Computational Ecology: Population Viability Analysis 9/15/2020

Data: Figure 5 (plus 10 years)

Make data vectors, calculate lambda, and put together dataframe with all necessary data.

census

The census period; an index from 1 to 39 of how many years of data have been collected.

census <- 1:39

year t

The year: 1959 to 1997 (Dennis et al use 1959-1987)

year.t   <- 1959:1997

Population size

Population size is recorded as the number of females with …

females.N <- c(44,47,46,44,46,
               45,46,40,39,39,
               42,39,41,40,33,
               36,34,39,35,34,
               38,36,37,41,39,
               51,47,57,48,60,
               65,74,69,65,57,
               70,81,99,99)

Population growth rate: example

Population growth rate is…

Enter the population size for each year

females.N.1959 <- 44
females.N.1960 <- 47

Calculate the ratio of the 2 population sizes

lambda.59_60 <- females.N.1960/females.N.1959

Access the population sizes by using bracket notation rather than hard coding

# Access the data
females.N[1]
#> [1] 44
females.N[2]
#> [1] 47

# store in objects
females.N.1959 <- females.N[1]
females.N.1960 <- females.N[2]

# confirm the output
females.N.1960/females.N.1959
#> [1] 1.068182

Calculate lambda using bracket notation

lambda.59_60 <- females.N[2]/females.N[1]

The first year of data is 1959. What is lambda for 1958 to 1959?

females.N[1]
#> [1] 44
lambda.58_59 <- females.N[1]/females.N[ ] #can't be done; there is no date for 1958

Population growth rate: vectorized

Calculate lambda from 1958 to 1959

# 2nd and 3rd year
females.N[2:3]
#> [1] 47 46
# 1st and 2nd year
females.N[1:2]
#> [1] 44 47
# lambda from 1958 to 1959
females.N[2:3]/females.N[1:2]
#> [1] 1.0681818 0.9787234

This is similar t the previous code chunk, just using all of the data (no need to describe)

length(females.N)
#> [1] 39
females.N[2:39]/females.N[1:38]
#>  [1] 1.0681818 0.9787234 0.9565217 1.0454545 0.9782609 1.0222222 0.8695652
#>  [8] 0.9750000 1.0000000 1.0769231 0.9285714 1.0512821 0.9756098 0.8250000
#> [15] 1.0909091 0.9444444 1.1470588 0.8974359 0.9714286 1.1176471 0.9473684
#> [22] 1.0277778 1.1081081 0.9512195 1.3076923 0.9215686 1.2127660 0.8421053
#> [29] 1.2500000 1.0833333 1.1384615 0.9324324 0.9420290 0.8769231 1.2280702
#> [36] 1.1571429 1.2222222 1.0000000

TASK What does this do? Briefly describe in 1 to 2 sentences why I am using length().

Calculate all lambda, using length() to avoid hard coding

len <- length(females.N)
females.N[2:len]/females.N[1:len-1]
#>  [1] 1.0681818 0.9787234 0.9565217 1.0454545 0.9782609 1.0222222 0.8695652
#>  [8] 0.9750000 1.0000000 1.0769231 0.9285714 1.0512821 0.9756098 0.8250000
#> [15] 1.0909091 0.9444444 1.1470588 0.8974359 0.9714286 1.1176471 0.9473684
#> [22] 1.0277778 1.1081081 0.9512195 1.3076923 0.9215686 1.2127660 0.8421053
#> [29] 1.2500000 1.0833333 1.1384615 0.9324324 0.9420290 0.8769231 1.2280702
#> [36] 1.1571429 1.2222222 1.0000000

TASK What does this do? Briefly describe in 1 to 2 sentences what is different about this code chunk from the previous one.

Work the same as the previous one, this one is to calculate all the lambda. The difference is that instead of assigning the length to a vector, this chunk combines the code in one line

females.N[2:length(females.N)]/females.N[1:length(females.N)-1]
#>  [1] 1.0681818 0.9787234 0.9565217 1.0454545 0.9782609 1.0222222 0.8695652
#>  [8] 0.9750000 1.0000000 1.0769231 0.9285714 1.0512821 0.9756098 0.8250000
#> [15] 1.0909091 0.9444444 1.1470588 0.8974359 0.9714286 1.1176471 0.9473684
#> [22] 1.0277778 1.1081081 0.9512195 1.3076923 0.9215686 1.2127660 0.8421053
#> [29] 1.2500000 1.0833333 1.1384615 0.9324324 0.9420290 0.8769231 1.2280702
#> [36] 1.1571429 1.2222222 1.0000000

Negative indexing

Make a short vector to play with; first 10 years

females.N[]
#>  [1] 44 47 46 44 46 45 46 40 39 39 42 39 41 40 33 36 34 39 35 34 38 36 37 41 39
#> [26] 51 47 57 48 60 65 74 69 65 57 70 81 99 99
females.Ntemp <- females.N[]

Check - are there 10 numbers

females.Ntemp
#>  [1] 44 47 46 44 46 45 46 40 39 39 42 39 41 40 33 36 34 39 35 34 38 36 37 41 39
#> [26] 51 47 57 48 60 65 74 69 65 57 70 81 99 99

TASK What does this do? Briefly describe what the [-1] is doing.

Drop the first element of females.Ntemp

females.Ntemp[-1]
#>  [1] 47 46 44 46 45 46 40 39 39 42 39 41 40 33 36 34 39 35 34 38 36 37 41 39 51
#> [26] 47 57 48 60 65 74 69 65 57 70 81 99 99

TASK How many lambdas can I calculate using the first 10 years of data?

9 lamdas

females.Ntemp[2:10]/females.Ntemp[1:9]
#> [1] 1.0681818 0.9787234 0.9565217 1.0454545 0.9782609 1.0222222 0.8695652
#> [8] 0.9750000 1.0000000

“Negative indexing” allows you to drop a specific element from a vector.

TASK Drop the the first element

females.Ntemp[-1]
#>  [1] 47 46 44 46 45 46 40 39 39 42 39 41 40 33 36 34 39 35 34 38 36 37 41 39 51
#> [26] 47 57 48 60 65 74 69 65 57 70 81 99 99

TASK Drop the second element

females.Ntemp[-2]
#>  [1] 44 46 44 46 45 46 40 39 39 42 39 41 40 33 36 34 39 35 34 38 36 37 41 39 51
#> [26] 47 57 48 60 65 74 69 65 57 70 81 99 99

TASK

How do you drop the 10th element? Type in the code below.

females.Ntemp[-10]
#>  [1] 44 47 46 44 46 45 46 40 39 42 39 41 40 33 36 34 39 35 34 38 36 37 41 39 51
#> [26] 47 57 48 60 65 74 69 65 57 70 81 99 99

TASK How do you access the last element? Do this in a general way without hard-coding.

females.Ntemp[length(females.Ntemp)]
#> [1] 99

TASK How do DROP the last element? Do this in a general way without hard-coding. By general, I mean in a way that if the length of the vector females.Ntemp changed the code would still drop the correct element.

females.Ntemp[-length(females.Ntemp)]
#>  [1] 44 47 46 44 46 45 46 40 39 39 42 39 41 40 33 36 34 39 35 34 38 36 37 41 39
#> [26] 51 47 57 48 60 65 74 69 65 57 70 81 99

TASK Calculate the first 9 lambdas.

#44 47 46 44 46 45 46 40 39 39 

#44 47 46 44 46 45 46 40 39 39 
#   44 47 46 44 46 45 46 40 39 39 

#47 46 44 46 45 46 40 39 39 numerator
#44 47 46 44 46 45 46 40 39 denominator
lambda.i <- females.Ntemp[-1]/females.Ntemp[-10]

Converting between these 2 code chunks would be a good test question : )

lambda.i <- females.Ntemp[-1]/females.Ntemp[-length(females.Ntemp)]

Calcualte lambdas for all data

TASK

Below each bulleted line describe what the parts of the code do. Run the code to test it.

What does females.N[-1] do? Drop the first element of females.N
What does females.N[-length(females.N)? Drop the last element of females.N

TASK Calculate lambdas for all of the data

females.N[-1]
#>  [1] 47 46 44 46 45 46 40 39 39 42 39 41 40 33 36 34 39 35 34 38 36 37 41 39 51
#> [26] 47 57 48 60 65 74 69 65 57 70 81 99 99
females.N[-length(females.N)]
#>  [1] 44 47 46 44 46 45 46 40 39 39 42 39 41 40 33 36 34 39 35 34 38 36 37 41 39
#> [26] 51 47 57 48 60 65 74 69 65 57 70 81 99

lambda.i <- females.N[-1]/females.N[-length(females.N)]

Finish putting together dataframe

Create special columns

TASK

What does this code do? Why do I include NA in the code? (I didn’t cover this in lecture, so just type 1 line - your best guess. “I don’t know” is fine.)

Drop the last element in lambda.i??

lambda.i <- c(lambda.i,NA)

TASK

Check the help file; what type of log does log() calculate (I forgot to put this question on the test!)

natural log

lambda_log <- log(lambda.i)

Assemble the dataframe

bear_N <- data.frame(census,
                year.t,
                females.N,
                lambda.i, 
                lambda_log)

TASK

List 3 functions that allow you to examine this dataframe.

1.head() 2.tail() 3.summary()

Examing the population growth rates

Plotting the raw data

TASK

Plot a time series graph of the number of bears (y) versus time (x)
Label the y axis “Population index (females + cubs)”
Label the x axis “Year”
Change the plot to type = “b” so that both points and dots are shown.

plot(females.N ~ year.t, data = bear_N, type = "b")

plot(females.N ~ year.t, data = bear_N, 
     type = "b",
     ylab = "Population index (females + cubs)",
     xlab = "Year")
abline(v = 1970)
abline(v = 1987, col = "red")

How do we determine if a population is likely to go extinct?

To model population dynamics we randomly pull population growth rates out of a hat.

hist(bear_N$lambda.i)

hat_of_lambdas <- bear_N$lambda.i

is.na(hat_of_lambdas)
#>  [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
#> [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
#> [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
#> [37] FALSE FALSE  TRUE

any(is.na(hat_of_lambdas) == TRUE)
#> [1] TRUE

Drop the NA

length(hat_of_lambdas)
#> [1] 39
hat_of_lambdas[39]
#> [1] NA
hat_of_lambdas[-39]
#>  [1] 1.0681818 0.9787234 0.9565217 1.0454545 0.9782609 1.0222222 0.8695652
#>  [8] 0.9750000 1.0000000 1.0769231 0.9285714 1.0512821 0.9756098 0.8250000
#> [15] 1.0909091 0.9444444 1.1470588 0.8974359 0.9714286 1.1176471 0.9473684
#> [22] 1.0277778 1.1081081 0.9512195 1.3076923 0.9215686 1.2127660 0.8421053
#> [29] 1.2500000 1.0833333 1.1384615 0.9324324 0.9420290 0.8769231 1.2280702
#> [36] 1.1571429 1.2222222 1.0000000
hat_of_lambdas[-length(hat_of_lambdas)]
#>  [1] 1.0681818 0.9787234 0.9565217 1.0454545 0.9782609 1.0222222 0.8695652
#>  [8] 0.9750000 1.0000000 1.0769231 0.9285714 1.0512821 0.9756098 0.8250000
#> [15] 1.0909091 0.9444444 1.1470588 0.8974359 0.9714286 1.1176471 0.9473684
#> [22] 1.0277778 1.1081081 0.9512195 1.3076923 0.9215686 1.2127660 0.8421053
#> [29] 1.2500000 1.0833333 1.1384615 0.9324324 0.9420290 0.8769231 1.2280702
#> [36] 1.1571429 1.2222222 1.0000000

na.omit(hat_of_lambdas)
#>  [1] 1.0681818 0.9787234 0.9565217 1.0454545 0.9782609 1.0222222 0.8695652
#>  [8] 0.9750000 1.0000000 1.0769231 0.9285714 1.0512821 0.9756098 0.8250000
#> [15] 1.0909091 0.9444444 1.1470588 0.8974359 0.9714286 1.1176471 0.9473684
#> [22] 1.0277778 1.1081081 0.9512195 1.3076923 0.9215686 1.2127660 0.8421053
#> [29] 1.2500000 1.0833333 1.1384615 0.9324324 0.9420290 0.8769231 1.2280702
#> [36] 1.1571429 1.2222222 1.0000000
#> attr(,"na.action")
#> [1] 39
#> attr(,"class")
#> [1] "omit"

hat_of_lambdas <- hat_of_lambdas[-length(hat_of_lambdas)]

hist(hat_of_lambdas)

Random Sampling of Lambda

Pull a random lambda from hat_of_lambda and same it to an object lambda_rand.t

# pulled a lambda from the hat
sample(x = hat_of_lambdas, 
       size = 1,        #size = ...
       replace = TRUE)  #replace = TRUE aka recycle = TRUE
#> [1] 1.212766

# pull lambda from hat and save to object
lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)

Get initial population size

Get the first and last 6 lines, the summary, and the dimension of the dataframe bear_N

head(bear_N)
#>   census year.t females.N  lambda.i  lambda_log
#> 1      1   1959        44 1.0681818  0.06595797
#> 2      2   1960        47 0.9787234 -0.02150621
#> 3      3   1961        46 0.9565217 -0.04445176
#> 4      4   1962        44 1.0454545  0.04445176
#> 5      5   1963        46 0.9782609 -0.02197891
#> 6      6   1964        45 1.0222222  0.02197891
tail(bear_N)
#>    census year.t females.N  lambda.i lambda_log
#> 34     34   1992        65 0.8769231 -0.1313360
#> 35     35   1993        57 1.2280702  0.2054440
#> 36     36   1994        70 1.1571429  0.1459539
#> 37     37   1995        81 1.2222222  0.2006707
#> 38     38   1996        99 1.0000000  0.0000000
#> 39     39   1997        99        NA         NA
summary(bear_N)
#>      census         year.t       females.N        lambda.i     
#>  Min.   : 1.0   Min.   :1959   Min.   :33.00   Min.   :0.8250  
#>  1st Qu.:10.5   1st Qu.:1968   1st Qu.:39.00   1st Qu.:0.9452  
#>  Median :20.0   Median :1978   Median :44.00   Median :1.0000  
#>  Mean   :20.0   Mean   :1978   Mean   :49.79   Mean   :1.0281  
#>  3rd Qu.:29.5   3rd Qu.:1988   3rd Qu.:57.00   3rd Qu.:1.1038  
#>  Max.   :39.0   Max.   :1997   Max.   :99.00   Max.   :1.3077  
#>                                                NA's   :1       
#>    lambda_log      
#>  Min.   :-0.19237  
#>  1st Qu.:-0.05639  
#>  Median : 0.00000  
#>  Mean   : 0.02134  
#>  3rd Qu.: 0.09874  
#>  Max.   : 0.26826  
#>  NA's   :1
dim(bear_N)
#> [1] 39  5

N.1997 <- 99

One run of population simulation

pull a ramdom lambda from the hat and assign it to N.1998

1.22807*99
#> [1] 121.5789
lambda_rand.t*N.1997
#> [1] 86.81538
N.1998 <- lambda_rand.t*N.1997

Simulation the hard way

pull 8 lambdas without using a for loop

#1997 to 1998
lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
N.1998 <- lambda_rand.t*N.1997

#1998 to 1999
lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
N.1999 <- lambda_rand.t*N.1998

#1999 to 2000
lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
N.2000 <- lambda_rand.t*N.1999

#2000 to 2001
lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
N.2001 <- lambda_rand.t*N.2000

#2001 to 2002
lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
N.2002 <- lambda_rand.t*N.2001

#2002 to 2003
lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
N.2003 <- lambda_rand.t*N.2002

#2003 to 2004
lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
N.2004 <- lambda_rand.t*N.2003

#2004 to 2005
lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
N.2005 <- lambda_rand.t*N.2004

Plot population change

plot the population from 1997 to 2004

year <- seq(1997, 2004)
N.rand <- c(N.1998,N.1999,N.2000,N.2001,N.2002,N.2003,N.2004,N.2005)
df.rand <- data.frame(N.rand, year)
plot(N.rand ~ year, data = df.rand, type = "b")

Predict the Population Change in 50 Years

Plot the populations manually, which means we need to chagne the value of t for 50 times

# Initial conditions

N.1997 <- 99
N.initial <- 99

# Explore xlim argument
plot(N.1997 ~ c(1997))

plot(N.1997 ~ c(1997), 
     xlim = c(1997, 1997+50))


#xlim and ylim
plot(N.1997 ~ c(1997), 
     xlim = c(1997, 1997+50), 
     ylim = c(0, 550))

# for loop the hard way
#
N.current <- N.initial

# this is where the for loop would be
t <- 1
  
  # grab a lambda
  lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
  
  # determine population size
  N.t <- N.current*lambda_rand.t
  
  # update year.t
  year.t <- 1997+t
  
  # plot the population
  # points() updates an existing graph
  points(N.t ~ year.t)

  
  # update n.current
  N.current <- N.t

plot the populations using for loop


# make a new plot with starting pop size
plot(N.1997 ~ c(1997), xlim = c(1997, 1997+50), ylim = c(0, 550))

#start at 1997s pop size
N.current <- N.1997

# start of for loop
for(t in 1:50){
  
  # grab a lambda
  lambda_rand.t <- sample(x = hat_of_lambdas, 
                          size = 1,
                          replace = TRUE)
  
  # determine population size
  N.t <- N.current*lambda_rand.t
  
  # update year.t
  year.t <- 1997+t
  
  # points update in existing graph
  points(N.t ~ year.t)
  
  # update N.current
  N.current <- N.t
}

goofy r plotting code

par(mfrow = c(3,3), 
    mar = c(1,1,1,1))

**simulate population change in 50 years, but this time nine graphs are put together in a 3*3 graph for comparison**

plot(N.1997 ~ c(1997), xlim = c(1997, 1997+50), ylim = c(0, 550))
N.current <- N.1997
for(t in 1:50){
  
  lambda_rand.t <- sample(x = hat_of_lambdas, 
                          size = 1,
                          replace = TRUE)
  
  N.t <- N.current*lambda_rand.t
  
  year.t <- 1997+t
  
  points(N.t ~ year.t)
  
  N.current <- N.t
}

simulate population change in 50 years for 100 runs

**set up a 10*10 graphs to fit 100 graphs**

par(mfrow = c(10,10), mar = c(0,0,0,0), xaxt = "n", yaxt = "n")

**Run the simulation MANUALLY for 100 times and get a 10*10 graph**


plot(N.1997 ~ c(1997), xlim = c(1997, 1997+50), ylim = c(0, 550), cex = 0.5)
abline(h = N.1997)
abline(h = 0, col = "red")
N.current <- N.1997
for(t in 1:50){
  lambda_rand.t <- sample(x = hat_of_lambdas, size = 1,replace = TRUE)
  N.t <- N.current*lambda_rand.t
  year.t <- 1997+t
  points(N.t ~ year.t, cex = 0.5)
  
  N.current <- N.t
}

Computational Ecology: Population Viability Analysis 9/15/2020

Nathan Brouwer

9/15/2020

Data: Figure 5 (plus 10 years)

census

year t

Population size

Population growth rate: example

Population growth rate: vectorized

Negative indexing

Calcualte lambdas for all data

Finish putting together dataframe

Create special columns

Assemble the dataframe

Examing the population growth rates

Plotting the raw data

How do we determine if a population is likely to go extinct?

Random Sampling of Lambda

Get initial population size

One run of population simulation

Simulation the hard way

Plot population change

Predict the Population Change in 50 Years

simulate population change in 50 years for 100 runs