A System Model for Crop Yield Potential as a 3 Function of Herbicide Weed Control over Time

−𝑎�22 +� � ∙� −𝑎�22−�𝐻 �� 2 + 2 𝑎𝐻 − 𝑏 ⎭⎬ ⎫ (3.13) Herbicide resistance in species is becoming more prevalent 1997,


3.1
Modelling is an all-encompassing term, which suggests anything from building a scaled representation of the system through to developing a rigorous mathematical analysis of the system's internal relationships. The ultimate purpose of the model is to produce a representation of the system (Wilson, 1988), which is easily understood, behaves in a similar way to the real system and can be more easily manipulated than the real system, in order to understand how things may change as inputs or time vary, which is referred to as "sensitivity analysis".
Often the system model is represented by mathematical relationships. There are two main strategies for determining these mathematical relationships. These are the empirical approach, which usually requires a statistical analysis of data from two or more system parameters (Walpole & Myers, 1972) to evaluate the degree of cause and effect between these parameters, or the deterministic analysis, which employs a range of clearly defined and rigorous mathematical procedures to define the relationship between two or more system parameters.
In terms of mathematical effort, empirical relationships are generally easier to develop, but can not be stretched beyond the sampling range of the original data (Walpole & Myers, 1972) from which the relationship was inferred. On the other hand, deterministic models are much harder to derive, requiring a sound knowledge of mathematic disciplines such as vectors, algebra and calculus, but the resulting equations tend to be much more robust than empirical relationships. Deterministic models can be extrapolated to explore more extreme cases.
System analysis can be applied to most agricultural systems to better understand their operation and optimise performance. System analysis usually includes the development of transfer functions (Åström & Murray, 2012;Smith, 1976). Transfer functions are mathematical equations, involving various input variables or matrices, which relate the system's output to these system inputs. In the case of an agricultural cropping system the key output from the system is potential crop yield. Some crop ecology studies have demonstrated that competition from weeds can reduce the potential yield of some crops by 35% to 55% (Cathcart & Swanton, 2003;Mondani, Golzardi, Ahmadvand, Ghorbani, & Moradi, 2011).
Modern no-till cropping depends on herbicides for weed management; therefore, herbicide applications are an important system input. Unfortunately, herbicide resistance in many weed species is also becoming wide spread (Heap, 1997) and multiple herbicide resistances in several economically important weed species has been widely reported (Owen, Walsh, Llewellyn, & Powles, 2007). In time, herbicide resistant weeds may result in significant yield reductions and grain contamination; therefore, this chapter derives a deterministic system transfer function, relating herbicide input to potential crop yield, in the presence of herbicide resistance, based on various ecological models published in literature.

Derivation of Crop System Transfer Function for Herbicide 3.2 Weed Management
The effect of weed damage on crop yields can be described by equation (3.1) (Schmidt & Pannell, 1996). The nomenclature of some parameters used in this chapter is outlined in Table 3.1, which is at the end of the chapter.
In equation (3.1), D(R) is the damage function caused by a weed density of R, which represents the number of weeds that are recruited from the seed bank (plants m -1 of row). The Damage function can be described by the following equation (Cousens, Brain, O'Donovan, & O'Sullivan, 1987): Substituting equation (3.1) into equation (3.2) yields:

Herbicide Weed Management 3.3
Weed infestations will be made up of some resistant weeds (R R ) and some weeds that can be easily controlled sby herbicides (R S ), where the total weed population is the sum of these two components (i.e. R R -(1 -S) and R S = S ). A typical kill function for a herbicide treatment is (Bosnić & Swanton, 1997): Substituting all this into equation (3.4), and realising that the herbicide treatment will not affect the resistant weeds, yields: 12 Swanton, 1997):

Substituting all this into equation
(3.4), and realising that the herbicide treatment will not affect the resistan yields: The recruitment of seedlings from the seed bank can be described by the following equation (Neve, Norsworthy, Smith, & Zelaya, 2011): The recruitment of seedlings from the seed bank can be described by the follow Norsworthy, Smith, & Zelaya, 2011): The portion of the population that is resistant to herbicide treatment will change generation depending on the selection pressure being applied by the herbicide tr work by Gubbins and Gilligan (1999), if there is a relatively constant selection p herbicide resistance from generation to generation, then the following relationsh � � = − This partial differential equation can be solved by integration to give: Substituting this into equation (3.7) and simplifying yields: Substituting this into equation (3.5) yields: The recruitment of seedlings from the seed bank can be described by the following equation (Nev Norsworthy, Smith, & Zelaya, 2011): Substituting this into equation The portion of the population that is resistant to herbicide treatment will change from generation generation depending on the selection pressure being applied by the herbicide treatments. Based o work by Gubbins and Gilligan (1999), if there is a relatively constant selection pressure (a) towar herbicide resistance from generation to generation, then the following relationship will hold: This partial differential equation can be solved by integration to give: Substituting this into equation (3.7) and simplifying yields: The portion of the population that is resistant to herbicide treatment will change from generation to generation depending on the selection pressure being applied by the herbicide treatments. Based on work by Gubbins and Gilligan (1999), if there is a relatively constant selection pressure (a) towards herbicide resistance from generation to generation, then the following relationship will hold: The recruitment of seedlings from the seed bank can be described by th Norsworthy, Smith, & Zelaya, 2011): Substituting this into equation The portion of the population that is resistant to herbicide treatment wi generation depending on the selection pressure being applied by the he work by Gubbins and Gilligan (1999), if there is a relatively constant se herbicide resistance from generation to generation, then the following r � � = − This partial differential equation can be solved by integration to give: Substituting this into equation (3.7) and simplifying yields: This partial differential equation can be solved by integration to give: The recruitment of seedlings from the seed bank can be described by the Norsworthy, Smith, & Zelaya, 2011): The portion of the population that is resistant to herbicide treatment will generation depending on the selection pressure being applied by the herb work by Gubbins and Gilligan (1999), if there is a relatively constant sel herbicide resistance from generation to generation, then the following re � � = − This partial differential equation can be solved by integration to give: Substituting this into equation (3.7) and simplifying yields: Substituting this into equation (3.7) and simplifying yields: 13 The recruitment of seedlings from the seed bank can be described by the following equation (Neve, Norsworthy, Smith, & Zelaya, 2011): Substituting this into equation (3.5) yields: The portion of the population that is resistant to herbicide treatment will change from generation to generation depending on the selection pressure being applied by the herbicide treatments. Based on work by Gubbins and Gilligan (1999), if there is a relatively constant selection pressure (a) towards herbicide resistance from generation to generation, then the following relationship will hold: This partial differential equation can be solved by integration to give: Substituting this into equation (3.7) and simplifying yields: There is also evidence that herbicides have a toxic effect on the crop as well. Using the study by Yin et al. (2008) as a guide, and assuming that the toxicity of the herbicide on a crop can be expressed as a polynomial of the form There is also evidence that herbicides have a toxic effect on the crop as well. Using the study by Yin et al. (2008) as a guide, and assuming that the toxicity of the herbicide on a crop can be expressed as a polynomial of the form Loss=aH 2 -bH, equation (3.10) can be modified to become:

14
(3.10) can be modified to become: The seed bank will be dynamic depending on factors such as natural seed mortality, immigration of seeds into the area from other locations via various vectors, emigration of seeds out of the area to other locations via various vectors, the onset of dormancy that prevents germination in the current season, and the breaking of dormancy from previous seasons in the seed bank.

Sensitivity Analysis
The development of transfer functions does not always provide accurate prediction but to provides insight into system behaviours as input parameters change. The sensitivity of the output to these changes can be assessed by differentiating the transfer function equations with respect to the input parameter of interest and assessing the magnitude of the resulting differential equation. For example, the sensitivity of the crop to herbicide weed control is given by differentiating equation (3.12) with respect to the herbicide dose, H: Herbicide resistance in many weed species is becoming more prevalent (Heap, 1997(Heap, , 2008. Thornby and Walker (2009)   The seed bank will be dynamic depending on factors such as natural seed mortality, immigration of seeds into the area from other locations via various vectors, emigration of seeds out of the area to other locations via various vectors, the onset of dormancy that prevents germination in the current season, and the breaking of dormancy from previous seasons in the seed bank.
14 (3.10) can be modified to become: The seed bank will be dynamic depending on factors such as natural seed mortality, immigration of seeds into the area from other locations via various vectors, emigration of seeds out of the area to other locations via various vectors, the onset of dormancy that prevents germination in the current season, and the breaking of dormancy from previous seasons in the seed bank.

Sensitivity Analysis
The development of transfer functions does not always provide accurate prediction but to provides insight into system behaviours as input parameters change. The sensitivity of the output to these changes can be assessed by differentiating the transfer function equations with respect to the input parameter of interest and assessing the magnitude of the resulting differential equation. For example, the sensitivity of the crop to herbicide weed control is given by differentiating equation (3.12) with respect to the herbicide dose, H: Herbicide resistance in many weed species is becoming more prevalent (Heap, 1997(Heap, , 2008. Thornby and Walker (2009)

Sensitivity Analysis 3.4
The development of transfer functions does not always provide accurate prediction but to provides insight into system behaviours as input parameters change. The sensitivity of the output to these changes can be assessed by differentiating the transfer function equations with respect to the input parameter of interest and assessing the magnitude of the resulting differential equation. For example, the sensitivity of the crop to herbicide weed control is given by differentiating equation (3.12) with respect to the herbicide dose, H: (3.10) can be modified to become: The seed bank will be dynamic depending on factors such as natural seed mortality, immigration of seeds into the area from other locations via various vectors, emigration of seeds out of the area to other locations via various vectors, the onset of dormancy that prevents germination in the current season, and the breaking of dormancy from previous seasons in the seed bank.

Sensitivity Analysis
The development of transfer functions does not always provide accurate prediction but to provides insight into system behaviours as input parameters change. The sensitivity of the output to these changes can be assessed by differentiating the transfer function equations with respect to the input parameter of interest and assessing the magnitude of the resulting differential equation. For example, the sensitivity of the crop to herbicide weed control is given by differentiating equation (3.12) with respect to the herbicide dose, H: Herbicide resistance in many weed species is becoming more prevalent (Heap, 1997(Heap, , 2008. Thornby and Walker (2009)   Herbicide resistance in many weed species is becoming more prevalent (Heap, 1997(Heap, , 2008. Thornby and Walker (2009)  Selection pressure for genetic traits depends on the initial efficacy of the herbicide to remove susceptible individuals from the population, leaving only the resistant individuals to reproduce. This is reinforced by the adoption of a single herbicide over a long period to sustain the selection pressure on the population.
The transfer function developed in equation (3.12) can also provide some insight in the rate of change of yield potential as a function of weed population generations, hence providing some insights into herbicide resistance. Differentiating equation (3.12) with respect to the generations of weeds gives: is reinforced by the adoption of a single herbicide over a long period to sustain the selection pressure on the population.
The transfer function developed in equation (3.12) can also provide some insight in the rate of change of yield potential as a function of weed population generations, hence providing some insights into herbicide resistance. Differentiating equation (3.12) with respect to the generations of weeds gives: Timeliness of herbicide application is another important consideration in weed management. Herbicide application can be delayed for several reasons, but often it is associated with inclement weather conditions such as wind and rain, both of which impede the opportunity to spray herbicides safely and effectively. If weeds become well established before the crop canopy closes, yield losses can be expected. The sensitivity of yield potential to timeliness can be evaluated by differentiating equation (3.12), with respect to t:

Examples
Equation (3.12) was coded into a simple cropping system model using the MatLab® software platform. Using data published by Bosnić and Swanton (1997) and Yin et al. (2008) for Rimsulfuron herbicide and assuming: an initially same small resistant population (i.e. S o =0.9999); a seed mortality rate of 10% each year; and a slightly positive selection coefficient of (a = 0.002) for herbicide resistance (Baucom & Mauricio, 2004), the system transfer function was used to analyse the effect of a single herbicide application on crop yield potential. The transfer function was also (3.14) Timeliness of herbicide application is another important consideration in weed management. Herbicide application can be delayed for several reasons, but often it is associated with inclement weather conditions such as wind and rain, both of which impede the opportunity to spray herbicides safely and effectively. If weeds become well established before the crop canopy closes, yield losses can be expected. The sensitivity of yield potential to timeliness can be evaluated by differentiating equation (3.12), with respect to t: is reinforced by the adoption of a single herbicide over a long period to sustain the selection pressure on the population.
The transfer function developed in equation (3.12) can also provide some insight in the rate of change of yield potential as a function of weed population generations, hence providing some insights into herbicide resistance. Differentiating equation (3.12) with respect to the generations of weeds gives: Timeliness of herbicide application is another important consideration in weed management. Herbicide application can be delayed for several reasons, but often it is associated with inclement weather conditions such as wind and rain, both of which impede the opportunity to spray herbicides safely and effectively. If weeds become well established before the crop canopy closes, yield losses can be expected. The sensitivity of yield potential to timeliness can be evaluated by differentiating equation (3.12), with respect to t:

Examples
Equation (3.12) was coded into a simple cropping system model using the MatLab® software platform. Using data published by Bosnić and Swanton (1997) and Yin et al. (2008) for Rimsulfuron herbicide and assuming: an initially same small resistant population (i.e. S o =0.9999); a reinforced by the adoption of a single herbicide over a long period to sustain the selection ressure on the population. he transfer function developed in equation (3.12) can also provide some insight in the rate of ange of yield potential as a function of weed population generations, hence providing some sights into herbicide resistance. Differentiating equation (3.12) with respect to the generations of eeds gives: (3.14) imeliness of herbicide application is another important consideration in weed management. erbicide application can be delayed for several reasons, but often it is associated with inclement eather conditions such as wind and rain, both of which impede the opportunity to spray herbicides fely and effectively. If weeds become well established before the crop canopy closes, yield losses n be expected. The sensitivity of yield potential to timeliness can be evaluated by differentiating quation (3.12), with respect to t: 3.15) .5 Examples quation (3.12) was coded into a simple cropping system model using the MatLab® ftware platform. Using data published by Bosnić and Swanton (1997)

3.5
Equation (3.12) was coded into a simple cropping system model using the MatLab® software platform. Using data published by Bosnić and Swanton (1997) and Yin et al. (2008) for Rimsulfuron herbicide and assuming: an initially same small resistant population (i.e. S o =0.9999); a seed mortality rate of 10% each year; and a slightly positive selection coefficient of (a = 0.002) for herbicide resistance (Baucom & Mauricio, 2004), the system transfer function was used to analyse the effect of a single herbicide application on crop yield potential. The transfer function was also used to forecast the long-term crop yield potential, if only a single herbicide type was used during this time. Figure 3.1 shows the expected crop yield response as a function of the herbicide's application rate. Based on the parameters used in this example, there is an optimal active ingredient application rate (i.e. where The transfer function also predicts that significant herbicide resistance will occur within 15 generations (Figure 3.2), as was also predicted by Thornby and Walker (2009). This is apparent when looking at how the relative crop yield potential reduces along the generations axis in Figure 3.2. After 15 to 20 years of using the same herbicide control system, the model outlined in equation (3.12) suggest that further herbicide application will be ineffectual. Herbicide rotations can forestall the development of a resistant population; however several weed species have developed multiple resistance to several herbicide groups (Owen et al., 2007).
It is possible to visualise the influence of both herbicide application and generational change in a response surface, as shown in Figure 3.3.
The sensitivity of crop yield potential to timeliness can be assessed from equations (3.12) and (3.15). Figure 3.4 depicts the influence of time between crop emergence and weed emergence over crop yield potential.

3.6
A growing herbicide resistance problem is already evident in most Australian cropping systems (Broster & Pratley, 2006;Gill & Holmes, 1997). There is evidence that glyphosate resistance has already developed in some weed populations (Broster & Pratley, 2006) and multiple herbicide resistances has been widely reported in several weed species (Kuk, Burgos, & Talbert, 2000;Owen et al., 2007;Walsh, Powles, Beard, Parkin, & Porter, 2004;Yu, Cairns, & Powles, 2007); therefore significant crop yield losses can be expected into the future as weed become more resistant to herbicide management strategies. Alternative weed management strategies that are compatible with no-till cropping systems need to be developed. The next chapter will discuss some of the non-chemical weed control strategies that have been considered in recent time.     Is the percentage yield loss as weed density approaches ∞ (= 38.0 (Bosnić & Swanton, 1997)) c Is the speed of light (m s -1 ) or the rate at which I approaches zero as t approaches ∞ (= 0.017 (Bosnić & Swanton, 1997)) d Is the slope of the seed bank recruitment curve at t o D b Fraction of the seed population from previous seasons breaking dormancy (Note: this is expressed as a fraction of the initial seed bank population W o ) D o Fraction of the seed population developing dormancy (Note: this is expressed as a fraction of the initial seed bank population W o ) E m Seed emigration from the area of interest g Is the generational number H Is the herbicide's active ingredient dose (kg ha -1 ) I Is the percentage yield loss as the weed density tends towards zero (= 0.38 (Bosnić & Swanton, 1997)) Im Seed immigration into the area of interest N Is the natural death rate for the whole population (Note: this is expressed as a fraction of the initial seed bank population W o ) S o Is the initial frequency of plants in the population that are susceptible to herbicide treatment S s Viable seed set per plant from surviving volunteers in the weed population t Is the time difference between crop emergence and weed emergence t o Is the time for 50% germination of the viable seed bank W Is the viable seed bank Y o Is the theoretical yield with no weed infestations λ Is an estimate of weed sensitivity to the herbicide