Computations for growth models

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Computations for growth models
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Plotting exponential growth/decline curve

• Matlab commands
• clear all
• figures for exponential growth/decline
• tim00.120
• r0.1
• N_0100
• NN_0exp(rtim)
• plot(tim,N,'k')

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Using Matlab function ode45 for computing
solution to logistic equation

• Matlab commands
• clear all
• environmental capacity
• K500
• Initial population size
• N01
• growth rate
• r0.1
• time scale
• tim00.1100

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• analytic solution to logistic differential
  equaiton
• for ki1length(tim)
• N(ki)K/(1(K/N0-1)exp(-rtim(ki)))
• end
• Now we compute the same using ode45
• and the function 'lgf' which defines logistic
  model
• Tode,Nodeode45('lgf',0 100,1)
• compare two curves
• plot(tim,N,Tode,Node,'or')

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• Matlab function lgf used in the above procedure
  ode45
• function lgf
• function dNlgf(T,N)
• r0.1
• K500
• dNrN(1-N/K)

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Simulink block diagram for logistic model with
delay in environmental limiting action
7
1
s
Integrator
Scope
0.5
Gain
Product
simout
To Workspace
-K-
Transport
Gain1
Delay
1
Constant
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Fitting model parameters to data

• Method of least squares, find (estimate) X

Size(B) n x 1, Size(A) n x k, Size(X) k x
1, n gt k
T matrix transposition
Matlab function to compute X XA\B
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Data on world population

• http//www.census.gov/ipc/www/world.html

10
Fitting exponential model to data on world
population - 1

• Data written in Matlab format
• clear all
• popdt-10000 1 -8000 5 -6500
  5 -5000 5
• -4000 7 -3000 14 -2000
  27 -1000 50
• -500 100 -400 162 -200
  150 1 170
• 200 190 400 190 500
  190 600 200
• 700 207 800 220 900
  226 1000 254
• 1100 301 1200 360 1250
  400 1300 360
• 1340 443 1400 350 1500
  425 1600 545
• 1650 470 1700 600 1750
  629 1800 813
• 1850 1128 1900 1550 1910
  1750 1920 1860
• 1930 2070 1940 2300 1950
  2400
• timpopdt(,1)
• populpopdt(,2)

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• Atim ones(size(tim))
• Blog(popul)
• our estimated parameters
• PARA\B
• now let us see how our estimates compare to
  data
• for kk1length(B)
• ppl(kk)PAR(1)tim(kk)PAR(2)
• end
• plot(tim,ppl,tim,log(popul),'or')
• pause
• or
• plot(tim,exp(ppl),tim,popul,'or')

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Fitting exponential model to data on world
population - 2

• Use of Matlab function fmins to minimize over
  arguments
• Matlab commands
• find parameters estimates
• QPARfmins('wp_fun',0.0006 5)
• now let us see how our estimates compare to
  data
• for kk1length(tim)
• qppl(kk)exp(QPAR(1)tim(kk)QPAR(2))
• end
• plot(tim,log(qppl),tim,log(popul),'or')
• pause
• or
• plot(tim,qppl,tim,popul,'or')

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More data

• USA population 1790 2000
• http//fisher.lib.virginia.edu/census/
• Chinese children growth charts
• http//www.fwcc.org/growthchart.html