1. Demography

2. Introduction to Life Tables
Logistic model approach provides a simple, general model, but important details missing.
In a life table we incorporate the vital statistics of the species. In particular, age-specific birth and death rates.
We will be able to analyze potential growth rates of a population.
We will be able to estimate the life expectancy of an individual of known age.

3. Recall assumptions r and K are constant
Environment will not change (physical and biotic)
Populations will not evolve
2. All individuals are identical
Ages are equal
Sizes are equal
Genders are equal
3. Populations are closed
No immigration
No emigration

4. Life Span Determinate versus indeterminate growth
Determinate versus indeterminate growth
Fish and trees vs mice and butterflies;
important when mortality correlated with size
Importance of physical environment.
Slow growth where temperature low or nutrients scarce

5. Survivorship of the lapwing
Plotted on a semi-log plot. The straight line indicates a constant probability of death, regardless of age.

6. Life Tables x age nx Sx dx lx gx px mx qx Nx Tx ex kx
200 1
100 2 50 3 20 4
x = age at beginning of the time interval
Sx = Number of survivors at start of interval

7. Survivorship x age nx Sx dx lx gx px mx qx Nx Tx ex kx
200
100
1.00 1 50
0.50 2 30
.25 3 20
.10 4
dx = Number dying during interval
lx = Proportion surviving at start of age interval = nx/no

8. Age-specific survival and mortality ratesx
age nx Sx dx lx gx px mx qx Nx Tx ex kx
200
100
1.00
0.50 1 50 2 30
.25
.40
.60 3 20
.10 4
gx = sx = px = age specific survival rate (probability of surviving to the end of the period) = nx+1/nx = lx+1/lx = (nx-dx)/nx
mx = qx = age specific mortality rate (probability of dying before the end of the period) = (nx-(nx+1)) / nx = dx/nx = 1 - gx

9. Calculation of age-specific life expectancynx Sx dx lx gx px mx qx Nx Tx ex kx
200
100
1.00
0.50
150
270
1.35 1 50 75
120
1.20 2 30
.25
.40
.60 35 45
0.90 3 20
.10 10 4
Nx = Mean number of individuals alive during the period = (nx+nx+1)/2
Tx = sum of the time to be lived by the nx currently alive
ex = age specific life expectancy

10. Exponential mortality rate
age nx Sx dx lx gx px mx qx Nx Tx ex kx
200
100
1.00
0.50
150
270
1.35
.693 1 50 75
120
1.20 2 30
.25
.40
.60 35 45
0.90
.916 3 20
.10 10 … 4
kx = Exponential mortality = ln gx
gx = e-kt

11. Data?

12. Types of mortality curves
Three basic types of survivorship curves
Semilog plot is Negative skewed = Type I
Survivorship starts out high but then drops off quickly at older age
Semilog plot is diagonal = Type II Survivorship is constant with age
Semilog plot is Positively skewed = Type III
Survivorship very low for young individuals, but increases greatly with age
Typically a blend with initial high drop, then diagonal, then senescence period.

13. Typical birds (Turdus) -- Type II survivorship

14. Survivorship in Sceloporus virgatus

15. Thar
Introduced in New Zealand
Type I

16. Dall sheep in Alaska - Type I

17. Deer in southern California – two contrasting habitats

18. Roe deer in Denmark with marked differences in survivorship rates between genders

19. Mortality rates of grey seals.
Observe the burden of being a mature male.

20. Density dependence in age-specific mortality rates of the African Buffalo.

21. Comparison of human survivorship in Rome and North Africa in the first four centuries A.D.

22. Comparisons of human life expectancy in various European populations

23. Survivorship in various British populations

24. Age-specific mortality in natural whitefish populations

25. Survivorship in Atlantic Mackeral

26. Survivorship in the sawfly, an insect species with a complex life history

27. Types of lifetable studies
Age specific (horizontal or dynamic or cohort) life table; one cohort followed
Time specific (vertical or static); one point in time – a snapshot view
Composite age specific; remains of dead
Three not equivalent unless stable population.

28. Age specific fecundity of human females

29. Gross Reproductive Ratex Sx lx bx Fx
lxbx
200
1.0 1
100 .5 2 50
.25 3 20 .1 4
GRR=5
Fecundity at age x = bx
Gross reproductive rate (GRR) = Σ bx

30. Net Reproductive Rate x Sx lx bx Fx lxbx
200
1.0 1
100 .5
0.5 2 50
.25 3 20 .1 40
0.2 4
GRR=5
NRR=1.2
lxbx = Number of female individuals born during interval x.
Net Reproductive Rate (average number)=Ro=lxbx=( Fx)/no

31. Projection of Population Sizex Sx lx bx Fx
lxbx gx N0 N1 N2 N3 N4 N5
200
1.0 .5 10 50 1
100
0.5 5 2
.25 .4 3 20 .1 40
0.2 4
GRR=5
NRR=1.2
Population at time t and age x = Nt(x)
Where x>1, Nt(x) = (N(t-1)(x-1))(sx-1)
Where x=1, Nt(1) = i=1,∞ (N(t-1)(i))(bi)

32. x Sx lx bx Fx lxbx gx N0 N1 N2 N3 N4 N5 200 1.0 .5 10 50 23 1 100 0.5
200
1.0 .5 10 50 23 1
100
0.5 5 25 2
.25 .4
2.5 3 20 .1 40
0.2 4
GRR=5
NRR=1.2

33. x nx lx bx Fx lxbx gx N0 N1 N2 N3 N4 N5 200 1.0 .5 10 50 23 34 38.5 1
200
1.0 .5 10 50 23 34
38.5 1
100
0.5 5 25
11.5 17
19.2 2
.25 .4
2.5
12.5
5.75
8.5 3 20 .1 40
0.2 4
2.3
GRR=5
NRR=1.2

34. G = [(lxbxx)]/[(lxbx)] = 2.1/1.2 = 1.75 = (xFx)/Fx nx lx bx Fx
lxbx
lxbxx
200
1.0 1
100 .5
0.5 2 50
.25 3 20 .1 40
0.2
0.6 4
GRR=5
NRR=1.2
G = cohort generation length = mean period elapsing between birth of parent and birth of offspring.
 G = [(lxbxx)]/[(lxbx)] = 2.1/1.2 = 1.75 = (xFx)/Fx
 (The time it takes) / (number produced)

35. Calculation of r
Recall Nt = Noert;
set t=G  NG/No = erG ~ Ro
ln(NG/No) = lnRo ~ rG
r ~ ln(Ro)/G = ln(1.2)/1.75 = .104
recall G = (xlxbx)/ (lxbx) = (xlxbx)/ Ro
r ~ (RolnRo)/ (xlxbx) 

37. Leslie matricies

38. n(t+1) = An(t)

42. Reproductive value
vx = [erx/lx][  (e-rxlyby)] ; where summation over range y = x → ∞

43. Stage and size structured populations

46. Teasel

49. Applications? Population projections? Sensitivity analysis?
Conservation strategies?

50. Elephant seals: McMahon et al. 2005
The dynamics of animal populations are determined by several key demographic parameters, which vary over time with resultant changes in the status of the population. When managing declining populations, the identification of the parameters that drive such change are a high priority, but are rarely achieved for large and long-lived species.
Southern elephant seal populations in the South Indian and South Pacific oceans have decreased by as much as 50% during the past 50 yr. The reasons for these decreases remained unknown. This study used a projected stochastic Leslie-matrix model based on long-term demographic data to examine the potential role of several life-history parameters in contributing to the declines. The models simulated the observed population trends that were independently derived from annual abundance surveys.

51. The models simulated the observed population trends that were independently derived from annual abundance surveys. Small changes in survival and fecundity had dramatic effects on population growth rates. At Macquarie Island for example, a small change (ca. 5%) in survival and fecundity rates resulted in the population reverting from a decreasing one to a population that increased. The vital rates that had the greatest impact on fitness were, in order of importance: (1) juvenile survival, (2) adult survival, (3) adult fecundity and (4) juvenile fecundity, Population viability analysis (PVA) for each of the 2 decreasing populations revealed that there was a high probability of the Marion Island population becoming extinct within the next 150 yr, while the probability of extinction at Macquarie Island was low.