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ESCI 433/533  tree succession example   1/18/2017





> mat.tree <- matrix(c(0.05, 0.20, 0.25, 0.50,

+                        0.05, 0.3, 0.50, 0.15,

+                        0.10, 0.1, 0.45, 0.35,

+                        0.0, 0.18, 0.22, 0.6),

+               byrow=T, nrow=4, ncol=4)

> mat.tree

     [,1] [,2] [,3] [,4]

[1,] 0.05 0.20 0.25 0.50

[2,] 0.05 0.30 0.50 0.15

[3,] 0.10 0.10 0.45 0.35

[4,] 0.00 0.18 0.22 0.60



> # Enter current numbers of canopy trees 

> gen1 <- c(100, 20, 60, 24)



> # Calculate abundances in next generation as Matrix * Vector(current abundances)

> gen2 <- mat.tree %*% gen1

> gen2

     [,1]

[1,] 36.0

[2,] 44.6

[3,] 47.4

[4,] 31.2



> # Third generation abundances = Matrix * 2nd generation abundances

> gen3 <- mat.tree %*% gen2

> gen3

       [,1]

[1,] 38.170

[2,] 43.560

[3,] 40.310

[4,] 37.176



> # Repeat until proportions do not change



> # Alternatively, calculate steady state proportions directly:

> climax <- eigen(mat.tree)

> climax$vectors

       [,1]                  [,2]                  [,3]           [,4]

[1,] 0.5+0i -0.0459667-0.2354884i -0.0459667+0.2354884i -0.84023777+0i

[2,] 0.5+0i  0.8561036+0.0000000i  0.8561036+0.0000000i -0.45100051+0i

[3,] 0.5+0i -0.1436178+0.3004680i -0.1436178-0.3004680i  0.29971029+0i

[4,] 0.5+0i -0.2240353-0.2200454i -0.2240353+0.2200454i  0.02779865+0i



> # Only first column is ecologically relevant (no imaginary trees)

> climax$vectors[,1]

[1] 0.5+0i 0.5+0i 0.5+0i 0.5+0i



> # Steady state fractions = squared eigenvector components

> (climax$vectors[,1])^2

[1] 0.25+0i 0.25+0i 0.25+0i 0.25+0i

> 



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



# Commands without R prompts and results:



mat.tree <- matrix(c(0.05, 0.20, 0.25, 0.50,

                       0.05, 0.3, 0.50, 0.15,

                       0.10, 0.1, 0.45, 0.35,

                       0.0, 0.18, 0.22, 0.6),

              byrow=T, nrow=4, ncol=4)

mat.tree



# Enter current numbers of canopy trees 

gen1 <- c(100, 20, 60, 24)



# Calculate abundances in next generation as Matrix * Vector(current abundances)

gen2 <- mat.tree %*% gen1



# Third generation abundances = Matrix * 2nd generation abundances

gen3 <- mat.tree %*% gen2



# Repeat until proportions do not change



# Alternatively, calculate steady state proportions directly:

climax <- eigen(mat.tree)



# Only first column is ecologically relevant (no imaginary trees)

climax$vectors[,1]



# Steady state fractions = squared eigenvector components

(climax$vectors[,1])^2