the importance of tillage depth in relation to seedling emergence in stale seedbeds
TRANSCRIPT
e c o l o g i c a l m o d e l l i n g 2 0 1 ( 2 0 0 7 ) 536–546
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The importance of tillage depth in relation to seedlingemergence in stale seedbeds
Angelique Lamour, Lambertus A.P. Lotz ∗
Plant Research International, Wageningen University and Research Centre, P.O. Box 16, 6700 AA Wageningen, The Netherlands
a r t i c l e i n f o
Article history:
Received 28 October 2005
Received in revised form
19 September 2006
Accepted 22 October 2006
Published on line 14 December 2006
Keywords:
Weeds
Seed bank
Seed distribution
Seedling emergence
Seed production
Mechanical weeding
a b s t r a c t
Stale seedbeds can be used in arable farming to reduce the density of weeds in the crop.
This type of tillage before crop sowing can contribute to a successful weed management in
systems where no herbicides are used, e.g. organic farming. The population dynamics of
weeds in response to stale seedbeds is, however, hardly understood, and therefore possibly
not optimised. The relevance of tillage depth in relation to seedling emergence is explored
with a simple, deterministic model. We systematically examined the effect of seedling emer-
gence and subsequent weed control on weed population dynamics, starting by considering
the seed bank as one soil layer, and continued by considering a depth-structured seed bank.
Whether the widely used tillage regime consisting of shallow tillage, again shallow tillage,
and ultimately deep tillage, is preferred above the regime we propose (i.e. deep, shallow and
ultimately shallow tillage) depends on the proportion emergence specific for each soil layer,
the proportion of seeds that is moved from one layer to the other, and the seed distribution
in the soil. A case study based on characteristics of the population dynamics of the weed
Polygonum persicaria showed that the proposed tillage regime could give reductions in weed
Tillage
Soil disturbance
Stale seedbed
density of up to 32% compared to the conventional tillage regime of stale seedbeds. The
tillage regime that we propose requires techniques that restrict soil compaction to fixed
traffic lanes, giving large zones favourable for crop growth. Because of ongoing processes
in precision farming technology with respect to controlled traffic systems, this regime has
s to
requires subsequent mechanical weeding, or hand weeding
realistic opportunitie
1. Introduction
Weed problems can be enormous in organic farming. Theworst annual species are those having a high seed produc-tion and/or those producing seeds over a prolonged periodof time. To reduce weed pressure it is important to preventinflow of weed seeds from the environment. Also, the seedbank size, influenced by previous crops, should be minimised(Kropff et al., 1996; Kebreab and Murdoch, 2001). In organic
farming prevention or reduction of seed production is relevantsince curative chemical control is not allowed. For exam-ple, weed seed production in a narrow spaced crop may be∗ Corresponding author.E-mail address: [email protected] (L.A.P. Lotz).
0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2006.10.015
become widely used in the future.
© 2006 Elsevier B.V. All rights reserved.
less compared to that in a widely spaced crop (Mertens andJansen, 2002). Mechanical weeding is an important manage-ment component. However, mechanical weeding not only killsa proportion of weed seedlings, but also stimulates a newcohort of seedlings. Namely, if soil is disturbed by tillage, e.g.ploughing or harrowing, seeds in the seed bank of a wide vari-ety of species are stimulated to germinate, e.g. when they areexposed to a light flash before reburial (Chancellor, 1985). This
of most remaining seedlings. In many countries hand weed-ing is expensive, and a shortage of available labour causesproblems.
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e c o l o g i c a l m o d e l l i n
Although not widely used, weed management may includehe application of stale seedbeds, usually managed beforeeedbed preparation of a late-sown crop. The principle is torepare a shallow ‘seedbed’ by disturbing the upper soil layer,owever, not followed by sowing and therefore termed a staleeedbed. After emergence of seedlings mechanical weedings applied, for example shallow harrowing, including uproot-ng of weeds followed by soil covering (Kurstjens and Kropff,001). Due to the absence of a crop this mechanical weedingan be aggressive and therefore very efficient. A subsequentohort of seedlings is again mechanically controlled. Theserocesses may be repeated several times, ultimately followedy seedbed preparation and sowing. Thereby, the density ofeeds is reduced, resulting in less seedling emergence whenhe crop is present. A prerequisite for a successful applica-ion is a late-sown crop, which allows germination of weedst suitable soil temperatures. In early spring soil temperaturesre often too low for various weed species.
The population dynamics of weeds in response to staleeedbeds is hardly understood, and therefore stale seedbedsre possibly not optimised nor widely used at present. Ques-ions pertain to how seedling emergence from the seed bankffects the weed population and how tillage affects seedlingortality and emergence. In this paper, we answer these ques-
ions through use of simple mechanistic models that capturessential features of the system. The present model is notesigned to predict the population dynamics of a particularpecies as many weed population models do (Mortimer etl., 1989; Cousens and Mortimer, 1995; Kebreab and Murdoch,001), but seeks to describe and explain ways in which tillageegimes affect the weed population. Simulation models pre-icting weed population dynamics are particularly useful ifarameter values of life-cycle processes are known for theystem under consideration. However, data on how manyeedlings emerged from which depth and at what time arecarce (Mohler, 1993). The ability of a seed to germinate andmerge varies with depth (Cussans et al., 1996; Grundy etl., 1996; Vleeshouwers, 1997; Kremer and Lotz, 1998), and asillage operations redistribute seeds in the soil (Cousens andoss, 1990; Mead et al., 1998), a weed seed population should
e considered as being structured by the depth at which seedsre located. In this paper we explore how tillage depth affectseedling emergence from a depth-structured seed bank, andhow that the application of stale seedbeds can be improved.
Nowadays, stale seedbeds in the Netherlands are preparedy shallow tillage (several times), which consists of hoeing inombination with shallow harrowing. Seedbed preparation,owever, requires deeper tillage by rotary harrowing, withepth of tillage being crop-specific. Deeper tillage involveseavier machinery, which can not be used under wet con-itions because wheel traffic then leads to soil compaction,
eading to limited crop growth and reduced yield (Soane andan Ouwerkerk, 1994; Ball and Ritchie, 1999). The machin-ry used for stale seedbed preparation is far less heavy and,herefore, can be used to stimulate weed emergence wheneedbed preparation is not yet desired because of inappropri-
te weather conditions.Stale seedbeds are not optimised as shallow tillage reduceshe seed density in the upper soil layer, but subsequent mixingf soil layers for seedbed preparation (deep tillage) replen-
1 ( 2 0 0 7 ) 536–546 537
ishes the seed density in the upper layer. From an ecologicalpoint of view it would be more beneficial to start with deeptillage, and use shallow tillage to reduce the seed density inthe upper soil layer, after which the crop can be sown withoutmixing of soil layers. A prerequisite is that soil compactionduring deep tillage is reduced, something that is enabled byrecent developments in innovative machinery (Hamza andAnderson, 2005). Alternative types of running gear and vehi-cles to minimise the soil compaction impact of wheel traffic onthe cropping system include: (1) reduction of ground contactpressure; (2) reduction of axle load; (3) use of controlled trafficsystems such as gantries; (4) use of tracks. Enhanced machin-ery, termed controlled traffic, allows crops to be grown in2.4 m wide, traffic-free cropped strips with all wheels runningon uncropped, permanent wheel tracks (Dickson and Ritchie,1996). The wheel tracks are permanent as the machinery issupported by ‘Global Positioning Systems’. This seems promis-ing for optimisation of stale seedbeds, since this innovativetechnique restricts soil compaction and therefore allows deeptillage early in the season, with subsequent shallow tillagebefore crop sowing. In the traffic-free zones soil structurebecomes more loose and less dependent of weather condi-tions.
In this paper we use models to study how emergence ofseveral cohorts of seedlings, and subsequent reduction inseedling numbers by mechanical weeding affect the weedpopulation, where we focus on tillage depth. For a systematicexamination of the effect of stale seedbeds on weed popu-lation dynamics, i.e. seedling emergence and mortality, westart to consider the seed bank as one soil layer from whichseeds germinate and seedlings emerge (Section 2). There-after, the soil is divided into several layers, each with itsown value for seedling emergence, giving a depth-structuredseed bank (Section 3). Tillage may now include mixing of soillayers. We hypothesise that a change in sequence of tillagedepth affects seedling mortality and emergence, i.e. we startwith deep tillage for preparation of a stale seedbed followedby shallow tillage, where the last shallow tillage includesseedbed preparation. We explore under which conditions thistillage regime is preferred with respect to both seedling andseed density. The resulting benefits are highlighted in a casestudy.
2. The seed bank considered as one soillayer
In this section we consider the seed bank as one soil layer. Westudy how emergence of seedlings and subsequent weed con-trol affect the weed population. In the following example weshow the emergence of various seedling cohorts. See Table 1for an overview of symbols for parameters and states. Thefirst cohort of seedlings (X1) emerges at moment = 1 (Table 2),where the value for the proportion of seeds emerging fromthe seed bank equals �1. Seedling cohort 1 is controlled byundefined mechanical weed control, causing a reduction in
seedlings at moment = 2. Weed control stimulates emergenceof a second cohort of seedlings with a proportion emergenceof �2 at moment = 3. Thereafter (moment = 4), both cohortsare reduced by mechanical weed control. Analogously, a third
538 e c o l o g i c a l m o d e l l i n g 2 0 1 ( 2 0 0 7 ) 536–546
Table 1 – Symbols for parameters and states
Symbol Description
n Cohort number (n ∈ [1, 2, 3, 4])L Soil layer (L ∈ [1, 2, 3, 4, 5, 6, 7, 8, 9, 10])�n Proportion emergence of weed seedlings of cohort n (0 ≤ �n ≤ 1)�L Proportion emergence of weed seedlings from layer L (0 ≤ �L ≤ 1)� Proportion mortality caused by mechanical weed control (0 ≤ � ≤ 1)k Number of times mechanical weed control is applied (k ∈ [1, 2, 3])Xn Number of weed seedlings of cohort n per unit areaS Number of seeds in the seed bank per unit area
umbeumbe
1]3]
>
3]5]
>
However, not all emerged seedlings are killed by subsequentmechanical weed control. Of all seedlings that survive untilsowing of the crop, the seedlings that emerged early arelarger in size than the seedlings that emerged late. Conse-
v Nw N
cohort emerges at moment = 5, after which mechanical weedcontrol reduces the density of seedlings of all three cohorts atmoment = 6. Finally, a fourth seedling cohort is stimulated toemerge at moment = 7.
Inherent in the consideration of only one soil layer is that:(1) seedling emergence has a single value, i.e. independentof the depth of seed placement and (2) mechanical weedingdisturbs the whole soil layer.
The number of controlled seedlings per unit area atmoment = 7 depends of course on the number of emergedseedlings per unit area, but the fraction controlled seedlingsdoes not depend on seedling emergence. The fraction con-trolled seedlings is calculated by (1 − (1 − �)k), with � being theproportion mortality caused by mechanical weed control and
k is the number of times mechanical weed control is applied.We assume that mechanical weed control either kills a weedseedling or not, without considering half cut seedlings with areduced fitness and limited seed production. Also, we assumethat � is not density-dependent, i.e. the efficacy of mechanicalweeding is independent of weed density. Although small-sizedseedlings may be more easily controlled than seedlings of alarger size (Kurstjens et al., 2000), we assume that � is not
X1[moment =X2[moment =X2[moment =X3[moment =
dependent on seedling size.Although the proportion seedling emergence may increase
with rising soil temperatures (O’Connor and Gusta, 1994),giving �1 < �2 < �3 < �4, or may increase to an optimum tem-
Table 2 – Description of events at each moment whenthe seed bank is considered as one soil layer
Moment Description
0 Initial condition1 Emergence of cohort 12 Mortality caused by first mechanical weed control3 Emergence of cohort 24 Mortality caused by second mechanical weed control5 Emergence of cohort 36 Mortality caused by third mechanical weed control7 Emergence of cohort 4
r of seeds in the first soil layer per unit arear of seeds in the second soil layer per unit area
perature (Blackshaw, 1990), we assume that the proportionemergence remains fixed, i.e. �1 = �2 = �3 = �4, which keeps themodel simple. During the limited time span of stale seedbeds,soil conditions are rather stable, and therefore a fixed propor-tion emergence is a reasonable assumption. Also, we assumethat seed mortality is negligible during this limited time span.
If the proportion emergence remains fixed, the seed banksize determines the number of seedlings that will emerge perunit area. Since the seed bank size decreases after emergenceof cohort 1, the number of emerged seedlings of cohort 2 perunit area will be lower than that of cohort 1. Analogously, thedensity of emerged seedlings will be lower for cohort 3 thanfor cohort 2, resulting in decreasing amplitudes of emergedseedlings in time (Fig. 1):
1,X1[moment = 1]X3[moment = 5]
> 1,X1[moment = 1]X4[moment = 7]
> 1
1,X2[moment = 3]X4[moment = 7]
> 1,X3[moment = 5]X4[moment = 7]
> 1
(1)
where Xn is the number of seedlings per unit area of cohort n(n = 1–4).
The higher the proportion emergence, the more the seedbank size decreases, which makes a high � value favourable.
Fig. 1 – The number of seedlings per unit area, calculatedfor each moment, shown for four cohorts of seedlings. Theproportion mortality caused by mechanical weed control is0.70, and the proportion seedling emergence is 0.2.
e c o l o g i c a l m o d e l l i n g 2 0 1 ( 2 0 0 7 ) 536–546 539
Table 3 – Derivation of number of seeds (S) and weed seedlings (X) per cohort n (Xn, n ∈ [1, 2, 3, 4]) per unit area as a resultof seedling emergence (�1 = �2 = �3 = �4 = �) and mortality caused by mechanical weed control (�) at each moment involved
Moment Description S X1 X2 X3 X4
0 Initial condition S0 – – – –1 Emergence of cohort 1 (1 − �)S0 �S0 – – –2 Weed control of cohort 1 (1 − �)S0 (1 − �)�S0 – – –3 Emergence of cohort 2 (1 − �)2S0 (1 − �)�S0 �(1 − �)S0 – –4 Weed control of cohort 1 + 2 (1 − �)2S0 (1 − �)2�S0 (1 − �)�(1 − �)S0 – –5 Emergence of cohort 3 (1 − �)3S (1 − �)2�S (1 − �)�(1 − �)S �(1 − �)2S –
(1 − �
(1 − �
q1(aastacpo
Tv
⎧⎪⎨⎪⎩Fst
t
<X
X4
Enawgoft
0
6 Weed control of cohort 1 + 2 + 3 (1 − �)3S0
7 Emergence of cohort 4 (1 − �)4S0
uently, they will be more competitive to the crop (Lotz et al.,995), and will produce more seeds, given they reach maturityThompson et al., 1991). Therefore, a higher � value also has
negative aspect in the fact that more large weed seedlingsre present after the last mechanical weed control just beforeowing (i.e. at moment = 7). Formulas were derived showinghe number of weed seedlings per unit area explicitly (Table 3),s a result of seedling emergence (�) and mechanical weedontrol (�). Of all seedlings surviving until crop sowing it isreferred to have few large ones, therefore we focus on ratiosf seedling densities of various cohorts at this moment = 7:
X1
X2= (1 − �)
(1 − �),
X1
X3= (1 − �)2
(1 − �)2,
X1
X4= (1 − �)3
(1 − �)3,
X2
X3= (1 − �)
(1 − �),
X2
X4= (1 − �)2
(1 − �)2,
X3
X4= (1 − �)
(1 − �)
(2)
hese ratios can be smaller or larger than 1, depending on thealues for � and �:
(1 − �
1 − �
)k
< 1 where k ∈ [1, 2, 3] if � > �(1 − �
1 − �
)k
> 1 where k ∈ [1, 2, 3] if � < �
(3)
avourable ratios are thus achieved when the proportion ofeedlings that is mechanically controlled exceeds the propor-ion of seedlings that emerges from the seed bank.
For a given value of �, these ratios are smaller for a lowhan a high value of �:
X1[low �]X2[low �]
<X1[high �]X2[high �]
,X1[low �]X3[low �]
<X1[high �]X3[high �]
,X1[low �]X4[low �]
X2[low �]X3[low �]
<X2[high �]X3[high �]
,X2[low �]X4[low �]
<X2[high �]X4[high �]
X3[low �]X4[low �]
<X3[high �]X4[high �]
q. (4) shows that in the case where � is low the relativeumber of large weeds (in relation to smaller ones) per unitrea is smaller than in the case where � is high. Therefore,ith respect to the size of seedlings, a low proportion emer-
ence is preferred. Logically, also with respect to the numberf seedlings per unit area, a low proportion emergence is pre-erred. However, this conflicts with the aim of a stale seedbedo exhaust the density of seeds in the soil.
0 0 0
)3�S0 (1 − �)2�(1 − �)S0 (1 − �)�(1 − �)2S0 –)3�S0 (1 − �)2�(1 − �)S0 (1 − �)�(1 − �)2S0 �(1 − �)3S0
1[high �][high �]
, (4)
3. The depth-structured seed bank
In the previous section we showed that, in the case wherethe soil is considered as one soil layer, it is favourable tohave a high proportion of seedlings mechanically controlled(preferably exceeding the proportion of seedlings that emergesfrom the seed bank). In this section we continue the system-atic examination of seedling emergence on weed populationdynamics by dividing the soil into several layers, and specify-ing the proportion seedling emergence per soil layer (�L withL the soil layer). We carry on with the introduced example, butconsider a soil profile consisting of two layers. The density ofseeds at moment = 0 (i.e. S0) equals v0 in layer 1 and w0 in layer2 (Table 1), represented by following vector:
S0 =(
v0
w0
)(5)
For cohort 1 (n = 1) the number of seedlings per area emerg-ing from both layers then equals �1v0 + �2w0. Subsequentmechanical weed control (three times as in Section 2) may nowinclude mixing of soil layers (Table 4). We consider two depths,termed deep or shallow tillage, for which we develop matricesrepresenting the appropriate soil mixing. Deep tillage involvesmixing of both soil layers, whereby some seeds will move toanother layer, and other seeds will stay in the same layer. Theprobability to stay in layer 1 equals a, and to move to layer 2equals (1 − a). The probability to stay in layer 2 equals b, andto move to layer 1 equals (1 − b), giving following matrix (Md):
Md =(
a 1 − b
1 − a b
)(6)
In the case where a = b = 0.5, homogeneous mixing of both soil
layers takes place, resulting in equal seed densities in both soillayers.We define the depth of shallow tillage to be equal to thedepth of the first soil layer, resulting in no movement of seeds
540 e c o l o g i c a l m o d e l l i n g
Table 4 – Description of events at each moment whenthe seed bank is depth-structured
Moment Description
0 Initial condition1 Emergence of cohort 12 Mortality caused by first mechanical weed
control3 Mixing of soil layers by first mechanical weed
control4 Emergence of cohort 25 Mortality caused by second mechanical weed
control6 Mixing of soil layers by second mechanical
weed control7 Emergence of cohort 38 Mortality caused by third mechanical weed
control
from layer 1 to layer 2. Seeds should not be moved to a layerin which they experience a higher proportion emergence thanin the layer they were before. It shows that the seed distribu-
9 Mixing of soil layers by third mechanical weedcontrol
10 Emergence of cohort 4
to other layers, which gives a matrix (Ms) equal to the identitymatrix.
Ms =(
1 00 1
)(7)
Multiplication of matrix Md or Ms by a vector of seed numberspresently at each layer will give a vector of seed numbers ateach layer after tillage.
For each mechanical weed control (three times) we choosedeep or shallow tillage, giving various possibilities. The widelyused tillage regime consisting of shallow tillage, again shallowtillage, and ultimately deep tillage is termed ‘regime SSD’. Wehypothesise that a change in sequence of tillage depth affectsseedling mortality and emergence. Therefore, the proposedtillage regime consists of deep tillage followed by shallowtillage twice, which is termed ‘regime DSS’. Below, we focuson regimes SSD and DSS, and explore which tillage regimeis preferred under which conditions. Firstly, we calculate thedensity of weed seedlings of each cohort that is present atthe last moment before sowing (moment = 10), and explorewhich tillage regime is preferred in terms of least density ofweed seedlings. Then we explore the density of seeds that isstill present in the soil at moment = 10, as emergence of theseseeds after sowing of the crop may be harmful, and explorewhich tillage regime is preferred in terms of least density ofseeds still present in the soil.
3.1. Calculation of X1 at moment = 10
The first cohort of seedlings (X1) at moment = 10, that is thelast moment before sowing (Table 4), is equal for both tillage
Table 5 – Number of seedlings of the second cohort per unit areafter deep tillage
Number of seedlings emerged from l
After shallow tillage �1(1 − �1)v0
After deep tillage �1a(1 − �1)v0 + �1(1 − b)(1 − �2)w0
2 0 1 ( 2 0 0 7 ) 536–546
regimes since it represents ‘natural’ emergence, i.e. before anyweed control is applied, reduced by mechanical weeding (�),three times applied during both regimes:
(1 − �)3[�1v0 + �2w0] (8)
3.2. Calculation of X2 at moment = 10
After emergence of the first seedling cohort, tillage is appliedeither shallow or deep, by which a second cohort is stimu-lated to emerge. In the case where shallow tillage is applied(regime SSD), no mixing of soil layers takes place, and thesecond cohort emerges both from layer 1 and layer 2. Due toemergence of the first cohort the density of seeds in layer 1 isreduced to (1 − �1)v0 of which proportion �1 emerges (Table 5).Analogously, proportion �2 of (1 − �2)w0 emerges from layer 2.
In the case where deep tillage is applied (regime DSS)mixing of soil layers is involved. Of the seeds in layer 1, i.e.(1 − �1)v0, a fraction a remains in layer 1 of which proportion �1
emerges. Of the seeds in layer 2, i.e. (1 − �2)w0, a fraction (1 − b)moves to layer 1 of which proportion �1 emerges. Analogouscalculations hold for seedlings emerging from layer 2.
In both tillage regimes the second cohort of seedlings istwice mechanically controlled. The difference (DSS − SSD) inX2 at moment = 10 can be derived from Table 5, which is afterrewriting:
(1 − �)2[(a − 1)(1 − �1)v0 + (1 − b)(1 − �2)w0][�1 − �2] (9)
If this difference is smaller than zero, the proposed tillageregime DSS is preferred, giving:
{(1 − b)(1 − �2)w0 < (1 − a)(1 − �1)v0 if �1 > �2
(1 − b)(1 − �2)w0 > (1 − a)(1 − �1)v0 if �1 < �2(10)
For �1 > �2 tillage regime DSS is preferred (with respect toX2 at moment = 10) if the number of seeds (per unit area) thatis moved from layer 2 to layer 1 is smaller than the number ofseeds (per unit area) that is moved from layer 1 to layer 2 as aresult of deep tillage. For �1 < �2 DSS is preferred if the numberof seeds (per unit area) that is moved from layer 2 to layer 1 islarger than the number of seeds (per unit area) that is moved
tion in the soil at moment = 0 (i.e. the values for v0 and w0)carries over into the weed density at moment = 10, and there-fore jointly determines whether tillage regime SSD or DSS ispreferred.
a emerged from layers 1 and 2 after shallow tillage, and
ayer 1 Number of seedlings emerged from layer 2
�2(1 − �2)w0
�2(1 − a)(1 − �1)v0 + �2b(1 − �2)w0
e c o l o g i c a l m o d e l l i n g 2 0 1 ( 2 0 0 7 ) 536–546 541
Table 6 – Number of seedlings of the third cohort per unit area emerged from layers 1 and 2 after shallow + shallowtillage, and after deep + shallow tillage
Number of seedlings emerged from layer 1 Number of seedlings emerged from layer 2
b)(1
3
Aitt(iXr
(
Ip
{
Franlifu
r(�
le
3
AsciT
3.5. Calculation of seeds at moment = 10
Seeds that are still present in the soil at moment = 10 may beharmful to the crop when they germinate after sowing of that
After shallow + shallow tillage �1(1 − �1)2v0
After deep + shallow tillage �1a(1 − �1)2v0 + �1(1 − �1)(1 −
.3. Calculation of X3 at moment = 10
fter emergence of the second seedling cohort, shallow tillages applied both in tillage regime SSD and in DSS, by which ahird cohort is stimulated to emerge. No mixing of soil layersakes place. Emergence from layers 1 and 2 can be calculatedTable 6). In both tillage regimes the third cohort of seedlingss once mechanically controlled. The difference (DSS − SSD) in
3 at moment = 10 can be derived from Table 6, which is afterewriting:
1 − �)[(a − 1)(1 − �1)v0 + (1 − b)(1 − �2)w0]
× [�1(1 − �1) − �2(1 − �2)] (11)
f this difference is smaller than zero, tillage regime DSS isreferred, giving:
(1 − b)(1 − �2)w0 < (1 − a)(1 − �1)v0 if �1(1 − �1) > �2(1 − �2)(1 − b)(1 − �2)w0 > (1 − a)(1 − �1)v0 if �1(1 − �1) < �2(1 − �2)
(12)
or �1(1 − �1) > �2(1 − �2) tillage regime DSS is preferred (withespect to X3 at moment = 10) if the number of seeds (per unitrea) that is moved from layer 2 to layer 1 is smaller than theumber of seeds (per unit area) that is moved from layer 1 to
ayer 2 as a result of deep tillage. For �1(1 − �1) < �2(1 − �2) DSSs preferred if the number of seeds (per unit area) that is movedrom layer 2 to layer 1 is larger than the number of seeds (pernit area) that is moved from layer 1 to layer 2.
With respect to X3, the conditions are different than withespect to X2. Emergence of the second cohort from layer 1stimulated by the first weed control) is low for a low value of
1, but the more seedlings of the second cohort emerge, theess seedlings of the third cohort emerge. The same holds formergence from layer 2. Clearly, there is a trade-off (Fig. 2).
.4. Calculation of X4 at moment = 10
fter emergence of the third cohort, tillage is applied either
hallow (regime DSS) or deep (regime SSD), by which a fourthohort is stimulated to emerge (Table 7). This fourth cohorts not mechanically controlled in the seedbed of the crop.he difference (DSS − SSD) in X4 at moment = 10 gives afterTable 7 – Number of seedlings of the fourth cohort per unit areashallow + shallow + deep tillage, and after deep + shallow + shall
Number of seedlings emerged
After shallow + shallow + deep tillage �1a(1 − �1)3v0 + �1(1 − b)(1 − �2)After deep + shallow + shallow tillage �1a(1 − �1)3v0 + �1(1 − �1)2(1 − b
�2(1 − �2)2w0
− �2)w0 �2(1 − �2)(1 − a)(1 − �1)v0 + �2b(1 − �2)2w0
rewriting:
[�2(a − 1)(1 − �1)v0 + �1(1 − b)(1 − �2)w0][(1 − �1)2 − (1 − �2)2]
(13)
If this difference is smaller than zero, tillage regime DSS ispreferred, giving:
{�1(1 − b)(1 − �2)w0 < �2(1 − a)(1 − �1)v0 if �1 < �2
�1(1 − b)(1 − �2)w0 > �2(1 − a)(1 − �1)v0 if �1 > �2(14)
Whether tillage regime SSD or DSS is preferred at the momentjust before sowing of the crop is not unambiguous, as isexplained below. As the frequency of tillage goes up, the levelof mathematical complexity increases. This makes it difficultto give the model a physical or biological interpretation, andshows the value of deterministic modelling by increasing themodel complexity step by step.
Fig. 2 – For all combinations of possible �1 and �2 valuesthere is indicated whether �1(1 − �1) is smaller or largerthan �2(1 − �2) as in Eq. (12).
emerged from layers 1 and 2 afterow tillage
from layer 1 Number of seedlings emerged from layer 2
3w0 �2(1 − a)(1 − �1)3v0 + �2b(1 − �2)3w0
)(1 − �2)w0 �2(1 − �2)2(1 − a)(1 − �1)v0 + �2b(1 − �2)3w0
542 e c o l o g i c a l m o d e l l i n g 2 0 1 ( 2 0 0 7 ) 536–546
Table 8 – Number of seeds per unit area in layers 1 and 2 after shallow + shallow + deep tillage, and afterdeep + shallow + shallow tillage
Number of seeds in layer 1 Number of seeds in layer 2
�1)(1�1)3(
After shallow + shallow + deep tillage a(1 − �1)4v0 + (1 −After deep + shallow + shallow tillage a(1 − �1)4v0 + (1 −
crop in the same season, or in other crops in following years.The number of seeds per unit area (Table 8) is the fraction ofseeds that, at moment = 10, does not contribute to emergenceof the fourth seedling cohort, and is therefore closely relatedto Table 7. The difference (DSS − SDD) in seed density can noweasily be derived for soil layer 1, for soil layer 2, and for thetotal soil profile (not shown).
3.6. Interpretation of X1, X2, X3, X4 and seeds atmoment = 10
Whether tillage regime SSD or DSS is preferred at the momentjust before sowing of the crop depends on three facts: (1) theproportion emergence specific for each soil layer (i.e. the val-
Fig. 3 – Three seed distributions for 100 seeds per unit area withdiscrete layers of 0.5 cm each. Uniform seed distribution (a); mos
− b)(1 − �2)3w0 (1 − �2)(1 − a)(1 − �1)3v0 + b(1 − �2)4w0
1 − b)(1 − �2)w0 (1 − �2)3(1 − a)(1 − �1)v0 + b(1 − �2)4w0
ues for �1 and �2); (2) the proportion of seeds that is movedfrom one layer to the other (i.e. the values for a and b); (3) theseed distribution in the soil (i.e. the values for v0 and w0). Thisis in terms of least density of weed seedlings, and also in termsof least seed density.
4. Case study
In this section we explore theoretically whether tillage regime
SSD or DSS would be more beneficial for a case study validin the Netherlands. The case study involves the growingof maize in arable soil. One of the worse annual weeds inthis agro-ecosystem includes Polygonum persicaria, showingin soil up to 6 cm depth, where the soil is divided into 12t seeds in upper layers (b); most seeds in deeper layers (c).
g 2 0
apad(so(an1omA0
pswebFrL
t
F(p(u
e c o l o g i c a l m o d e l l i n
bundant emergence in April/May. The emergence rate of P.ersicaria depends on temperature, soil penetration resistance,nd depth of seed placement (Vleeshouwers, 1997). We useata provided by this author for common spring temperatures
5 ◦C), when stale seedbeds are usually applied, and for highoil penetration resistances (1.0 MPa). With respect to depthf seed placement, we explore three distributions of seeds
Fig. 3). The first includes a uniform distribution of seeds overll soil layers that is, e.g. created after repeated ploughing withegligible seed input (Van Esso et al., 1986; Cousens and Moss,990). The second represents a soil that had many seeds falln the surface in the recent past, and the third a soil bearingost seeds in deeper layers as a result of tillage regime choice.soil profile is created consisting of twelve soil layers, each
.5 cm thick.In the Netherlands, seedbed preparation of maize may be
receded by several stale seedbeds. In this case study two staleeedbeds are prepared by shallow tillage, where mechanicaleed control (Table 4) is applied after 5 days. If we define the
mergence rate per day, provided by Vleeshouwers (1997), toe �*, then the emergence rate per 5 days equals 1 − (1 − �*)5.or the twelve soil layers L of our soil profile the emergence
ates per 5 days then read 0.99 for L = 1, 0.50 for L = 2, 0.18 for= 3–6, and 0 for L = 7–12.In maize 3-cm deep stale seedbeds are prepared by shallowillage that likely mixes all soil layers involved homogeneously,
ig. 4 – Number of weeds per unit area specified for four cohortsi.e. SSD), and after tillage regime deep–shallow–shallow tillage (iroportion seedling emergence per 5 days is specified per soil la
L = 7–12). The proportion mortality caused by mechanical weed cpper layers (b); most seeds in deeper layers (c).
1 ( 2 0 0 7 ) 536–546 543
resulting in equal seed densities. Since the soil profile in thiscase study consists of 12 layers, each of 0.5 cm thick, shallowtillage affects the first 6 soil layers. Shallow tillage is repre-sented by the following 12-dimensional square matrix (Ms):
Ms =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
16
16
16
16
16
16
0 0 0 0 0 0
16
16
16
16
16
16
0 0 0 0 0 0
16
16
16
16
16
16
0 0 0 0 0 0
16
16
16
16
16
16
0 0 0 0 0 0
16
16
16
16
16
16
0 0 0 0 0 0
16
16
16
16
16
16
0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 0
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(15)
0 0 0 0 0 0 0 0 0 0 0 1
The growing of maize requires a 6-cm deep seedbed. Seedbedsare prepared by deep tillage, usually rotary harrowing that
(X1–X4) after tillage regime shallow–shallow–deep tillage.e. DSS) for three seed distributions (specified in Fig. 3). Theyer L and equals 0.99 (L = 1), 0.50 (L = 2), 0.18 (L = 3–6), and 0ontrol is 0.70. Uniform seed distribution (a); most seeds in
i n g
544 e c o l o g i c a l m o d e l llikely mixes all soil layers involved homogeneously. Since the
soil profile in this case study consists of 12 layers, each of0.5 cm thick, deep tillage affects all 12 soil layers and is rep-resented by a 12-dimensional square matrix of which each ofthe 144 elements has value 1/12.Fig. 5 – Number of seeds per unit area specified per soil layer L (L(i.e. SSD) in (a)–(c), and after tillage regime deep–shallow–shallow(specified in Fig. 3). Uniform seed distribution (a and d); most seeand f). Weed seedling emergence and mortality caused by mecha
2 0 1 ( 2 0 0 7 ) 536–546
The density of weed seedlings of each cohort is calculated
after tillage regime SSD and DSS (Fig. 4). The fourth cohortcontributes most to the total number of weeds since thiscohort is not mechanically controlled. Generally, harrowingcontrols 40–70% of the weeds (Dierauer and Stoppler-Zimmer,= 1–12) after tillage regime shallow–shallow–deep tillagetillage (i.e. DSS) in (d)–(f) for three seed distributions
ds in upper layers (b and e); most seeds in deeper layers (cnical weed control during the tillage regimes is as in Fig. 4.
g 2 0
1iStot
jattssrc(
5
Odetstalds
asvtejltoipssdwwwwa(
adcl(i‘u
r
e c o l o g i c a l m o d e l l i n
994), where we used 70% as default value for seedling mortal-ty by mechanical weeding (�). Changing from tillage regimeSD to the proposed tillage regime DSS is beneficial for allhree seed distributions, showing decreases in total numberf weeds per unit area of 31, 27, and 32% (Fig. 4a–c, respec-ively).
Also, the density of seeds still present in each soil layerust before crop sowing is calculated after tillage regime SSDnd DSS (Fig. 5). With respect to soil layers 1–6, the proposedillage regime DSS is better than tillage regime SSD for allhree seed distributions. This is in terms of least number ofeeds still present in the soil after the applied tillage regime,o those seeds that may emerge after sowing of the crop. Withespect to soil layers 7–12 this is vice versa, however, P. persi-aria does not usually emerge from depths larger than 3–4 cmVleeshouwers, 1997).
. Conclusions and discussion
ur goal has been to show how tillage regimes differing inepth affect the weed population. We have used simple mod-ls and examined various sequences of shallow and deepillage from which we conclude that the application of staleeedbeds is promising under controlled traffic. This innova-ive tillage technique restricts soil compaction and thereforellows deep tillage early in the season, with subsequent shal-ow tillage before crop sowing. In the conventional system theeep tillage is required to re-compact the soil shortly beforeowing.
We did a systematic examination of seedling emergencend subsequent weed control on weed population dynamics,tarting by considering the seed bank as one soil layer. Thealues for seedling emergence (�) and mechanical weed con-rol (�) determined whether fewer or more seedlings of thearlier cohort were present after the last mechanical weedingust before sowing. Seedlings of early cohorts are potentiallyarger at maturity with higher seed productions compared tohose of later cohorts. It is shown that relatively few seedlingsf the earlier cohort were present at the time of crop sow-
ng for low proportions emergence. However, the higher theroportion emergence, the more the seed bank decreased inize, which made a high � value favourable with respect toeed density. Although early cohorts may have large seed pro-uctions in practice, a farmer can specifically control largeeed plants, e.g. by hand weeding. Whether the survivingeeds will really contribute to a marked population growthill depend on the maximum seed production per plant,hich varies between about 200 seeds per plant for Polygonum
viculare to 500,000 seeds per plant for Chenopodium albumunpublished results).
The examination continued systematically by consideringdepth-structured seed bank, where we examined how tillageepth affected the population of seedlings and seeds. Weonsidered the widely used tillage regime consisting of shal-ow tillage, again shallow tillage, and ultimately deep tillage
termed ‘regime SSD’), and proposed a tillage regime consist-ng of deep tillage followed by shallow tillage twice (termedregime DSS’). The tillage regimes were assumed to result in aniform distribution of seeds over all soil layers involved. This1 ( 2 0 0 7 ) 536–546 545
differed from the well-known ‘ploughing’ matrix (Cousens andMoss, 1990) where soil layers up to 30 cm depth are involved,and seeds are heterogeneously distributed. However, given thetillage depth and the type of harrowing, we believe that ourassumption is reasonable.
The presented model is based on a number of simplifica-tions. For example, the model assumes that � is not dependenton seedling size, although it is a fact that weed control effi-cacy decreases for larger seedlings (Kurstjens et al., 2000).The model also assumes that during the time span of thestale seedbeds, soil conditions are stable. Emergence, how-ever, may be strongly stimulated by rainfall or irrigation (Bondand Baker, 1990; Benvenuti, 2003). On the one hand thismay under- or overestimate seedling densities. On the otherhand simplicity enables biological interpretation of analyticalsolutions. It is this biological interpretation that challengesecologists using recently developed sophisticated methods toanalyse complex ecological data, e.g. with respect to inva-sive species establishment (Gevrey et al., 2006; Leung andDelaney, 2006). Therefore, we suggest that the present simpleapproach should be integrated with these more sophisticatedapproaches like self-organising maps linked to more empiricalwork (Gevrey et al., 2006)
Whether tillage regime SSD or DSS is preferred dependedon: (1) the proportion emergence specific for each soil layer;(2) the proportion of seeds that is moved from one layer tothe other; (3) the seed distribution in the soil. Information onthese aspects determines whether the effect of a stale seedbedcan be optimised. Interpretation of the algebra requires min-imal knowledge of parameter values. A case study in theNetherlands, with information on the required parameter val-ues, showed that the proposed tillage regime DSS could givehigh reductions in weed density (27–32%) by using innovativemachinery. As technology proceeds (Hamza and Anderson,2005), this regime has realistic opportunities to become widelyused.
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