Models

“A mathematical model simplifies the real world, identifies the critical components, environmental parameters, describes the state of knowledge and the art of management, identifies the knowledge gaps, organizes available data, determines research model”

• Impact of PPN on crop yield varies with biogeographic location, cropping sequence & intensity, cultivar selection, soil texture & nematode community structure.

• Nematodes and their damage are frequently not obvious

• Necessary to understand their biology, ecology, interaction with other organisms.

• Nematode management has economic, environmental and social cost components.

Why modelling?

Steps of modelling

1.Definition of problem and its diagnosis

2. Identification of the components

3.Quantification of relations

4.Environmental and other modifying factors

5.Validation of model

Patterns of Population Dynamics

A measure of the reproduction of a nematode species is given by the reproduction rate

RI defined as the ratio between the density of the nematode population in the soil at the end of the crop cycle the final population density (Pf), and that present at sowing or planting, usually called the initial population density (Pi).

RI = Pf/Pi (Cook and Noel, 2002).

Relation between Pf and Pi

Best model describing the relationship between initial (Pi) nematode density at planting of a host crop and final (Pf) density at harvest, proposed by

Seinhorst (1966, 1967a, b, 1970, 1986a)

Steps to derive at Pf based on Seinhorst model

Step 1

The final nematodepopulation (Pf ) on a given crop inoculated at planting with a low population density (Pi ) would be

Pf = a Pi

Where a = multiplication rate

Step 2

Pf = a Pi xy x = proportion of Pi that will affect the cropY= food availability

The nematode's final population density (Pf) cannot surpass a certain limit (ceiling) There will be an initial nematode level (Pi) at which the final nematode population (Pf) will remain unchanged (the equilibrium density or maintenance density) (E).

Therefore, at the end of a growing cycle, the maximum density that the final nematode population could reach is the ceiling.

If the proportion x of Pi that infects the host plant is less than one, at the end of the crop cycle in the soil there will be a residual portion of the inoculum at planting (Pi)

(1-x)Pi

Step 3

If, of the proportion x only an amount xy will reproduce, because of limited food availability,

(x-xy)Pi

Step 4

Step 5the proportion will not be affected by the host plant and will (theoretically) behave as in the absence of a host. Therefore, a proportion s (= that remaining in the soil in the absence of a host)

(x-xy)Pi = sx (1-y)Pi

Step 6

Pf = xyaPi E

(a-1)Pi +E+(1-x)Pi+sx(1-y)Pi

Models of Nematode Crop loss

1. Brown2. Oostenbrink3. Seinhorst 4. Elston5. Critical point model6. Multiple pest model7. Others

1. Brown model

The most simple equation describes a linear regression curve,

Brown (1969): E(Y) = a + b.P i

E(Y) is the expected yield, Pi is the nematode density before planting (numbers of encysted eggs and juveniles per unit soil, g or ml), 'a' presents the yield at density zero {E(Y)=a} and 'b' the rate of decrease in yield per unit increase of nematode density

2. Oostenbrink model

E(Y)=a + b.(log Pi)

A log linear regression curve was introduced by Oostenbrink (1966) to describe nematode damage relationships with the equation:

Seinhorst model

Seinhorst (1965) introduced the concept of a nematode threshold density below which no yield loss exists and named it the tolerance limit (T),

also defined the minimum yield (m), a value below which yield will not further decrease with increasing nematode density.

Based on this concept he developed the equation

y = m + (1 - m)zPi-T

Contd.,

E(Y) = Ymax if Pi<T

E(Y) = Ymax {m + (1-m).zPi-T} if Pi>T

Where ymax is the yield at Pi=0

Elston model

Elston et al. (1991) introduced an inverse linear regression curve to describe the relation between yield and preplanting initial nematode density, formulated as:

E(Y) = Ymax{1- (l-m)Pi/(c + Pi)}

C= rate at which increasing Pi decreses expected tuber yield

m= the minimum yield as fraction of the yield without PCN infestation

EXAMPLE AND RELATIONS

To demonstrate similarities and differences of the four equations, the corresponding curves were calculated for nematode densities from 0 to 260 eggs per g soil.

The highest yield (Ymax=yield at nematode density zero) was set at 50 tons per ha,

and the minimum yield (m) was set at 15 tons per ha, which was to be obtained in any case at an initial PCN density of 260 eggs per g soil

Four equation taking the values results in

These assumptions yield the following equations:

1.Brown: E(Y) =50-0.1346*Pi

2. Oostenbrink: E(Y) = 50 - 14.482.1og Pi

3. Seinhorst: E(Y) = 50 "{0.3+ (1 - 0.3)'0.965 Pi}

4.Elston: E(Y) =50" {1-Pi/(111.45 +Pi)]

T= 0, m=0.3 and m=0

Graphical representation of the model

Brown (1), Oostenbrink (2), Seinhorst (3) and Elston et al. (4)

Critical point model

Crop rotation results in a discontinuous system and the continuous models cannot operate.

For multispecifc communities, yield relationship experiments can be conducted with a range of nematode communities at different densities in varying sets of environmental condition

Nematode multiplication rates Pf/Pi and overwinter survivorship (Pi2/Pfl) for Meloidogyne incognita were both adequately described by negative exponential models,

y=m+(1- m)ce-bPi

The seinhorst equation y = m + (1 - m)zPi-T becomes

Where Z-T = c and Z = e-b

Parameters of a negative exponential model for overwinter survival of Meloidogyne incognita: f = acPi -b such that Pi2/Pfl is minimum

Interaction between nematodes and soilbrone plant pathogens are recognized

When the multiple pests are two or more nematode species, the densities of both can be measeured and expressed

Duncan and Ferris (1982) derived a two nematode version which was validated for M. incognita and M. javanica on cowpea

Multiple model

Multiple model

Systems involving plants infected and model of plant yield as influenced by two more than one nematode species are frequent.

y = m'+ (1-m')c’z1P11z2P12,

~ for y < 1.0 and y = 1.0

And for y > 1.0, where m' = m1 + (m2-m1)(1-y2) /[(1-y1) + (1-y2] and c' = (z1-T1+ z2-T2)/2

M. ]avanica, z = .999, m = 0.62, T = 10 (range y = 0.54-0.99); M. incognita,z= .994,m = 0.86,T = 20 (range y = =0.83-1.07).

Predicted relationship between relative dry bean yield of Vigna sinensis and Pl of Meloidogyne iavanica and M. incognita

Other models

1. y = 1-[(1-y1)+ (l-y2) • • • + (1-yn)]

Bookbinder and Bloom, 1980

Multiple-point model

2. % Loss = 5.3788 + 5.5260X2 - 0.3308X3 + 0.5019X4

Certain disease models

1. Single-point models.

% Loss = -25.53 + 27.17 1n XX= ¾ tillering stage PDI

2. Yield (t/ha) = 234.0 - 1.706XX= blight free days

3. % Loss = f (Disease-crop stage)

Romig and Calpouzos (1970)

Olofsson (1968)

Teng and Gaunt (1980)