Growth functions - AGRON 590 MG: Crop-Soil Modeling

IntroductionGrowth functions

Growth functionsAGRON 590 MG: Crop-Soil Modeling

Fernando E. Miguez

Iowa State University

Oct 15, 2010

Fernando E. Miguez Growth functions


Modeling Leaf-level photosynthesisExample of a non-linear function: Rectangular hyperbola

A =φQ+Asat −

√(φQ+Asat)2 − 4θφQAsat

2θ−Rd

where

φ = apparent quantum efficiency

Q = Quantum flux

Asat = saturated rate of photosynthesis

θ = curvature

Rd = dark respiration



General function

y = f (x)

where

y is the response

f is an undefined functionx is the explanatory variable



A simple functionSimple Linear Model

y = f(x)

y = a+ b× x

a = 5

b = 3

x [0, 20]

simple linear model

x

y

0

20

40

60

0 5 10 15 20



Extending the linear model

y = β0+β1x+β2x2

x

y

50

100

150

200

250

0 5 10 15 20

beta0 = 5, beta1 = 0.5, beta2 = 0.5beta0 = 5, beta1 = 15, beta2 = −0.5



A simple function and its rate of change

W = f (t)

dW

dt



Example: exponential growth

dW

dt= µW

How do we integrate this?




dW

dt= µ W

Separate variables




dW

dt= µ W

1WdW = µ dt




dW

dt= µ W

1WdW = µ dt∫

1WdW =

∫µ dt




dW

dt= µ W

1WdW = µ dt∫

1WdW =

∫µ dt

lnX = µt




dW

dt= µ W

1WdW = µ dt∫

1WdW =

∫µ dt

lnW = µ t

exp(lnW ) = exp(µ t)




dW

dt= µ W

1WdW = µ dt∫

1WdW =

∫µ dt

lnW = µ t

W = exp(µ t)




dW

dt= µ W

1WdW = µ dt∫

1WdW =

∫µ dt

lnW = µ t

W = exp(µ t)

When t = 0, W = 1 but we can add a parameter for the initialcondition

W = W0 exp(µ t)



Tools in RFor linear models

Go to R for linear models



Tools in RFor non-linear models

nls selfStart nlme



Example in RFor non-linear models

Go to R



Logistic function

y =A

1 + exp((xmid − x)/scal)

A = asymptotexmid = midpointscal = ’scale’ or spread

x

y

0

5

10

15

20

25

30

0 10 20 30 40 50 60


Growth functions - AGRON 590 MG: Crop-Soil Modeling

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