2. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
1. Life Table
- static: look at one point in time and survival for one time period

3. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
1. Life Table

4. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
1. Life Table
Why λ ? (discrete breeding season and discrete time intervals)

5. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
1. Life Table
- dynamic (or “cohort”) – follow a group of individuals through their life
Song Sparrows Mandarte Isl., B.C. (1988) x nx lx dx qx Lm ex
115
1.00 90
0.78
70.0
1.01 1 25
0.22 6
0.24
22.0
1.86 2 19
0.17 7
0.37
15.5
1.29 3 12
0.10 10
0.83
7.0
0.75 4
0.02
0.50
1.5 5
0.01
0.5 -

6. Age classes (x): x = 0, x = 1, etc.
Initial size of the population: nx, at x = 0. x nx lx dx qx Lm ex
115 1 2 3 4 5 6

7. Age classes (x): x = 0, x = 1, etc.
Initial size of the population: nx, at x = 0.
Number reaching each birthday are subsequent values of nx x nx lx dx qx Lm ex
115 1 25 2 19 3 12 4 5 6

8. Age classes (x): x = 0, x = 1, etc.
Initial size of the population: nx, at x = 0.
Survivorship (lx): proportion of population surviving to age x. x nx lx dx qx Lm ex
115
1.00 1 25
0.22 2 19
0.17 3 12
0.10 4
0.02 5
0.01 6

9. Age classes (x): x = 0, x = 1, etc.
Initial size of the population: nx, at x = 0.
Survivorship (lx): proportion of population surviving to age x.
Mortality: dx = # dying during interval x to x+1.
Mortality rate: mx = proportion of individuals age x that die during interval x to x+1. x nx lx dx mx Lm ex
115
1.00 90
0.78 1 25
0.22 6
0.24 2 19
0.17 7
0.37 3 12
0.10 10
0.83 4
0.02
0.50 5
0.01 -

10. Survivorship Curves:
Describe age-specific probabilities of survival, as a consequence of age-specific mortality risks.

11. Age classes (x): x = 0, x = 1, etc.
Initial size of the population: nx, at x = 0.
Survivorship (lx): proportion of population surviving to age x.
Number alive DURING age class x: Lm = (nx + (nx+1))/2 x nx lx dx qx Lm ex
115
1.00 90
0.78
70.0 1 25
0.22 6
0.24
22.0 2 19
0.17 7
0.37
15.5 3 12
0.10 10
0.83
7.0 4
0.02
0.50
1.5 5
0.01
0.5 -

12. Age classes (x): x = 0, x = 1, etc.
Initial size of the population: nx, at x = 0.
Survivorship (lx): proportion of population surviving to age x.
Number alive DURING age class x: Lm = (nx + (nx+1))/2
Expected lifespan at age x = ex
- T = Sum of Lm's for age classes = , > than age (for 3, T = 9)
- ex = T/nx (number of individuals in the age class) ( = 9/12 = 0.75)
- ex = the number of additional age classes an individual can expect to live. x nx lx dx qx Lm ex
115
1.00 90
0.78
70.0
1.01 1 25
0.22 6
0.24
22.0
1.86 2 19
0.17 7
0.37
15.5
1.29 3 12
0.10 10
0.83
7.0
0.75 4
0.02
0.50
1.5 5
0.01
0.5 -

13. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
1. Life Tables
2. Age Class Distributions

14. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
1. Life Tables
2. Age Class Distributions
When these rates equilibrate, all age classes are growing at the same single rate – the intrinsic rate of increase of the population (rm)

15. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
Net Reproductive Rate = Σ(lxbx) = 2.1
Number of daughters/female during lifetime.
If it is >1 (“replacement”), then the population has the potential to increase multiplicatively (exponentially).

16. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
Generation Time – T = Σ(xlxbx)/ Σ(lxbx) = 1.95

17. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
Intrinsic rate of increase:
rm (estimated) = ln(Ro)/T = 0.38
Pop growth dependent on reproductive rate (Ro) and age of reproduction (T).

18. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
Intrinsic rate of increase:
rm (estimated) = ln(Ro)/T = 0.38
Pop growth dependent on:
reproductive rate (Ro) and age of reproduction (T).
Doubling time = t2 = ln(2)/r = 0.69/0.38 = 1.82 yrs

19. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
Northern Elephant Seals: <100 in 1900  150,000 in r = 0.073, λ = 1.067
Malthus: All populations have the capacity to expand exponentially

20. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
D. Life History Redux x lx bx
lxbx
xlxbx
1.0 1
0.5 20 10 2 -
R = 10
T = 1
r = 2.303
Net Reproductive Rate = Σ(lxbx) = 10
Generation Time – T = Σ(xlxbx)/ Σ(lxbx) = 1.0
rm (estimated) = ln(Ro)/T = 2.303

21. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
D. Life History Redux
- increase fecundity, increase growth rate (obvious) x lx bx
lxbx
xlxbx
1.0 1
0.5 20 10 2 -
R = 10
T = 1
r = 2.303 x lx bx
lxbx
xlxbx
1.0 1
0.5 22 11 2 -
R = 11
T = 1
r = 2.398

22. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
D. Life History Redux
- increase fecundity, increase growth rate (obvious)
- decrease generation time (reproduce earlier) – increase growth rate x lx bx
lxbx
xlxbx
1.0 1
0.5 22 11 2 -
R =11
T = 1
r = 2.398 x lx bx
lxbx
xlxbx
1.0 2 1
0.5 20 10 -
R = 12
T = 0.833
r = 2.983

23. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
D. Life History Redux
- increase fecundity, increase growth rate (obvious)
- decrease generation time (reproduce earlier) – increase growth rate
- increasing survivorship – DECREASE GROWTH RATE (lengthen T) x lx bx
lxbx
xlxbx
1.0 1
0.5 22 11 2 -
R = 11
T = 1
r = 2.398 x lx bx
lxbx
xlxbx
1.0 1
0.5 20 10 2 3 -
R = 20
T = 1.5
r = 2.00

24. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
D. Life History Redux
- increase fecundity, increase growth rate (obvious)
- decrease generation time (reproduce earlier) – increase growth rate
- increasing survivorship – DECREASE GROWTH RATE (lengthen T)
- survivorship adaptive IF:
- necessary to reproduce at all
- by storing E, reproduce disproportionately in the future x lx bx
lxbx
xlxbx
1.0  0 1
0.9 2
0.8
200
160
320 3
0.7
100 70
210 4 -
R = 230
T = 2.30
r = 2.36
Original r = 2.303

25. III. Population Growth – change in size through time
Calculating Growth Rates
B. The Effects of Age Structure
C. Growth Potential
D. Life History Redux
E. Limits on Growth: Density Dependence
Robert Malthus

26. Premise:
- as population density increases, resources become limiting and cause an increase in mortality rate, a decrease in birth rate, or both...
r > 0
DEATH
BIRTH
RATE
r < 0
DENSITY

27. Premise:
- as population density increases, resources become limiting and cause an increase in mortality rate, a decrease in birth rate, or both...
r > 0
RATE
r < 0
DEATH
BIRTH
DENSITY

28. Premise:
- as population density increases, resources become limiting and cause an increase in mortality rate, a decrease in birth rate, or both...
r > 0
r < 0
DEATH
RATE
BIRTH
DENSITY

29. As density increases, successful reproduction declines
And juvenile suvivorship declines (mortality increases)

30. Lots of little plants begin to grow and compete
Lots of little plants begin to grow and compete. This kills off most of the plants, and only a few large plants survive.

31. K Premise: Result: There is a density at which r = 0 and DN/dt = 0.
THIS IS AN EQUILIBRIUM....
r = 0
DEATH
RATE
BIRTH
DENSITY K

33. Premise
2. Result
3. The Logistic Growth Equation:
Exponential:
dN/dt = rN N t

34. Premise
2. Result
3. The Logistic Growth Equation:
Exponential: Logistic:
dN/dt = rN dN/dt = rN [(K-N)/K] K N N t t

35. Premise
2. Result
3. The Logistic Growth Equation (Pearl-Verhulst Equation, ):
Logistic:
dN/dt = rN [(K-N)/K]
When a population is very small (N~0), the logistic term ((K-N)/K) approaches K/K (=1) and growth rate approaches the exponential maximum (dN/dt = rN). K N t

36. Premise
2. Result
3. The Logistic Growth Equation:
Logistic:
dN/dt = rN [(K-N)/K]
As N approaches K, K-N approaches 0; so that the term ((K-N)/K) approaches 0 and dN/dt approaches 0 (no growth). K N t

37. Premise
2. Result
3. The Logistic Growth Equation:
Logistic:
dN/dt = rN [(K-N)/K]
Should N increase beyond K, K-N becomes negative, as does dN/dt (the population will decline in size). K N t

38. Premise
2. Result
3. The Logistic Growth Equation:
Logistic:
dN/dt = rN [(K-N)/K] [(N-m)/K]
Minimum viable population size add (N-m)/N K N m t