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+# Matrix operators for population models
+# place in R 'Resources/R/' folder
+
+## maximum lambda function
+max.lambda <- function(x) Re((eigen(x)$values)[1]) ## where 'x' is a Leslie matrix
+
+## Maximum r function
+max.r <- function(x) log(Re((eigen(x)$values)[1])) ## where 'x' is a Leslie matrix
+
+## Stable stage distribution
+stable.stage.dist <- function(x) ((x %*% (Re((eigen(x)$vectors)[,1])))/(sum((x %*% (Re((eigen(x)$vectors)[,1]))))))[,1]
+
+## Generation length function
+R.val <- function(X,age.max) ## reproductive value (R0) where X = Leslie matrix; age.max = maximum age of females
+{		
+		## define the transition matrix
+		T <- X[1:age.max,1:age.max]
+		T[1,1:(age.max)] <- 0
+
+		## define the fertility matrix
+		F <- X[1:age.max,1:age.max]
+		diag(F[2:age.max,1:(age.max-1)]) <- 0
+
+		## define the identity matrix
+		I <- matrix(data<-0,nrow<-age.max,ncol<-age.max)
+		diag(I) <- 1
+
+		## define the fundamental matrix
+		library(MASS)
+		N.fund <- ginv(I-T)
+
+		## define the reproductive matrix
+		R <- F %*% N.fund
+
+		## define R0 (number of female offspring produced per female during lifetime)
+		R0 <- Re((eigen(R)$values)[1])
+		
+		## output
+		print("number of female offspring produced per female during its lifetime")
+		print("_________________________________________________________________")
+		print(R0)
+
+}
+
+## Mean generation time function
+G.val <- function (X,age.max) ## where X is a Leslie Matrix
+{	
+		G <- (log(R.val(X,age.max)))/(log(Re((eigen(X)$values)[1])))
+		print("mean generation time")
+		print("____________________")
+		print(G)
+}