Veit and Lewis 1996. Am. Nat. 148(2):255-274

Allee Effect & Demographic Stochasticity?

At each time step, 50% chance of birth, 50% chance of death

1

100

Time

1

0

0)(,12

2,2

100

99

100

101 EP

5.0)(,12

2,2

1

0

1

2 EP

Allee Effect & Demographic Stochasticity

• What is mean lambda for an infinitely large population?• 1 because MM mates with FF.

• What is mean lambda for population size = 2?• 0.5 because MM and FF do not mate.

At each time step, 50% chance of male, 50% chance of femaleEach breeding pair produces two offspring, then die.

M,MM,FF,MF,F

M,F

Sea Turtle Management

Eggs, Hatchlings

Novice Breeders

Juveniles

Mature Breeders

Sub-Adults

Sea Turtle Management

Eggs, Hatchlings

Novice Breeders

Juveniles

Mature Breeders

Sub-Adults

Snow Goose Management

Eggs, Hatchlings

Juveniles

Breeders (Age > 2)

DeathsAge Frequency Survival Mortality Mortality

rate Survival rate (1- qx)

x fx lx dx qx px

0 (to 1) 205 1.00 .533 .533 .467

1 (to 2) 96 .467 .006 .013 .987

2 (to 3) 94 .461 .028 .061 .939

3 (to 4) 89 .433 .046 .106 .894

4 (to 5) 79 .387 .056 .145 .855

5 (to 6) 68 .331 .062 .187 .813

12 6 .029

lx = fx / f0 dx = lx - lx+1 qx = dx / lx

Survivorship Graphs

http://www.sciencemag.org/cgi/content/full/291/5508/1491

Births

• Mean number of female (or reproductive) offspring produced per female (or reproductive) individual.

• Represents mean offspring produced for females that have survived to year “x”.

Age bx

0 (to 1) 0

1 (to 2) 1.1

2 (to 3) 3.5

3 (to 4) .7

Births

• What would the birth schedule look like for Pacific salmon?

Age bx

0 (to 1)

1 (to 2)

2 (to 3)

3 (to 4)

Births

• What would a possible birth schedule look like for Pacific salmon?

Age bx

0 (to 1) 0

1 (to 2) 0

2 (to 3) 0

3 (to 4) 1000

R0

• “Net reproductive rate”

• Mean number of female offspring produced per female over her lifetime

• Mean number of reproductive offspring produced per reproductive individual over its lifetime

Can We Calculate R0 From bx

• Mean number of female offspring produced per female golden lion tamarin.

Age bx

0 (to 1) 0

1 (to 2) 1.1

2 (to 3) 3.5

3 (to 4) .7

Can We Calculate R0 From bx

• Mean number of female offspring produced per female golden lion tamarin.

• NO! Represents mean offspring produced for females that have survived to year “x”.

Age bx

0 (to 1) 0

1 (to 2) 1.1

2 (to 3) 3.5

3 (to 4) .7

R0

xx

x

x

blR

b

l

0

fecundity

ipsurvivorsh

Calculate R0 for a population of tamarins in which 50% of the females survive to the breeding season each year, starting one year after birth, and then produces 6 offspring (3 females) per year. This continues until the end of their 3rd breeding season, at which time all survivors die of old age.

x lx bx lxbx

0

1

2

3

4

R0

Calculate R0 for a population of tamarins in which 50% of the females survive to the breeding season each year, starting one year after birth, and then produces 6 offspring (3 females). This continues until the end of their 3rd breeding season, at which time all survivors die of old age.

x lx bx lxbx

0 1

1 .5

2 .25

3 .125

4 0

R0

Calculate R0 for a population of tamarins in which 50% of the females survive to the breeding season each year, starting one year after birth, and then produces 6 offspring (3 females). This continues until the end of their 3rd breeding season, at which time all survivors die of old age.

x lx bx lxbx

0 1 0

1 .5 3

2 .25 3

3 .125 3

4 0 0

R0

Calculate R0 for a population of tamarins in which 50% of the females survive to the breeding season each year, starting one year after birth, and then produces 6 offspring (3 females). This continues until the end of their 3rd breeding season, at which time all survivors die of old age.

x lx bx lxbx

0 1 0 0

1 .5 3 1.5

2 .25 3 .75

3 .125 3 .375

4 0 0 0

R0

Calculate R0 for a population of tamarins in which 50% of the females survive to the breeding season each year, starting one year after birth, and then produces 6 offspring (3 females). This continues until the end of their 3rd breeding season, at which time all survivors die of old age.

x lx bx lxbx

0 1 0 0

1 0.5 3 1.5

2 0.25 3 0.75

3 0.125 3 0.375

4 0 0 0

R0 = 2.625

Growth Rates

• Population increases if r > 0, lambda >1, R0 > ?

• Population is constant if r = 0, lambda =1, R0 = ?

• Population declines if r < 0, lambda < 1, or R0 < ?

R0 vs. λ

• R0 defines population growth per generation.

• λ defines population growth for a given time step.

• R0 = λ where 1 time step = 1 generation.

Mean Generation Time (G) and r

• Mean age of all reproducing females

• Mean age of mothers of all newborn females

• Mean time between birth of a mother and all her offspring

xx

xx

bl

xblG

G

Rr

)ln( 0