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

I will talk to you about it. When I tell you about it, you will know. As we speak of better opportunities, we can speculate about

Population size

Population size is recorded as the number of females with cubs.

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 change in population over a year.

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? can’t be calculated

females.N[1]
#> [1] 44
lambda.58_59 <- females.N[1]/females.N[ ]

Population growth rate: vectorized

TASK

Briefly describe (1-2 sentence) what this code is doing. It prints the second and third index or the first and second index.

females.N[2:3]
#> [1] 47 46
females.N[1:2]
#> [1] 44 47

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(). calculates number of items in the data set vector (generalizing)

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. It’s the same as the code above (just calculating length inside a function)


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[1:10]
#>  [1] 44 47 46 44 46 45 46 40 39 39
females.Ntemp <- females.N[seq(1:10)]

Check - are there 10 numbers

females.Ntemp
#>  [1] 44 47 46 44 46 45 46 40 39 39

TASK

What does this do? Briefly describe what the [-1] is doing. It drops the first element.

females.Ntemp[-1]
#> [1] 47 46 44 46 45 46 40 39 39

TASK How many lambdas can I calculate using the first 10 years of data? 9 lambdas

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

TASK Drop the second element

females.Ntemp[-2]
#> [1] 44 46 44 46 45 46 40 39 39

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

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

females.Ntemp[length(females.Ntemp)]
#> [1] 39

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

TASK Calculate the first 9 lambdas.

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
What does females.N[-length(females.N)]? drop the last element

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.) I don’t know

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 logarithms

?log
#> starting httpd help server ... done
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.

mean()
var()

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", ylab="Population Index (females+cubs)", xlab="Year")

census <- 1:39
year.t   <- 1959:1997
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)
lambda.i <- females.N[-1]/females.N[-length(females.N)]
lambda.i <- c(lambda.i,NA)
lambda_log <- log(lambda.i)
bear_N <- data.frame(census,
                year.t,
                females.N,
                lambda.i, 
                lambda_log)

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

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

ADD 1-2 sentences note here

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

hist(bear_N$lambda.i) #variation of population growth rate (lambda)

hat_of_lambdas <- bear_N$lambda.i

NA Nan Inf -Inf NULL

is.na() tells you true/false and mentions if something is missing

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)]#use this one: general form(not the one above)
#>  [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() #handy function

na.omit(hat_of_lambdas)#gets rid of null values
#>  [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)

Lambda Random Sampling

ADD 1-2 sentences note here This pulls a random lambda from the hat and saves to object lambda_rand.t

# pulled a lambda from the hat

sample(x = hat_of_lambdas, 
       size = 1,
       replace = TRUE) #replace = TRUE (recycling data)
#> [1] 1.25

lambda_rand.t <- sample(x = hat_of_lambdas, 
                        size = 1,
                        replace = TRUE)
lambda_rand.t
#> [1] 0.9782609

Receive Initial Population Size

ADD 1-2 sentences note here Sets up values for population simulations

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 Round of Population Simulation

ADD 1-2 sentences note here Predicting bear values for the next year.

1.22807*99
#> [1] 121.5789
lambda_rand.t*N.1997
#> [1] 96.84783
N.1998 <- lambda_rand.t*N.1997

Simulation(hard way)

Worse than excel

ADD 1-2 sentences note here Calculates predictions for bear populations in future sequentially using random lambda values one by one

#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 Changes

ADD 1-2 sentences note here

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")

ADD TITLE HERE

ADD 1-2 sentences note here

# Initial Conditions

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

# xlim and ylim
plot(N.1997 ~ c(1997))

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

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

# not a for loop for loop
#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 pop size
  N.t <- N.current*lambda_rand.t
  
  # find new year
  year.t <- 1997+t
  
  # plot the new pop point
  #points() updates exsisting graph
  points(N.t ~ year.t)

  
  # update current population size
  N.current <- N.t

ADD 1-2 sentences note here This is going to create a new plot to indicate bear population vs year.


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

#initialize conditions
N.current <- N.1997

# for loop from 1 to 50
for(t in 1:50){
  
  # pick random lambda
  lambda_rand.t <- sample(x = hat_of_lambdas, 
                          size = 1,
                          replace = TRUE)
  
  # find new bear population
  N.t <- N.current*lambda_rand.t
  
  # find new time
  year.t <- 1997+t
  
  # plots new point
  points(N.t ~ year.t)
  
  # update bear population
  N.current <- N.t
}

ADD 1-2 sentences note here Goofy R plotting code/magic

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

ADD 1-2 sentences note here plot(females.N ~ year.t, data = bear_N, plot(females.N ~ year.t, data = bear_N, plot(females.N ~ year.t, data = bear_N, plot(females.N ~ year.t, data = bear_N, N.current <- N.t

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
}

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?

Lambda Random Sampling

Receive Initial Population Size

One Round of Population Simulation

Simulation(hard way)

Plot Population Changes

ADD TITLE HERE