griz.3zone.mat.txt

				McLaughlin  rev 2/1/2017



Metapopulation matrix for set of three Grizzly Recovery Zones,

Northern Continental Divide (NCDE), Greater Yellowstone (GYE),

and Selway-Bitterroot (SB)



Metapopulation matrix from Hanski & Ovaskainen 2000. Nature 404:755-758.



Matrix M has elements for every pair of Recovery Zones; 



	m_ii = 0	

	m_ij = A_i*A_j exp(-alpha*d_ij)    (i<>j)

	

	where A_i = area of patch i

	      1/alpha = mean dispersal distance

	      d_ij = distance betw/ patches i and j



Assume mean dispersal distance = 20 km, (males =29.9km, females=9.8km)

Dispersal data from: 

	McLellan BN and FW Hovey 2001. Natal dispersal of grizzly bears.

	   Can.J.Zool. 79(5)838-844.



# ~~~~~~~~~~~~~~~~~~~~~~ R commands only ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



griz.zones.3name <- c("NCDE","GYE","Sel-Broot")

griz.zones.3name

griz.zones.3area <- c(24864, 24645, 14027)	# area in km^2

griz.zones.3area



# create metapop matrix elements, m_ij

alpha.griz <- 1/20	# mean dispersal distance = 20km

alpha.griz

 

gm.ncd.gye <- 24864*24645*exp(-1*alpha.griz*155)

gm.ncd.gye



gm.ncd.sb <- 24864*14027*exp(-1*alpha.griz*61)

gm.ncd.sb



gm.gye.sb <- 24645*14027*exp(-1*alpha.griz*200)

gm.gye.sb

 

# Assemble individual matrix elements, m_ij, into metapopulation matrix, M:



griz.mat.3zone <- matrix(c(0, gm.ncd.gye, gm.ncd.sb,

                           gm.ncd.gye, 0, gm.gye.sb,

                           gm.ncd.sb, gm.gye.sb, 0),

                         byrow=T, nrow=3, ncol=3)

griz.mat.3zone



# Calculate eigenvalues and eigenvectors:



griz.LM <- eigen(griz.mat.3zone)

griz.LM

# Metapopulation potential, lambda_M = dominant eigenvalue = 1.651961e+07



# Fractional (out of 1) contribution of NCDE, GYE, SB 

#	= square of 1st element of leading eigenvector

#    Calculate for all 3 grizzly zones 

griz.LM$vectors[,1]^2





# ~~~~~~~~~~~~~~~~ Results using commands above ~~~~~~~~~~~~~~~

> griz.zones.3name <- c("NCDE","GYE","Sel-Broot")

> griz.zones.3name

[1] "NCDE"      "GYE"       "Sel-Broot"

> griz.zones.3area <- c(24864, 24645, 14027)# area in km^2

> griz.zones.3area

[1] 24864 24645 14027

> 

> # create metapop matrix elements, m_ij

> alpha.griz <- 1/20# mean dispersal distance = 20km

> alpha.griz

[1] 0.05

>  

> gm.ncd.gye <- 24864*24645*exp(-1*alpha.griz*155)

> gm.ncd.gye

[1] 263947.5

> 

> gm.ncd.sb <- 24864*14027*exp(-1*alpha.griz*61)

> gm.ncd.sb

[1] 16517246

> 

> gm.gye.sb <- 24645*14027*exp(-1*alpha.griz*200)

> gm.gye.sb

[1] 15694.55

>  

> # Assemble individual matrix elements, m_ij, into metapopulation matrix, M:

> 

> griz.mat.3zone <- matrix(c(0, gm.ncd.gye, gm.ncd.sb,

+                            gm.ncd.gye, 0, gm.gye.sb,

+                            gm.ncd.sb, gm.gye.sb, 0),

+                          byrow=T, nrow=3, ncol=3)

> griz.mat.3zone

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

[1,]        0.0 263947.52 16517245.52

[2,]   263947.5      0.00    15694.55

[3,] 16517245.5  15694.55        0.00

> 

> # Calculate eigenvalues and eigenvectors:

> 

> griz.LM <- eigen(griz.mat.3zone)

> griz.LM

$values

[1]  1.651961e+07 -5.014729e+02 -1.651911e+07



$vectors

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

[1,] -0.70710110  0.0009495847  0.70711183

[2,] -0.01196964 -0.9998718920 -0.01062672

[3,] -0.70701115  0.0159780412 -0.70702188



> # Metapopulation potential, lambda_M = dominant eigenvalue = 1.651961e+07

> 

> # Fractional (out of 1) contribution of NCDE, GYE, SB 

> #= square of 1st element of leading eigenvector

> #    Calculate for all 3 grizzly zones 

> griz.LM$vectors[,1]^2

[1] 0.4999919600 0.0001432723 0.4998647677



> # NCDE = 50.00%, GYE = 0.01%, SB = 49.98%