Mortality matrices with the dynamic pool model • rFAMS

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Mortality Matrices for Dynamic Pool Models

Simulations with Dynamic Pool Models (DPM) in rFAMS (e.g., using dpmBH_minLL()) require inputs matrices that describe conditional natural and fishing mortalities for each age and year in the simulation. For example, if simulating a fish population with a maximum age (i.e., tmax) of 5 for 10 years (i.e., simyears) then a matrix is required that describe conditional natural mortality (i.e., $cm$ ) for age-0 fish in year 1, age-1 fish in year 1, age-2 fish in year 1, and so on through to age-5 fish in year 10.¹ It is unlikely and unrealistic to simulate different mortality rates for each age-year combination, so the matrix will often contain many repeated values. Below we describe some methods for entering these values into a matrix for use in DPM simulations.

The matrix required by the DPM functions in rFAMS is very specific. Each column corresponds to an age in the simulation, beginning with age-0 and ending with the maximum age.² The first column, for example, is $cm$ of age-0 fish for each year in the simulation. In contrast, each row corresponds to a year in the simulation. The first row, for example, is $cm$ for all ages for the first year of the simulation. Given these requirements, there will be tmax+1 columns and simyears rows in the matrix.

Examples

Values can be entered into a matrix in R with matrix(). The first argument to matrix() is a vector (i.e., the set) of values to be placed into the matrix. The values can be placed into the matrix by moving across the rows first and then down the columns (i.e., fill first row, then fill second row, etc.) by including byrow=TRUE. Alternatively, the values can be placed into the matrix by moving down columns first and then across rows (i.e., fill first column, then fill second column, etc.) by including byrow=FALSE. The function knows when to stop filling a row or column based on the requested number of rows in nrow= or columns in ncol=. Finally, the rows and columns can be named with a list in dimnames=, which will be demonstrated below.

For all examples below, assume that the maximum age is 5 and 10 years will be simulated with the DPM. These values are entered into constants below to facilitate working through the different examples. Note that tmax will usually be extracted from the life history parameters in lhparms with lhparms[["tmax"]] or lhparms$tmax.

tmax <- 5
simyears <- 10

Also note that all examples below are shown for conditional natural mortality ( $cm$ ). The same procedures would be used to make matrices of conditional fishing mortality ( $cf$ ) across ages and simulation years.

Example 1

As a first simple example, assume that $cm$ is a constant 0.3 for all ages and years. A vector with this value repeated for the product of tmax+1 and simyears (=60) is created to serve as the first argument to matrix().

tmp <- rep(0.3,(tmax+1)*simyears)
tmp
#>  [1] 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
#> [20] 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
#> [39] 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
#> [58] 0.3 0.3 0.3

Given that $cm$ is constant across all age-year combinations the values in this vector can be placed into the matrix either by row or by column, but is by row below. Note that the number of rows is defined by simyears.³

cm <- matrix(tmp,byrow=TRUE,nrow=simyears)
cm
#>       [,1] [,2] [,3] [,4] [,5] [,6]
#>  [1,]  0.3  0.3  0.3  0.3  0.3  0.3
#>  [2,]  0.3  0.3  0.3  0.3  0.3  0.3
#>  [3,]  0.3  0.3  0.3  0.3  0.3  0.3
#>  [4,]  0.3  0.3  0.3  0.3  0.3  0.3
#>  [5,]  0.3  0.3  0.3  0.3  0.3  0.3
#>  [6,]  0.3  0.3  0.3  0.3  0.3  0.3
#>  [7,]  0.3  0.3  0.3  0.3  0.3  0.3
#>  [8,]  0.3  0.3  0.3  0.3  0.3  0.3
#>  [9,]  0.3  0.3  0.3  0.3  0.3  0.3
#> [10,]  0.3  0.3  0.3  0.3  0.3  0.3

The matrix may be easier to read if the rows and columns are labeled as shown below with dimnames= (note that row names are given first in the list, followed by column names).⁴

cm <- matrix(tmp,byrow=TRUE,nrow=simyears,
             dimnames=list(paste0("yr_",1:simyears),paste0("age_",0:tmax)))
cm
#>       age_0 age_1 age_2 age_3 age_4 age_5
#> yr_1    0.3   0.3   0.3   0.3   0.3   0.3
#> yr_2    0.3   0.3   0.3   0.3   0.3   0.3
#> yr_3    0.3   0.3   0.3   0.3   0.3   0.3
#> yr_4    0.3   0.3   0.3   0.3   0.3   0.3
#> yr_5    0.3   0.3   0.3   0.3   0.3   0.3
#> yr_6    0.3   0.3   0.3   0.3   0.3   0.3
#> yr_7    0.3   0.3   0.3   0.3   0.3   0.3
#> yr_8    0.3   0.3   0.3   0.3   0.3   0.3
#> yr_9    0.3   0.3   0.3   0.3   0.3   0.3
#> yr_10   0.3   0.3   0.3   0.3   0.3   0.3

Example 2

Suppose that $cm$ varies by age but is constant across all years. Further suppose that the age variability is very simple in that age-0 fish have a $cm$ of 0.1 whereas all other ages are 0.2.

The $cm$ values for each age for one year are entered into a vector by combining (i.e., c()) a 0.05 for age-0 fish with a set of 0.2 values repeated (with rep()) for ages 1-tmax..

tmp <- c(0.05,rep(0.2,tmax))
tmp
#> [1] 0.05 0.20 0.20 0.20 0.20 0.20

This vector is then repeated for each simulation year for input into matrix().

tmp <- rep(tmp,simyears)
tmp
#>  [1] 0.05 0.20 0.20 0.20 0.20 0.20 0.05 0.20 0.20 0.20 0.20 0.20 0.05 0.20 0.20
#> [16] 0.20 0.20 0.20 0.05 0.20 0.20 0.20 0.20 0.20 0.05 0.20 0.20 0.20 0.20 0.20
#> [31] 0.05 0.20 0.20 0.20 0.20 0.20 0.05 0.20 0.20 0.20 0.20 0.20 0.05 0.20 0.20
#> [46] 0.20 0.20 0.20 0.05 0.20 0.20 0.20 0.20 0.20 0.05 0.20 0.20 0.20 0.20 0.20

Finally, these values are entered by row into a matrix.

cm <- matrix(tmp,byrow=TRUE,nrow=simyears,
             dimnames=list(paste0("yr_",1:simyears),paste0("age_",0:tmax)))
cm
#>       age_0 age_1 age_2 age_3 age_4 age_5
#> yr_1   0.05   0.2   0.2   0.2   0.2   0.2
#> yr_2   0.05   0.2   0.2   0.2   0.2   0.2
#> yr_3   0.05   0.2   0.2   0.2   0.2   0.2
#> yr_4   0.05   0.2   0.2   0.2   0.2   0.2
#> yr_5   0.05   0.2   0.2   0.2   0.2   0.2
#> yr_6   0.05   0.2   0.2   0.2   0.2   0.2
#> yr_7   0.05   0.2   0.2   0.2   0.2   0.2
#> yr_8   0.05   0.2   0.2   0.2   0.2   0.2
#> yr_9   0.05   0.2   0.2   0.2   0.2   0.2
#> yr_10  0.05   0.2   0.2   0.2   0.2   0.2

Example 3

Suppose that $cm$ does not vary by age, but that it does vary by year in a fairly patterned way – $cm$ is 0.2 for the first three years, 0.15 for the next four years, and 0.10 for the next three years. Here, the values for a column should be entered and then those values repeated for the other columns (as the values are constant across columns). The values for one column can either be entered directly (which is easy enough here) or by combining the results of multiple uses of rep() (based on the pattern for $cm$ described above).

tmp <- c(0.2,0.2,0.2,0.15,0.15,0.15,0.15,0.1,0.1,0.1)   # either this or next line works
tmp <- c(rep(0.2,3),rep(0.15,4),rep(0.1,3))             #   don't need both

For longer vectors such as this it may be worthwhile to make sure you have entered the required number of values.

length(tmp)==simyears    # Should be TRUE if simyears values entered
#> [1] TRUE

The values in this vector are now repeated for each age (i.e., tmax+1) for use in matrix(). Note byrow=FALSE here because the values should be placed by column first.

tmp <- rep(tmp,tmax+1)
cm <- matrix(tmp,byrow=FALSE,nrow=simyears,
             dimnames=list(paste0("yr_",1:simyears),paste0("age_",0:tmax)))
cm
#>       age_0 age_1 age_2 age_3 age_4 age_5
#> yr_1   0.20  0.20  0.20  0.20  0.20  0.20
#> yr_2   0.20  0.20  0.20  0.20  0.20  0.20
#> yr_3   0.20  0.20  0.20  0.20  0.20  0.20
#> yr_4   0.15  0.15  0.15  0.15  0.15  0.15
#> yr_5   0.15  0.15  0.15  0.15  0.15  0.15
#> yr_6   0.15  0.15  0.15  0.15  0.15  0.15
#> yr_7   0.15  0.15  0.15  0.15  0.15  0.15
#> yr_8   0.10  0.10  0.10  0.10  0.10  0.10
#> yr_9   0.10  0.10  0.10  0.10  0.10  0.10
#> yr_10  0.10  0.10  0.10  0.10  0.10  0.10

Example 4

Suppose that there is variability both by age and by year, but it is still fairly patterned - first four years are the same, next four years are the same, and last two years are the same. Here, $cm$ values by age are entered separately for the three blocks of years for which $cm$ does not change by age.

tmp1 <- c(0.1,0.20,0.20,0.15,0.15,0.10)
tmp2 <- c(0.1,0.15,0.15,0.15,0.15,0.15)
tmp3 <- c(0.1,0.15,0.15,0.10,0.10,0.10)

Each of these is then repeated by how many years are in the block where those $cm$ values don’t vary by age.

tmp1 <- rep(tmp1,4)
tmp2 <- rep(tmp2,4)
tmp3 <- rep(tmp3,2)

Those three vectors are then combined together for matrix().

tmp <- c(tmp1,tmp2,tmp3)
cm <- matrix(tmp,byrow=TRUE,nrow=simyears,
             dimnames=list(paste0("yr_",1:simyears),paste0("age_",0:tmax)))
cm
#>       age_0 age_1 age_2 age_3 age_4 age_5
#> yr_1    0.1  0.20  0.20  0.15  0.15  0.10
#> yr_2    0.1  0.20  0.20  0.15  0.15  0.10
#> yr_3    0.1  0.20  0.20  0.15  0.15  0.10
#> yr_4    0.1  0.20  0.20  0.15  0.15  0.10
#> yr_5    0.1  0.15  0.15  0.15  0.15  0.15
#> yr_6    0.1  0.15  0.15  0.15  0.15  0.15
#> yr_7    0.1  0.15  0.15  0.15  0.15  0.15
#> yr_8    0.1  0.15  0.15  0.15  0.15  0.15
#> yr_9    0.1  0.15  0.15  0.10  0.10  0.10
#> yr_10   0.1  0.15  0.15  0.10  0.10  0.10

Example 5

While it is unlikely to have information at a resolution fine enough to simulate different $cm$ for each age-year combination (with no discernible pattern), it is possible to use such information in rFAMS. Below values for each age-year combination are entered. Note that using similar digits and a new line after each age when creating the vector help with data entry as the entered vector will look similar to the desired matrix.

vals <- c(0.00,0.10,0.15,0.15,0.15,0.10,
          0.05,0.10,0.15,0.10,0.10,0.10,
          0.05,0.15,0.20,0.15,0.15,0.05,
          0.05,0.20,0.15,0.15,0.10,0.10,
          0.05,0.10,0.15,0.10,0.10,0.10,
          0.05,0.15,0.20,0.15,0.15,0.05,
          0.05,0.10,0.15,0.10,0.10,0.10,
          0.05,0.20,0.15,0.15,0.10,0.10,
          0.05,0.20,0.15,0.15,0.10,0.10,
          0.05,0.15,0.20,0.15,0.15,0.05)

It is a good idea to make sure the required number of values have been entered.

length(vals)==simyears*(tmax+1)
#> [1] TRUE

This vector is then given to matrix().

cm <- matrix(vals,byrow=TRUE,nrow=simyears,
             dimnames=list(paste0("yr_",1:simyears),paste0("age_",0:tmax)))
cm
#>       age_0 age_1 age_2 age_3 age_4 age_5
#> yr_1   0.00  0.10  0.15  0.15  0.15  0.10
#> yr_2   0.05  0.10  0.15  0.10  0.10  0.10
#> yr_3   0.05  0.15  0.20  0.15  0.15  0.05
#> yr_4   0.05  0.20  0.15  0.15  0.10  0.10
#> yr_5   0.05  0.10  0.15  0.10  0.10  0.10
#> yr_6   0.05  0.15  0.20  0.15  0.15  0.05
#> yr_7   0.05  0.10  0.15  0.10  0.10  0.10
#> yr_8   0.05  0.20  0.15  0.15  0.10  0.10
#> yr_9   0.05  0.20  0.15  0.15  0.10  0.10
#> yr_10  0.05  0.15  0.20  0.15  0.15  0.05