Long-term management of the parasitic weed Striga hermonthica: Strategy evaluation with a population model

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Crop Protection 26 (2007) 219–227 www.elsevier.com/locate/cropro

Long-term management of the parasitic weed Striga hermonthica: Strategy evaluation with a population model P.R. Westerman,1, A. van Ast, T.J. Stomph, W. van der Werf Group Crop and Weed Ecology, Wageningen University, P.O. Box 430, 6700 AK Wageningen, The Netherlands Received 14 April 2005; accepted 8 January 2006

Abstract To increase sorghum yields in areas in Africa that are heavily infested with the root parasite Striga hermonthica, crop varieties are being bred whose roots emit fewer exudates that stimulate S. hermonthica seeds to germinate. Because S. hermonthica has a persistent seedbank, it is important to anticipate the long-term effects of such breeding efforts on the seedbank dynamics. This study reports the results of analyses conducted with a population model for S. hermonthica based on existing and earlier published models and data. The essential innovation is an explicit modelling of density-dependent feedback, which was included at different points in the life cycle. Sensitivity analyses showed that density-dependence reduced the impact on the equilibrium seedbank density of life cycle parameters at stages preceding the density-dependent process. The implication is that intervention early in the parasite life cycle through, for instance, breeding for low exudate emission of the cereal host, carries the risks of maintaining or increasing S. hermonthica seedbanks, and selection for S. hermonthica populations responsive to the new varieties. Only crop varieties with very low production of germinationstimulant will be effective in the long run. The best breeding strategy is to select for crop varieties that inhibit S. hermonthica development or growth at stages later in the life cycle or that affect the parasite at multiple stages simultaneously. The most effective management strategy is to use control measures that cause a reduction in seed production, viability of newly produced seed, or seed survival in the soil, or to use a combination of measures that affect the parasites at multiple stages. Despite considerable knowledge gaps regarding the basic demography of S. hermonthica, the model proved useful in identifying points in the S. hermonthica life cycle that are of particular interest for designing intervention strategies. In-depth studies on the demography of S. hermonthica and on the location(s) of density-dependence in the parasite’s life cycle are needed. r 2006 Elsevier Ltd. All rights reserved. Keywords: Density-dependent feedback; Long-term effects; Striga hermonthica; Seedbank; Resistance

1. Introduction The obligate root parasite Striga hermonthica (Del.) Benth. is widespread in Africa and seriously endangers the production of cereal crops, such as sorghum, maize, and millet, in huge areas of the savannah zone. It is estimated that in Africa 21 million ha cropland are infested and another 44 million ha are endangered by the parasite, with damage amounting to 2.9 billion US dollars annually (Sauerborn, 1991). Crop losses are mainly caused by Corresponding author. Tel.: +1 515 294 6735; fax: +1 515 294 3163.

E-mail address: [email protected] (P.R. Westerman). Current address. Iowa State University, 2501 Agronomy Hall, Ames, IA 50011, USA. 1

0261-2194/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.cropro.2006.01.017

phytotoxic substances released by the parasite that affect crop growth shortly after attachment even at low levels of infestation (Press et al., 2001). Consequently, most of the damage to the crop occurs before S. hermonthica plants emerge above the soil surface. The clearing of emerged S. hermonthica plants by weeding or herbicides has therefore limited effect on yield (Press et al., 2001). Additional yield losses are caused by competition for assimilates, water and light between host and parasites. A pro-active approach, aiming at reduction of the seedbank, is needed. The seedbank of S. hermonthica is very persistent (Eplee and Westbrooks, 1990; Ransom and Odhiambo, 1994; Gbe`hounou et al., 1996; Van Delft et al., 1997; Van Ast et al., 2001). Farmers have a variety of options to reduce the seedbank of S. hermonthica, including hand and mechan-

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ical weeding, herbicide application, biological control, fallow, synthetic germination-stimulants, trap crops, catch crops, crop rotations, mixed cropping or intercropping with non-hosts, soil disinfectants, fumigation, solarization, or soil antagonists (Parker and Riches, 1993). Unfortunately, measures that are predominantly effective in the long-term are seldom adopted because the chief incentive to use control measures is maintaining or increasing crop yield in the current season, not food or income security in the future. In addition, the availability of equipment, materials and labour necessary to conduct control measures, the status of ownership of the land, and the level of knowledge and acceptance of methods by farmers may counteract long-term commitments. A focus on short-term control measures without attention for long-term consequences carries the risk of a gradual build-up of S. hermonthica infestation levels, eventually leading to complete crop failure (Carsky et al., 1996). Over the past decade, international research efforts have focussed on the development of Striga-resistant or tolerant varieties for enhancing crop yield. A promising breeding approach is based on reducing germination-stimulant production by crop roots, resulting in a reduction of the number of attached parasites per host plant. The prime objective of this study was to evaluate the likely consequences of such varieties on long-term parasite population dynamics. Because long-term trends are difficult to study via experimentation or observation, we used a demographic model that characterizes the parasite–host interaction and the life cycle of S. hermonthica to predict the sensitivity of seedbank density to Striga-resistant sorghum varieties and other control measures. Density-dependent feedback has to be operative in the parasite–host interaction, since crop plants cannot support infinite numbers of parasites. Fecundity of S. hermonthica is high (5000–84 000; Andrews, 1945; Liang, 1984; Smith et al., 1993), and in the case of unrestricted population growth, seedbank densities would increase to levels far beyond what is reported for S. hermonthica (190 500 seeds m2, Carsky et al., 1996; 205 200 seeds m2, Smith and Webb, 1996; 243 750 seeds m2, Van Delft et al., 1997; 882 000 seeds m2, Oswald and Ransom, 2001; 243 000 seeds m2, Sauerborn et al., 2003). It is important to know during which stage of the life cycle the densitydependent feedback takes place, because control measures affecting survival of individuals before the density-dependent feedback has eliminated the weaker competitors will be less effective in reducing population growth than those affecting post-competitive stages. Although it is common knowledge that the effectiveness of control measures is affected by the mode of action of density-dependent feedback, this has never been analysed quantitatively for S. hermonthica nor have the consequences of breeding strategies for long-term population dynamics been investigated. Density-dependent feedback can occur during any life cycle stage where host and parasite interact with each

other. Seed production is density-dependent in the parasite Orobanche crenata (Lo´pez-Granados and Garcı´ a-Torres, 1991), but apparently not so in S. hermonthica (Webb and Smith, 1996). Other experimental data supporting one or the other feedback mechanism are absent. Several authors have observed a maximum in the number of emerging parasites per host plant, which would suggest that densitydependent feedback is operating somewhere between seed germination and parasite emergence (Doggett, 1965; Smith and Webb, 1996; Webb and Smith, 1996; Van Delft et al., 1997; Van Ast et al., 2000). Population models for S. hermonthica so far have been equipped with densitydependent mechanisms at arbitrarily chosen stages in the life cycle (Kunisch et al., 1991; Smith et al., 1993; Smith and Webb, 1996). The long-term dynamics of populations, i.e., convergence to equilibrium, periodic or a-periodic oscillations, or chaotic behaviour, depend on the shape of the curve of the transition probability vs. the population density (Caswell, 2001). A concave curve for the density-dependent transition probability as a function of the number of individuals in the preceding stage usually leads to a stable equilibrium density in the long run (Mortimer et al., 1989). For S. hermonthica dynamics, it is important to consider the shape of the density-dependent function at field scale rather than at the level of an individual host plant and its associated parasites. At the field scale the responses of crop plants to locally highly variable S. hermonthica infestation levels (Van Delft et al., 1997) are averaged, and the resulting average relationship may deviate from that at the level of the individual host. The density-dependent relationship in S. hermonthica has not been characterized, neither at the level of the individual host plant nor at population level. So far, arbitrarily shaped density-dependent functions have been used in modelling analyses (Kunisch et al., 1991; Smith et al., 1993). Here, we adapted, extended, and used an existing model (Kunisch et al., 1991; Smith et al., 1993) to explore the consequences of density-dependence at various stages in the life cycle of S. hermonthica on the sensitivity of the seedbank density to variations in demographic parameters. The consequences of control measures that affect these parameters are discussed. As a first step in model development, we determined the most probable shape of the density-dependent response function at field scale. Results were used to evaluate the most probable consequences of low stimulant producing crop varieties on long-term population dynamics of S. hermonthica. 2. Model development 2.1. Basic model of S. hermonthica seedbank dynamics The model describes the seedbank dynamics of S. hermonthica at a yearly time step. We assumed continuous monocultures of sorghum. There is one crop per year and there is only one state variable: the spring seedbank density

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of viable S. hermonthica seeds in the soil, S. Intermediate processes that describe the life cycle of S. hermonthica are included to model the yearly multiplication factor. For a detailed description of the S. hermonthica life cycle the reader is referred to Kuiper (1996) and Parker and Riches (1993). The life cycle is divided into 10 steps (Fig. 1), each characterized by a transition probability, except for seed production, which is expressed as numbers of seeds per S. hermonthica plant. The processes and probabilities associated with the life cycle are: (1) conditioning and germination of S. hermonthica seeds in response to crop host roots (g), (2) spontaneous germination (Maass, 2001) and germination in response to non-host cues (n), (3) attachment (a), (4) establishment (b), (5) subsurface growth until seedling emergence (e), (6) the proportion that develops into above-ground vegetative plants (v), (7) the proportion that becomes reproductive (r), (8) seed production (s), (9) viability of newly produced seeds (l), and (10) survival of non-germinated seeds into the next season (1m; with m mortality of seeds in the seedbank). Seedbank dynamics from year t to year t+1 was modelled as (Fig. 1):

I t ¼ g a b e v r s l ð1  mÞ  S t ,  Ot ¼ ðg þ nÞ þ ð1  g  nÞ m  St , where St and St+1 are the number of viable S. hermonthica seeds in the seedbank in year t and t+1, respectively, while It denotes the newly produced seeds entering the seedbank and Ot denotes the seeds that germinate or die between year t and t+1. Non germinated seeds

IN SEEDBANK

1- g - n n

Seeds Seeds

NON HOST

g Viable seeds produced

Germinated seeds

li

CYCLE ON HOST

Seeds produced

3 M, smax

a Attached seedlings

b

s

Established parasitic seedlings

Reproductive plant

r

Vegetative plant

v 2

Parameter values were taken from the literature and are summarized in Table 1. If possible, parameter values for S. hermonthica in the presence of sorghum were used. If no estimates were available, a best guess value was assumed. In addition to the basic density-independent model, which used fixed proportions, a second model was formulated in which different parameters were entered as densitydependent variables (see below).

2.2. Determination of the shape of the density-dependent response curve at field scale Here, it was assumed that the proportion of seedling emergence depended on the number of established S. hermonthica seedlings, Nb, occurring in the same season. The shape of the function e ¼ f(Nb) at field scale was determined by using differently shaped functions for e ¼ f(Nb) at the individual host plant level in combination with spatial variability within a field. The modelling involved three hierarchical steps: 1. Seedbank dynamics beneath individual host plants was modelled as:  e ¼ emax  ðN max  N b Þ=ðN max þ cN b Þ ,

S tþ1 ¼ St þ I t  Ot ,

1- m

221

Emerged seedlings

e 1

K, emax

L, vmax

Fig. 1. Flow diagram illustrating the life cycle of Striga hermonthica, assuming a single cropping season per year. The encircled numbers indicate the point of impact of the three density-dependent variants of model 2. For explanation of the symbols, see text.

where emax is the maximum proportion of seedlings emerging, Nb the number of established parasites per host plant and Nmax the number of established parasites per host plant just resulting in zero proportion seedling emergence. The parameter c determines the shape of function e ¼ f(Nb), which can vary from convex to concave as a function of the value of c: Fconcave

for  1pco0

Flinear

for c ¼ 0

Fconvex

for c40

(modified after Kropff and Spitters, 1991). The shape parameter for density-dependent seedling emergence c was varied between 0.9 and 9 (0.9, 0.7, 0.5,0, 1, 5 and 9) (Fig. 2) and Nmax was set at 100 S. hermonthica per host plant. For all other parameters the values listed in Table 1 were used. 2. Spatial variation was included by running simulations for 10 000 individual host plants, considered a ‘field’, exposed to m seeds on average. Infestation levels per plant varied around m according to a log-normal distribution with CV ¼ 1. 3. For each of the seven density-dependent curves at the individual host plant level, the shape of the densitydependent relationship at the field scale was determined by varying the initial seedbank density, m, between 1 and 20 000 S. hermonthica seeds per host plant using 100 random values for m and plotting e versus the average number of established S. hermonthica plants per host plant, Nb.

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Table 1 Parameter description and values (rates per growing season) Parameter

Description

Dimensions

Value chosen

Range

References

1m

Prop. seed survival in the seedbank

Seed  seed1

0.4

0.3–1.0

g

Seedlings  seed1

0.1

Seedlings  seed1

0.2

No data available

a

Prop. germination due to crop roots Prop. germination due to nonhost roots Prop. attachments

Eplee and Westbrooks (1990), Ransom and Odhiambo (1994), Gbe`hounou et al. (1996), Van Delft et al. (1997), Van Ast et al. (2001) No data available

Seedlings  seedling1

0.20

0–0.15

b e

Prop. successful establishments Prop. seedling emergence

Seedlings  seedling1 Plants  seedling1

0.50 0.3

0.12–0.32

emax K

Max. prop. seedling emergence Max. emerging plants  host plant1

Plants  seedling1 Plants

1.0 50

6.6–39.8

v vmax L

Plants  plant1 Plants  plant1 Plants

0.34 1.0 20

r s

Prop. that reaches maturity Max. prop. vegetative plants Max. vegetative plants  host plant1 Prop. that produces seeds Seed production

Plants  plant1 Seed  plant1

0.76 7500

5000–84 000

smax

Max. seed production

Seed  plant1

15 000

45 000–200 000

M

Max. seed producing plants  host plant1 Prop. viable seeds

Plants

10

Seed  seed1

0.7

n

l

0.66

0.35–1.0

Dawoud et al. (1996), and calculateda No data available Doggett (1965); Van Ast et al.(2000) Doggett (1965) Doggett (1965); Smith and Webb, (1996); Webb and Smith (1996); Van Delft et al. (1997); Van Ast et al. (2000) Webb and Smith (1996) No data available No data available Calculatedb Andrews (1945); Liang (1984); Smith et al. (1993) Andrews (1945); Liang (1984); Parker and Riches, (1993) No data available Kuiper (1996); Gbe`hounou, (1998); Van Ast et al. (2001)

a Seeds germinate in an area of about 20 mm around the root tip. About 65% of these grow in the direction of the host root (Doggett, 1965; Kunisch et al., 1991). Radicles of S. hermonthica can reach a length of about 11 (10–15 mm) on agar-gel (Haussmann et al., 1996), so a maximum of 30% will reach the root. Consequently, the proportion attachments, a, is about 0.20 (¼ 0.3*0.65). b About 5% of the mature plants dies before seed production (Webb and Smith, 1996). Estimates of capsules containing seeds range from 68 (Webb and Smith, 1996) to 93% (Aigbokhan et al., 1998). Therefore, r was calculated at 0.76 (¼ 0.95*((0.93+0.68)/2)).

The calculations, done in Matlab 5.3 (Mathworks, Inc., Natick, MA, USA), showed that the density-dependent relationship e ¼ f(Nb) for a population of host plants was convex, except when the shape of the density-dependent relationship at plant level was highly concave (e.g., cp0.7) (Fig. 3). Had the degree of spatial aggregation been chosen higher (CV41) the form of the curve would have been convex in all cases. The Beverton–Holt function (see below) seems appropriate to describe the decrease in a proportion as a function of the density at field scale, and was, therefore, used in the remainder of the study. 2.3. Density-dependence at various stages in the life cycle of S. hermonthica Two models were constructed: – Model 1: the basic density-independent model, resulting in exponential growth of the seedbank density,

– Model 2: a model in which one stage was assumed density-dependent px ¼ f(Nx1), so the parameter for a stage (px) is a function of the numbers in the preceding stage (Nx1).

The transition probability of an individual from stage x to stage x+1, decreases with increasing Nx, the number of individuals in stage x, according to a Beverton–Holt function (e.g., Kunisch et al., 1991; Caswell, 2001): px ¼ pmax Y =ðY þ N x Þ, where pmax is the maximum proportion of individuals that can transfer from stage x to x+1, and pmax  Y is the maximum number that can enter stage x+1. px is at its maximum at low Nx and decreases with increasing numbers of Nx. The function was incorporated at different stages of

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– M and smax for model 2c, in which seed production was density-dependent, with smax, the maximum seed production per S. hermonthica plant, while s  Nr is the seed production per host plant.

Proportion emergence,e

1

0.8

0.4

Parameter values are listed in Table 1. The seedbank density in model 2 reaches a stable equilibrium seedbank density, S*, in the long-run, when St+1 ¼ St, or:

0.2

I t  Ot ¼ 0  ) g a b e v r s l ð1  mÞ S n  ðg þ nÞ þ ð1  g  nÞ m S n ¼ 0.

0.6

2.4. Sensitivity analysis

0 0

25

50

75

100

Number of established S.hermonthica shoots, Nb c = - 0.9;

c = -0.5;

c = 1;

c=9

c = 0;

Fig. 2. Shape of the density-dependent relationships for e as a function of Nb, describing competition among S. hermonthica at individual host plant level: e ¼ emax  {(NmaxNb)/(Nmax+cNb)}, with emax ¼ 1, and Nmax ¼ 100, for c ¼ 0.9, 0.5, 0, 1, and 9.

1.0 Proportion emergence,e

223

0.8

0.6

0.4

0.2

0.0 0

50

100

150

200

Number of established S.hermonthica shoots, Nb c = -0.9; c = 1;

c = -0.7; c = 5;

c = -0.5; c=9

c = 0;

Fig. 3. Relationship between the proportion seedling emergence, e, and the number of established S. hermonthica shoots, Nb, for host plant populations of 10 000 plants with S. hermonthica competing at seedling emergence according to a linear (c ¼ 0), convex (c40) or concave (co0) relationship. Calculations were repeated for 100 different initial values of seedbank density, m, randomly drawn from between 1 and 20 000 seeds per plant while seedbank density for each individual sorghum varied around m according to a log-normal distribution with CV ¼ 1.

the life cycle. The parameter designation of Y and pmax varied per variant: – K and emax for model 2a, in which seedling emergence was density-dependent, – L and vmax for model 2b, in which the proportion that develops into vegetative plants was density-dependent,

To compare the sensitivity of parameters measured on different scales, i.e. proportional parameters (range 0–1) vs. s (0–30 000), the relative change in S (model 1) or S* (model 2) as a result of a relative change in one of the parameters, x was used as a measure of model sensitivity: (DS/S)/(Dx/x) for model 1 and (DS*/S*)/(Dx/x) for model 2. The analytically derived solution for sensitivity yielded complex formulae, which contributed little to clarifying and understanding the relationship with individual parameters. As an alternative, an empirical approach was used, in which the sensitivity was calculated as the relative change in S after 10 years or in S* as a result of a 1%, 10% or 20% increase or decrease in one of the parameters, x, around the standard value of x. In addition, for each of the versions of model 2, S* is expressed as a function of the demographic parameters (e.g., S* ¼ f(g), S* ¼ f(a), S* ¼ f(b), S* ¼ f(e) etc.). Parameter sensitivity over a broad range of values was explored graphically by displaying S* as a function of the selected parameters. A steep slope indicates a strong response of S* to the parameter. 3. Results The effect of a 20% decrease in one of the parameter values on S is shown in Fig. 4. Ranking of the sensitivities was the same for a 1%, 10%, or 20% decrease or increase. A change in any of the parameters of model 1 had a very similar effect on S, except for n, the proportion germination induced by non-hosts. Spontaneous germination or germination induced by non-hosts is lethal. An increase in n will therefore result in a decrease in the seedbank density, and vice versa. Furthermore, S was slightly less responsive to a change in germination, g, and more responsive to seed survival (1m) than to the other parameters. As predicted, a 1%, 10%, or 20% reduction in one of the parameters of model 2 corresponding to demographic processes occurring before the density-dependent process had a small impact on S*, while a change in one of the parameters directly linked to the density-dependent process or processes occurring after the dependent process had a large, and equal, impact on S* (Fig. 4). Only the results for

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smax

v

emax

b

n

-0.5

(ΔS*/S*)/(Δx/x)

0.25 1-m

0.5 l

0.75

M

1.5

s

1.25

r

2.5

K

1.75

e

3.5

a

2.25

g

ΔS/S) /(Δx /x) (Δ

Sensitivity model 1

4.5

Sensitivity model 2

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224

- 0.25

Parameter model 1

model 2a

model 2c

Fig. 4. The sensitivity of S ((DS/S)/(Dx/x) model 1, left y-axis) or S* ((DS*/S*)/(Dx/x) model 2, right y-axis) to a 20% decrease in one of the parameters, x, compared to the standard value of x.

Model 2b

Model 2a 100000

80000

80000

60000

60000

S*

S*

100000

40000

40000

20000

20000

0

0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

Parameter value

0.6

g

n

a

g

v

r

l

vmax

r

b

emax

1-m

b

e

*

0.8

Parameter value n

1.0 a l 1-m

*

Fig. 5. The equilibrium seedbank density, S , as a function of demographic parameters in model 2a. For an explanation of parameter symbols, see Table 1.

Fig. 6. The equilibrium seedbank density, S , as a function of demographic parameters in model 2b. For an explanation of parameter symbols, see Table 1.

variants 2a and 2c are shown in Fig. 4, but the same was observed for variant 2b. Parameter sensitivity over a broad range of values was explored graphically by displaying S* as a function of selected parameters (e.g., S* ¼ f(g), S* ¼ f(a), S* ¼ f(b), S* ¼ f(e) etc.). Graphs for S* as a function of the parameters K, L, M, s and smax were not included in Figs. 5–7, because these parameters were measured on a different scale than the rest of the parameters. In all three versions of model 2 the equilibrium seedbank density, S*, showed an optimum for the proportion germination g at gE0.15, decreased gradually for g40.15, and strongly for go0.15

(Figs. 5–7). Values of go0.02 did not yield a positive equilibrium seedbank density, indicating that the S. hermonthica seedbank population went extinct. Similarly, for all other parameters of processes preceding the densitydependent process, S* responded strongly to changes in parameter values at the lower end of the scale (e.g.o0.3), while the S. hermonthica seedbank population went extinct at parameter valueso0.1. The equilibrium seedbank density, S*, responded in a linear fashion to a change in the parameter values of any of the density-dependent processes, K and emax (model 2a), L and vmax (model 2b), and M and smax (model 2c), as well as the parameters of

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Model 2c 250000

200000

S*

150000

100000

50000

0 0.0

0.2

0.4 g

0.6

Parameter value n

0.8

1.0 a

v

r

l

b

emax

1-m

Fig. 7. The equilibrium seedbank density, S*, as a function of demographic parameters in model 2c. For an explanation of parameter symbols, see Table 1.

processes occurring after the density-dependent process, except 1m. S* was more than proportionally sensitive to a decrease in seed survival in the soil, 1m. For the given set of parameter values, the S. hermonthica seedbank population will go extinct at fecundity parameters values so1100 (model 2a), so2500 (model 2b), or smaxo3300 (model 2c) seeds per reproductive S. hermonthica. 4. Discussion Reduction of germination-stimulant production by the crop, and thus the proportion of S. hermonthica seeds induced to germinate, is one of the main targets in current resistance breeding programs (Hess et al., 1992; Vogler et al., 1996; Ejeta et al., 2000; Haussmann et al., 2001). The sensitivity analyses in this study show that, although the strategy may be successful for increasing short-term crop yield, it may be a risky approach for long-term S. hermonthica control. Equilibrium seedbank densities were insensitive to changes in the proportion of seeds that germinated over most of the parameter range. The breeding strategy will only be successful in the long-term if the proportion germination of S. hermonthica seeds, g, can be reduced to very low values. The exact threshold value for the population to go extinct is unknown and depends on the values of the other parameters. For the current set of parameter values, g would not have to decrease much, as it was set at 0.1, but in reality this value may be considerably higher. Repeated use of very low germination-stimulant producing crop varieties would result in the local extinction of S. hermonthica. The use of crop varieties that allow germination in excess of the

225

threshold value for g will result in a continuation of high S. hermonthica seedbank densities. Those varieties will most likely select quickly for S. hermonthica populations more responsive to low doses of the stimulants, assuming survival and reproduction of S. hermonthica individuals that do germinate at lower exudate emission levels are not impaired. Because the exact location of density-dependent feedback in S. hermonthica is unknown, and indeed density-dependence may be operating on more than one stage, the best strategy for breeders would be to select for crop varieties that interact with S. hermonthica at stages later in the life cycle, or that affect the parasite at multiple growth stages. Obviously, this should come with a yield benefit to make cultivars interesting to farmers. The results of this study do not only have practical implications for breeders, but for all those involved in designing S. hermonthica management strategies. The effect of any control measure affecting S. hermonthica in a stage before the density-dependent stage(s) will be, in full or partially, compensated through the density-dependent mechanism itself, resulting in higher survival of or more abundant seed production per S. hermonthica individual later on in the life cycle. Consequently, a change in processes towards the end of the life cycle, will always have a larger impact on the level of the equilibrium density, S*, than a change in those at the beginning of the life cycle. The most effective control measures should be those causing a reduction in seed production, viability of newly produced seed, or seed survival in the soil. A large number of measures seems to aim at reducing seed survival, for example, the use of soil disinfectants, fumigants, synthetic germination-stimulants, below-ground biological control agents and solarization, but also traditional fallow, trap crops, catch crops, crop rotations, mixed cropping or intercropping with non-hosts (Abbasher and Sauerborn, 1992; Oswald and Ransom, 2001; Gbe`hounou and Adango, 2003; Hess and Dodo, 2004). In this context, the use of crops that induce germination of S. hermonthica seeds, but that are not hosts themselves, resulting in seed mortality is a promising strategy (Gacheru and Rao, 2001; Khan et al., 2002). In the light of the uncertainty regarding the location of the density-dependent feedback, a combination of measures affecting the parasites at multiple stages, such as in diverse crop rotations (Liebman and Dyck, 1993; Oswald and Ransom, 2001), would be a sensible approach for S. hermonthica management. The effectiveness of control measures is usually measured in terms of crop yield rather than in changes in life cycle parameters or seedbank density, and therefore the magnitude and ease with which parameters can be changed is unknown. This has practical implications, because the choice of the optimal point of impact is based not only on the impact on the seedbank, but also on a combination of sensitivity, costs, and the feasibility of a method under the given circumstances. Changes in some demographic parameter may be more feasible and easier to achieve, in terms of effort, money or genetic variation, than in others. For

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example, a large decrease in an insensitive parameter, such as germination or attachment, may be cheaper or easier to achieve than a small decrease in a sensitive parameter, such as seed production, or viability of newly produced seed. It is not clear if and how improved soil fertility, which stimulates crop growth and yield, and reduces S. hermonthica parasitism (Parker and Riches, 1993), influences demographic parameter values. Probably, the balance of power in inter-specific competition is changed in favour of the crop, but how this translates into demographic parameters is unknown. Increasing host plant density seems to do the opposite, as it increases S. hermonthica parasitism (Smith and Webb, 1996). Seed production and seed survival are probably two of the most variable parameters in the life cycle of S. hermonthica. The indeterminate growth of the parasite ensures extremely high seed production under favourable conditions. A short rainy season, early plant death or crop harvest, and (hand) weeding of mature S. hermonthica plants can strongly reduce seed production (Parker and Riches, 1993; Webb and Smith, 1996). Unfortunately, the reverse is also true: abundant seed production, or relaxing measures that would otherwise have caused seed mortality, will immediately be followed by a large increase in S. hermonthica infestation level. This may explain the devastating effects of yearly cultivation of grain crops, such as sorghum and millet, without the traditional fallow. The model used assumes parameter values that change in response to S. hermonthica density only. However, life cycle parameters may change in response to external factors, such as management and environmental conditions, as may the effectiveness of control measures. The current model can be adapted to include region- and season-specific analysis of demographic parameters by allowing parameters to change in response to, for example, temperature, soil humidity, organic matter and N content of the soil, crop rotation scheme, etc. Unfortunately, for the most part, the responses of S. hermonthica life cycle parameters to numerous external variables have not been quantified yet. This study illustrates the need for more in-depth studies on the demography of S. hermonthica, through both experimentation and modelling. Although various aspects of the biology of S. hermonthica have been studied intensely, our knowledge of the basic life cycle shows considerable gaps. Experiments that help to reveal the location and the shape of the density-dependence in S. hermonthica are needed. Although imperfect, the model investigated here proved useful in identifying stages in the S. hermonthica life cycle that are of particular interest for designing intervention strategies. Acknowledgements We thank T. van Mourik, M. Liebman, and three anonymous reviewers for their comments and suggestions. Financial support was provided by the European Commis-

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