Examples of Demography

Answer key below.*

For each case, draw a life cycle graph, find the transition probabilities and fertilities, create a transtion matrix, and add the transition probabilities to the life cycle graph. In addition, multiple the transition matrix by (100, 10) or (100, 20, 10) depending on the number of stages, and SHOW YOUR WORK!

[To help get myself started with each example, I like to make a bulleted list of information that I think might be relevant.]

Two stages

1.

Billy Joe studied Tribolium castaneum, the red flour beetle, which can be modeled in two stages, larvae and adults, with a sampling interval of five weeks. He couldn’t tell the females apart from the males, so he decided to assume the sex ratio was 50:50. Billy Joe set up a series of experimental chambers, and sampled them every 5 weeks. He found that that in a container of 800 larvae, about 400 died (as larvae or pupae), and 400 eventually became adults. From a container of flour that contained 50 adults, he sifted the flour and found 2500 eggs. Of those eggs, only 250 became larvae, and the others died or were eaten (by larvae or adults themselves). Placing the adults back in the container, he found that 95% of the adults survived for 5 weeks.

2.

Bobby Sue studied a biennial plant and decided to model it in two stages, rosettes and adults. The species is monoecious, which is a breeding system in which individuals have both male and female flowers. The plants set seed in July and the seeds overwinter and germinate in April, becoming rosettes that spring. The rosettes become adults, and flower and set seed the following summer (2nd year). Bobby Sue tagged all the plants in a given area and these totaled 500 rosettes and 100 flowering adults. She found that 80% of the rosettes survived to the next year. Of those that survived, 75% flowered. She also found 400 new rosettes. She did a couple experiments (in the field and in the greenhouse) and confirmed that seeds do not survive more than one year.

3.

Billy Mack was a researcher down in Texas (UT - Austin), with NSF funding to study the Southern cockroach, Billyann bahbisoozii. He decided to model its demography. He determined that it can be described in two stages, eggs and adults. He sampled populations each month, and marked individuals. He found that the probability of an egg hatching and becoming an adult is 0.5. He also found that only 5% of adult roaches die over a month. Females that survive a month lay, on average 5.263 eggs. After careful census and examination, he found that the sex ratio is 50:50.

Three stages

1.

Johnny studied the charismatic saguaro cactus (Carnegiea gigantea) for his thesis. He decided on three life history stages, based on size: 0-9.99 cm, 10-24.99 cm, and >25cm (small, medium and large). In a large number of sample plots, he tagged all individuals, in March, before any flowering. After a year, he found that out of 1000 small plants, 100 remained small, and 50 become medium. Of 100 medium plants he found 90 remained medium, and 6 became large. Of 50 large plants, he found 49 survived. He also found 50 new small plants. He also learned from the literature that only large cacti are capable of producing viable seeds, and all large individuals can do so.

2.

Frankie studied the timber rattlesnake (Crotalus horridus) for her thesis. She decided that three stages made the most sense: eggs, juveniles (0-95 cm), and adults (> 95 cm). She found that on average, females produced about 10 eggs every 5 years. She found that, on average, about 60% of the eggs hatched. She marked individuals in a population, and found that each year about 80% of juveniles survived and of those, about 10% of the juveniles reach adulthood. Adults have a 70% annual survival rate. She assumed an even sex ratio. She censused snakes and eggs each year during egg laying season.

3.

Frankie and Johnny were lovers, but they quarreled about their research. Johnny said that Frankie ought to use integral projection instead of a stage-based model. She told him to f— off and do it on his own damn population. She then shot him dead. Was she justified, or did Johnny have a point? Justify your answer.

4.

Rosa studied a perennial plant that takes several years to mature into a reproductive plant. She decided to model it in three stages: seeds, non-reproductives, and adults. Adults are defined merely as individuals that produce flowers. The species is dioecious, which is a breeding system in which individuals have either male or female flowers. The plants set seed in July and the seeds germinate immediately, becoming rosettes that overwinter. Rosa sampled the plants in June, and tagged all the plants in a given area. These totaled 400 non-reproductives and 100 flowering adults. The following year, she found that 300 of the non-reproductive individuals survived. However, of those that survived, 90 became adults. Of the 100 original adults, she found that 70 survived and flowered again, and 10 survived but did not flower. The following year, she also found 100 new non-reproductives. She did a couple experiments (in the field and in the greenhouse) and found that 90% of seeds remain viable in the seed bank and have an average germination rate of 30%. She also found that females that flower produce an average of 20 seeds.

ANSWER KEY

Two stages

1.

Billy Joe studied Tribolium castaneum, the red flour beetle, which can be modeled in two stages, larvae and adults, with a sampling interval of five weeks. He couldn’t tell the females apart from the males, so he decided to assume the sex ratio was 50:50. Billy Joe set up a series of experimental chambers, and sampled them every 5 weeks. He found that that in a container of 800 larvae, about 400 died (as larvae or pupae), and 400 eventually became adults. From a container of flour that contained 50 adults, he sifted the flour and found 2500 eggs. Of those eggs, only 250 became larvae, and the others died or were eaten (by larvae or adults themselves). Placing the adults back in the container, he found that 95% of the adults survived for 5 weeks.

Data:

larvae and adults
half of all larvae and adults are female.
larvae become adults (\(p = 0.5\)) or die.
Fertility = 2500 eggs / (50/2 adult females)
eggs become larvae with p = .1
Adults persist with p = 0.95 probabililty.

p11 <- 0
p21 <- (400/2) / (800/2)  
p22 <- 0.95
p12 <- 2500/2 / (50 / 2) * 0.1 # also equals 2500 / (100 / 2) * 0.1

A <- matrix( c(p11, p21, p12, p22), nrow=2)
rownames(A) <- colnames(A) <- c("larvae", "adults")
library(diagram)

## Loading required package: shape

plotmat(A, 2)

N.2s <- c(100, 10)
A %*% N.2s

##        [,1]
## larvae 50.0
## adults 59.5

## Work not shown....FAIL.

2.

Bobby Sue studied a biennial plant and decided to model it in two stages, rosettes and adults. The species is monoecious, which is a breeding system in which individuals have both male and female flowers. The plants set seed in July and the seeds overwinter and germinate in April, becoming rosettes that spring. The rosettes become adults, and flower and set seed the following summer (2nd year). Bobby Sue tagged all the plants in a given area and these totaled 500 rosettes and 100 flowering adults. She found that 80% of the rosettes survived to the next year. She also found 400 new rosettes. She did a couple experiments (in the field and in the greenhouse) and confirmed that seeds do not survive more than one year.

Data:

two stages
all adults make seeds
rosettes become adults at p = 0.80

p11 <- 0
p21 <- 0.8
p22 <- 0
p12 <- 400 / 100

A <- matrix( c(p11, p21, p12, p22), nrow=2)
rownames(A) <- colnames(A) <- c("rosette", "adult")
A

##         rosette adult
## rosette     0.0     4
## adult       0.8     0

library(diagram)
plotmat(A, 2)

N.2s <- c(100, 10)
A %*% N.2s

##         [,1]
## rosette   40
## adult     80

## Work not shown....FAIL.

3.

Billy Mack was a researcher down in Texas (UT - Austin), with NSF funding to study the Southern cockroach, Billyann bahbisoozii. He decided to model its demography. He determined that it can be described in two stages, eggs and adults. He sampled populations each month, and marked individuals. He found that the probability of an egg hatching and becoming an adult is 0.5. He also found that only 5% of adult roaches die over a month. Females that survive a month lay, on average 5.263 eggs. After careful census and examination, he found that the sex ratio is 50:50.

Data:

p11 <- 0
p21 <- 0.5
p22 <- 1-0.05
p12 <- 0.95 * 5.263

p11 <- 0
p21 <- 0.5
p22 <- 1-0.05
p12 <- 0.95 * 5.263 / 2
A <- matrix( c(p11, p21, p12, p22), nrow=2)
rownames(A) <- colnames(A) <- c("egg", "adults")
A

##        egg   adults
## egg    0.0 2.499925
## adults 0.5 0.950000

library(diagram)
plotmat(A, 2)

N.2s <- c(100, 10)
A %*% N.2s

##            [,1]
## egg    24.99925
## adults 59.50000

Three stages

Johnny studied the charismatic saguaro cactus (Carnegiea gigantea) for his thesis. He decided on three life history stages, based on size: 0-9.99 cm, 10-24.99 cm, and >25cm (small, medium and large). In a large number of sample plots, he tagged all individuals, in March, before any flowering. After a year, he found that out of 1000 small plants, 100 remained small, and 50 become medium. Of 100 medium plants he found 90 remained medium, and 6 became large. Of 50 large plants, he found 49 survived. He also found 50 new small plants. He also learned from the literature that only large cacti are capable of producing viable seeds, and all large individuals can do so.

Data:

3 stages
p11 = 100/1000
p21 = 50/1000
p22 = 90/100
p32 = 6/100
p33 = 49/50
F3 = 50 new recruits / 50 large adults

50 adults * 49/50 percent survival * 50 recruits per 49 adults If we assume mortality occurs which is the same as 50 recruits / 49 survivors * 49 survivors / 50 adults per capita fecunidty * probability of survival.

p11 = 100/1000
p21 = 50/1000
p22 = 90/100
p32 = 6/100
p33 = 49/50
F3 = 50 / 50

A31 <- matrix(c(p11, 0, F3,
                p21, p22, 0,
                0,   p32, p33), nrow=3, byrow=TRUE)
rownames(A31) <- colnames(A31) <- c("small", "medium", "large")
A31

##        small medium large
## small   0.10   0.00  1.00
## medium  0.05   0.90  0.00
## large   0.00   0.06  0.98

plotmat(A31, 3, curve=c(.3))

N.3s <- c(100, 20, 10)
A31 %*% N.3s

##        [,1]
## small    20
## medium   23
## large    11

2.

Frankie studied the timber rattlesnake (Crotalus horridus) for her thesis. She decided that three stages made the most sense: eggs, juveniles (0-95 cm), and adults (> 95 cm). She found that on average, females produced about 10 eggs every 5 years. She found that, on average, about 60% of the eggs hatched. She marked individuals in a population, and found that each year about 80% of juveniles survived and of those, about 10% of the juveniles reach adulthood. Adults have a 70% annual survival rate. She assumed an even sex ratio. She censused snakes and eggs each year during egg laying season.

Data:

3 stages, 50:50 sex ratio
p11 = 0
p21 = 0.6(0.8).9
p22 = 0.8(.9)
p32 = 0.8(0.1)
p33 = 0.7
F2 = 10 / 5 / 2 * .1
F3 = 10 / 5 / 2 * .7 ( we have to count males and females, but understand that fertility above refers to females. This fertility is “eggs” per individual of either sex, each year).

p11 = 0
p21 = 0.6 * 0.8*.9
p22 = 0.8 *.9
p32 = 0.8*0.1
p33 = 0.7
F2 = 10 / 5 / 2 * .8 * .1
F3 = 10 / 5 / 2  * .7

A32 <- matrix(c(p11, F2, F3,
                p21, p22, 0,
                0,   p32, p33), nrow=3, byrow=TRUE)
rownames(A32) <- colnames(A32) <- c("egg", "juvenile", "adult")
A32

##            egg juvenile adult
## egg      0.000     0.08   0.7
## juvenile 0.432     0.72   0.0
## adult    0.000     0.08   0.7

plotmat(A32, 3, curve=c(.3))

N.3s <- c(100, 20, 10)
A32 %*% N.3s

##          [,1]
## egg       8.6
## juvenile 57.6
## adult     8.6

3.

Frankie and Johnny were lovers, but they quarreled about their research. Johnny said that Frankie ought to use integral projection instead of a stage-based model. She told him to f— off and do it on his own damn population. She then shot him dead. Was she justified, or did Johnny have a point? Justify your answer.

[no answer provided here]

4

Data:

p32 = 90/400
p22 = (300 - 90) / 400
p33 = .7
p23 = .1
p11 = 0.9
p21 = 0.3
F3 = 20/2

p32 = 90/400
p22 = (300 - 90) / 400
p33 = .7
p23 = .1
p11 = 0.9
p21 = 0.3
F3 = 20/2


A3new <- matrix(c(p11, 0, F3,
                p21, p22, p23,
                0,   p32, p33), nrow=3, byrow=TRUE)
rownames(A3new) <- colnames(A3new) <- c("seed", "non-r", "adult")
A3new

##       seed non-r adult
## seed   0.9 0.000  10.0
## non-r  0.3 0.525   0.1
## adult  0.0 0.225   0.7

plotmat(A3new, 3, curve=c(.3))

N.3s <- c(100, 20, 10)
A3new %*% N.3s

##        [,1]
## seed  190.0
## non-r  41.5
## adult  11.5

eigen(A3new)

## eigen() decomposition
## $values
## [1] 1.605762+0.0000000i 0.259619+0.7391382i 0.259619-0.7391382i
## 
## $vectors
##               [,1]                    [,2]                    [,3]
## [1,] 0.95972087+0i  0.93274208+0.00000000i  0.93274208+0.00000000i
## [2,] 0.27266837+0i -0.10957177-0.33115763i -0.10957177+0.33115763i
## [3,] 0.06773345+0i -0.05973103+0.06894253i -0.05973103-0.06894253i

library(diagram)
library(primer)

## Loading required package: deSolve

## Loading required package: lattice

DemoInfo(A3new)

## $lambda
## [1] 1.605762
## 
## $SSD
## [1] 0.73817716 0.20972511 0.05209774
## 
## $RV
## [1]  1.00000  2.35254 11.30016
## 
## $Sensitivities
##           [,1]      [,2]       [,3]
## [1,] 0.4055302 0.1152161 0.02862078
## [2,] 0.9540260 0.2710504 0.06733153
## [3,] 4.5825559 1.3019599 0.32341937
## 
## $Elasticities
##            seed      non-r      adult
## seed  0.2272922 0.00000000 0.17823801
## non-r 0.1782380 0.08861927 0.00419312
## adult 0.0000000 0.18243113 0.14098824
## 
## $PPM
##       seed non-r adult
## seed   0.9 0.000  10.0
## non-r  0.3 0.525   0.1
## adult  0.0 0.225   0.7

Demography practice

Examples of Demography

Two stages

1.

2.

3.

Three stages

1.

2.

3.

4.

ANSWER KEY

Two stages

1.

2.

3.

Three stages

2.

3.

4