growth functions - agron 590 mg: crop-soil modeling
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IntroductionGrowth functions
Growth functionsAGRON 590 MG: Crop-Soil Modeling
Fernando E. Miguez
Iowa State University
Oct 15, 2010
Fernando E. Miguez Growth functions
IntroductionGrowth 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
Fernando E. Miguez Growth functions
IntroductionGrowth functions
General function
y = f (x)
where
y is the response
f is an undefined functionx is the explanatory variable
Fernando E. Miguez Growth functions
IntroductionGrowth functions
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
Fernando E. Miguez Growth functions
IntroductionGrowth functions
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
Fernando E. Miguez Growth functions
IntroductionGrowth functions
A simple function and its rate of change
W = f (t)
dW
dt
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Example: exponential growth
dW
dt= µW
How do we integrate this?
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Example: exponential growth
dW
dt= µ W
Separate variables
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Example: exponential growth
dW
dt= µ W
1WdW = µ dt
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Example: exponential growth
dW
dt= µ W
1WdW = µ dt∫
1WdW =
∫µ dt
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Example: exponential growth
dW
dt= µ W
1WdW = µ dt∫
1WdW =
∫µ dt
lnX = µt
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Example: exponential growth
dW
dt= µ W
1WdW = µ dt∫
1WdW =
∫µ dt
lnW = µ t
exp(lnW ) = exp(µ t)
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Example: exponential growth
dW
dt= µ W
1WdW = µ dt∫
1WdW =
∫µ dt
lnW = µ t
W = exp(µ t)
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Example: exponential growth
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)
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Tools in RFor linear models
Go to R for linear models
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Tools in RFor non-linear models
nls selfStart nlme
Fernando E. Miguez Growth functions
IntroductionGrowth functions
Example in RFor non-linear models
Go to R
Fernando E. Miguez Growth functions
IntroductionGrowth functions
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
Fernando E. Miguez Growth functions