assess the relationship between seed production and plant biomass at several times in the lifecycle of a plant (see Chapter 4).
The real question is: At what time-scale does ECOSEDYN need to simulate population dynamics? This depends on the objectives (weed management scenarios) and the time-scale over which the related regulating factors operate. Processes can be
represented at different time resolutions in ECOSEDYN only if they are unrelated. Otherwise the time-scale at the finest resolution that is required for one process will dictate the time-scale of the related processes as well. The weed management scenarios proposed focus on crop sowing time and crop cultivar (maturity time). Each of the cultural control factors has the potential to affect weed population dynamics. Studies exploring the relative time of emergence of crop and weed show that emerging a few days earlier can tip the balance in favour of either crop or weeds. It is unsure if the advantage of harvesting the crop a few days earlier can have the same effect as the crop emerging a few days earlier than the weeds. It is possible that when weed emergence is expressed on a weekly basis that the resolution of relative differences in time are being lost. The (smallest) time-scale at which ECOSEDYN operates is therefore on a daily time step.
2.4.2.3 Space
Spatially homogeneous models do not include an explicit representation of space whereas heterogeneous or spatially explicit models do, either in a discrete way such as cellular automata models (Wang et al., 2003) or in a continuous way as in diffusion equations (Haefner, 2005).
Weed and seedbank populations tend to have a patchy distribution at the field scale. This spatial heterogeneity is the result of uneven effects of biotic (humped dispersal curve), abiotic (soil properties) and management factors (cultivation, weed control) and their interactions (Blanco Moreno et al., 2004). Brain and Cousens (1990) showed mathematically that if weeds in a field had a patchy distribution, crop yield would be underestimated by assuming a random distribution, especially under high weed densities. In more aggregated weed distributions, on average each weed individual exerts more competition on surrounding weed plants and less on surrounding crop plants. However, unless weed density exceeded the economic threshold this was unlikely to have a major effect on crop yield, regardless of spatial distribution. On the other hand, Garrett and Dixon (1998) showed that for less competitive weeds, with aggregation at small scale, weed spatial pattern is important and large shifts of the weed threshold density may occur as a consequence.
Most population dynamic models have ignored the spatial distribution of weeds and their dispersal curves (Holst et al., 2007), but see (Gonzalez Andujar et al., 1999;
distribution can be simulated by dividing the field into a number of smaller areas where the density of weeds is allowed to vary according to a negative binomial distribution (Buckley et al., 2003).
Leaving spatial aspects out of a weed population-dynamics model can be justified under two assumptions. The first assumption is that ignoring the spatial distribution
within
The
the study area (more or less aggregated) is not fundamentally jeopardising the objective of the study. Fields where carrots are grown are subject to one of the most rigorous cultivation regimes present: among other operations the field is ridged and soil passes through a stone-and-clod separator (see Chapter 5). The net effect is that seeds are very unlikely to have a patchy distribution and therefore the need to account for increased intraspecific competition is reduced.
second assumption is that immigration to and emigration from the study area is negligible. The scale at which ECOSEDYN operates is the field level or lower. If substantial dispersal occurs between fields, or from the field margin to the field, possibly enhanced by different intensities of weed control, then ignoring such source- sink effects could lead to spurious predictions of population size (Perry and Gonzalez Andujar, 1993). For Bromus sterilis this assumption was invalidated (Theaker et al., 1995), as the field edges functioned as an important source of replenishment. However, Marshall (1989) found that only 30% of the common species in arable field boundaries also occurred in the crop, mostly within 2.5 meters from the edge. More specifically, field boundaries were a highly unfavourable habitat for T. inodorum
whilst S. media
There can be substantial seed dispersal between fields for wind-dispersed species (Dauer et al., 2007) and for such weeds dispersal ought to be taken into consideration in terms of the structure of the population dynamics model. Without cleaning the combine harvester before proceeding with the harvest of cereals in the next field, for some grass weeds, seed dispersal between fields is a realistic possibility (Ballaré et al., 1987; McCanny and Cavers, 1988). Apart from these two categories, inter-field dispersal of weeds has been shown to be trivial (Jones and Naylor, 1992; Theaker et
was found to some extent in the edges of the field but their prostrate growth and lack of wind-dispersal predict a short-tailed seed dispersal curve. Other studies have also concluded that the dispersal of seeds from plants in the field margins to the field is minor (Fogelfors, 1985; Blumenthal and Jordan, 2001). One of the suggested reasons for this is that the combine harvester usually harvests the crop-strip most prone to weed infestation along the field boundary in a parallel rather than a perpendicular way (Rew et al., 1996).
al., 1995; Bischoff, 2005). Neither S. media nor T. inodorum are wind-dispersed. Both have very small seeds (ca. 0.8-1.3 mm.) and therefore are unlikely to be moved great distances by a combine harvester.
In conclusion: spatial aspects only need to be included in a model to account for
inter-field dispersal of wind-dispersed species and a few other weeds that are known to be able to spread from the field margins. Many of the problematic weeds in field vegetable crops, including the two model weed species, do not belong to these categories. Therefore, spatial aspects on the horizontal plane were not included in ECOSEDYN.
2.4.2.4 Random events
Three forms of stochasticity can affect data on (weed) population dynamics (Lande et al., 2003): demographic stochasticity, extrinsic stochasticity and measurement error. Demographic stochasticity consists of random events related to births and death. It is cancelled out as being a relevant source of variation in this study because in general only small populations are sensitive to demographic stochasticity. Temperature and rainfall are extremely important in regulating the emergence patterns of weeds but are accounted for as driving variables.
Stochastic models are more complex than their deterministic counterparts and from a parsimony principle “there is little point in complicating a model just for the sake of it” (Cousens and Mortimer, 1995). The value of using a stochastic model over a deterministic model depends on how much information is available, on the shape of the probability distribution function of the parameter(s) in question (Cousens and Mortimer, 1995). In deciding on whether to make parameters stochastic a number of aspects need to be taken into consideration:
Objectives of the project; the use of a stochastic model is warranted over a deterministic model if the project objectives require an estimate of the variability of the system (Grant et al., 2000) or the time to extinction or eradication under a given management regime (Lande et al., 2003). Ignoring stochasticity may lead to an under estimation of the time to extinction of the population of the organism under study Understanding of the system and / or the mathematical representation of a model; an empirical model is better suited to describe the data than a mechanistic model if the
mechanistic a model, the lower the role of stochasticity. If collected data is subject to considerable variation without any knowledge to which process this variation can be attributed to, then a stochastic model may be more appropriate.
In conclusion: the aim of this model is not to estimate the time to eradication or
extinction, as referred to as one of the conditions for stochasticity. Whether enough knowledge is available about certain biological processes (e.g. dormancy, seed predation) is species-specific. The default position was not to include stochasticity in ECOSEDYN unless no general principles could be derived from the literature.