Case Study #3: Modeling the population dynamics of the sugar beet cyst Nematode, Heterodera schachtii

Schmidt, K., Sikora, R. A., and Richter, O. 1993. Modeling the population dynamics of the sugar beet cyst nematode Heterodera schachtii. Crop Protection 12: 490-496.

Heterodera schachtii, the sugar beet cyst nematode, causes significant crop damage. Female nematodes form cysts that, when mature, are able to over-winter in the soil. These cysts consist of the protective ectoderm of the female and several eggs. As the season progresses, the eggs hatch in the female to form J1 stage juveniles which then molt once and form J2 juveniles (Figure 1). When environmental conditions are conducive, J2 juveniles leave the cyst and find a plant root to feed on. The juveniles develop through J3 and J4 stages until adult males and females are formed. Adult males leave the roots and search for females. Adult females, when fertilized, form cysts.

Figure 1: Simple model of the sugar beet cyst nematode life cycle adapted from Schmidt et al. (2003).

The number of nematodes in a field is highly dependent on two things:

temperature and

the current and previous crop species.

Relatively hot years may result in several generations of nematodes whereas cool years may only allow one generation of reproduction. Though the sugar beet nematode is somewhat generalist and can survive to some degree on many different hosts, it is greatly favored by planting sugar beets. Crop rotation may be an effective way to reduce the population of the sugar beet cyst nematode. Knowledge of previous crop rotation and temperature may be used to forecast the amount of disease in a given sugar beet field over a season (Schmidt et al. 1993).

In the model developed by Schmidt et al. (1993), probabilities are calculated for movement between each of the stages of a nematode life cycle. First of all, an important factor for determining the number of nematodes in a given year is the number of nematodes in the soil in the spring (represented by x_{n}_{+1} where n+1 is the year). The number of nematodes in the spring is dependent on the number of nematodes that have developed during the previous year (L(x_{n})) multiplied by the probability of over-wintering (p_{ov}), x_{n}_{+1}= L(x_{n})*p_{ov}. The algorithm for determining the development of the nematode population over year n is given as follows:

where x_{n}_{,i} = density of juveniles and eggs of the i-th generation of the year n; J2= developmental stage J2; J3= developmental state J2, J3, J4 in root; A= developmental stage adults; F= fertility rate; p_{h} = probability of hatching; p_{p} = probability of root penetration; p_{d} = probability of adult development; p_{surv} = survival probability in one generation; 1-p_{h} = non-hatching probability (Figure 2).

Figure 2: Model of the sugar beet cyst nematode life cycle showing parameters (see text for description of parameters) adapted from Schmidt et al. (2003).

Before the model is more thoroughly described, please note that the θ parameter is a crop-specific parameter that may take on a different value for each type of crop grown. In a more complete model set forth by the authors, θ is calculated separately for each component of the life cycle (the crop specific parameter is not necessarily equal for all components of the life cycle). Here, only one θ parameter is considered for each crop. For a list of θ values for different crops, see Table 1.

Table 1. θ value for various crops

Crop

Parameter θ

Sugar beet

1

Winter Wheat

0.475

Winter Barley

0.444

Winter Rye

0.485

Summer Barley

0.535

Summer Wheat

0.576

Oats

0.546

Beans

0.394

Peas

0.354

Maize

0.394

Potato

0.374

The following equations provide more details of the model:

p_{h,i}(θ)=c/g_{i}*(1-e^{- θ})

where c = maximum hatching rate; g_{i} = generation-dependent weighting factor for hatching;

where J2_{n}_{,i} = nematode population of the generation i; p_{p0}= root infection probability; D_{r} = critical density for J2 juveniles (crop dependent); γ = form parameter:

where J3_{n}_{,i} = nematode population of the generation i that successfully penetrate the root; p_{p0}= maximum female development probability; D_{d} = critical density nematode stages (crop dependent); λ = form parameter;

F(A)θ=F_{0}* θ.

where F_{0} = fertility rate.

Several of the values for the parameters in the above equations are given in Table 2.

Table 2. Values of parameters

Parameter

Value

p_{ov}

0.6

p_{surv}

0.9

c

0.8

g_{i}

i

p_{p0}

0.7

D_{r}

5000

γ

4

p_{d0}

0.4

D_{d}

1600

λ

2.5

F_{0}

40

Information about crop rotations can be included in this model by changing the θ value. The effect of temperature is modeled through its effect on the numbers of generations per season. A large number of generations represents an environment favorable to the nematodes.

A sugar beet/winter wheat/winter barley crop rotation is common in Rhineland, Germany. Information about nematode generations per season and the form of rotation can be used to construct a model to forecast the changes in nematode density over time.

#First, the number of seasons is established s <- 10

#Now, an array representing the population of nematodes at the # end of each season is created; each value in the array is # set to 0; x <- c(1:s) x[1:s] <- 0

#Now, an array representing the population of nematodes at the # beginning of each season is created; each value in the # array is set to 0; the starting number of nematodes for the # first generation is set to 1000 xstart <- c(1:(s+1)) xstart[1:(s+1)] <- 0 xstart[1] <- 1000

#Here, an array representing the number of generations in each # season is created; for all seasons, the number of # generations is set to 2 g <- c(1:10) g[1:s] <- 2

#The crop specific parameters are set here; notice that sb # representing sugar beet is set to 1 whereas the parameter # is less than 1 for ww (winter wheat) and wb (winter barley); # these parameters correspond to Table 1 sb <- 1 ww <- 0.475 wb <- 0.444

#An array is created representing the crop rotation for the # model; here a sugar beet/winter wheat/winter barley # rotation is represented. crop <- c(sb, ww, wb, sb, ww, wb, sb, ww, wb, sb)

#Several of the parameters used in the model are given below; # these parameters correspond to Table 2 pov <- 0.6 psurv <- 0.9 c <- 0.8 ppo <- 0.7 dr <- 5000 y <- 4 pdo <- 0.4 dd <- 1600 L <- 2.5 f <- 40

#The rest of the model is given below as a set of loops to # calculate the values in the x and xstart arrays for(j in1:s){ x[j] <- xstart[j]* (c)*(1-exp(-crop[j]))* crop[j]*ppo*exp(-((xstart[j]/(crop[j]*dr))^y))* crop[j]*pdo*exp(-((xstart[j]/(crop[j]*dd))^L))* f*crop[j]+ xstart[j]*psurv*(1-((c)*(1-exp(-crop[j])))) if(g[j]>1)for(k in1:(g[j]-1)){ x[j] <- x[j]*(c/k)*(1-exp(-crop[j]))* crop[j]*ppo*exp(-((x[j]/(crop[j]*dr))^y))* crop[j]*pdo*exp(-((x[j]/(crop[j]*dd))^L))* f*crop[j]+ x[j]*psurv*(1-((c/k)*(1-exp(-crop[j])))) }; xstart[j+1] <- x [j]* pov }

#Here code is given in order to plot the resulting change in # nematode density at the end of each season over time (x) # (Figure 3) p <- c(1:10)

plot(p, x, xlab='Year', ylab='Nematodes/100 g Soil', xlim=c(0,10), ylim=c(0,5000),type='l')

With the resulting plot

Click to enlarge.

Nematode density at the end of each season over time in a sugar beet/winter wheat/winter barley rotation with two nematode generations per year.

Different crop rotations and number of nematode generations per year give different results. Observe the effect of reducing the number of nematode generations per season (g) to one.

Nematode density at the end of each season over time in a sugar beet/winter wheat/winter barley rotation with one nematode generation per year.

Observe the effect of utilizing a crop rotation common in Lower Saxony Germany: sugar beet/winter wheat/winter wheat/winter wheat. Here, the number of nematode generations per year (g) is set to one.

Nematode density at the end of each season over time in a sugar beet/winter wheat/winter wheat/winter wheat rotation with one nematode generation per year.

Suggested Exercise

There are numerous parameters in this model for which changes may have important effects on the model predictions. One of particular interest would be the initial nematode population. Throughout the example, the initial population was kept at 1000. What happens to the nematode population in each of the scenarios above if you change xstart[1] to 500? to 2000? to 5000? Examine each of the different situations with the new starting values.

In conclusion, information on both crop rotation and temperature (through its effect on the number of nematode generations in a season) can be used to model and forecast plant disease. In this model, land managers can input readily available temperature and crop information into a model that predicts nematode density. Thus, management decisions, such as whether or not to use a nematicide or whether or not to grow sugar beets in a particular year, can be made without constantly monitoring the nematode populations throughout fields.