2. Characterizing population dynamics
How fast does a population increase/decrease? (rate of change)
What will the population size be next year? (management)
Need to summarize Births & Deaths (age structure) of a population
Cohort Life Tables
follow one group from birth until the last one dies
Static Life Table
census population for abundance in each age/stage combined with estimates of survival and reproductive output by age/stage
Limitations?
Must be able to age/stage each organism (challenging, uncertainty)

3. Survivorship curves

4. Structured demographic models
Translate variation in individuals (measureable) to population dynamics (less measurable)
Structured around life-history schedules of different species or populations
Demography (vital rates by age, sex, stage, size, etc.) captures both the dynamics & structure of populations
Widely adopted to evaluate species vulnerability (Population Viability Analysis, late 1990’s )

5. Life history components
Maturity- age at 1st reproduction
Mosquito (14 days), Desert tortoise ( years)
Parity- # of episodes for reproduction
Sockeye salmon (1), White footed mouse (4-12)
Semelparity vs. Iteroparity (annual, perennial)
Fecundity- # offspring/episode
Elephant (1), Western Toad (8,000-15,000)
Aging/Senescence- survival/life span
- Fruit fly (35 d), Blue Whale (80-90 y)

6. Not all combinations are possible...
Log
2 eggs x 2 / yr
20d parental care
Lifespan 2-3yr
Log
1 egg/ ~2yr
9 mo. parental care
Lifespan >60yr

7. Life history of Cascades frogs
4 life stages:
Embryonic: lasts 1-3 weeks,
0-80 (~60)% survival
Larval (tadpole): lasts 8-12 weeks
~50% survival
Juvenile (metamorph): lasts 2-3 years
(10-40??)% annual survival
Adult: lasts years, offspring/yr
70-80% annual survival
Rana cascade

8. Life history diagram for Cascades frogs
Vital rates, also called transition probabilities
/yr
Embryos
Larvae
Juveniles
Adults x
1 - x
0.6
0.5
2-3 yr
4-7 yr
1 yr

9. Life history of elephants
5 life stages:
Yearling: 80% survival, lasts 1 year
Pre-reproductive: 98% survival, lasts 15 year
Early reproductive: 98% survival, lasts 5 years, offspring/yr
Middle reproductive: 95% survival, lasts 25 years, 0.3 offspring/yr
Post-reproductive: 80% survival, 5-25 years

10. Life history diagram for elephants
0.1/yr
0.08/yr
0.8
0.98
0.95
Post-reproductive
Pre-reproductive
Middle age
Early Repro
Yearling x x x
1- x
1- x
1- x
1 yr
15 yr
5 yr
25 yr
5-25 yr

11. Other ways to depict life cycle
Age-based
Stage-based
Size-based

12. Life histories Bubble diagrams summarize average life history events
with fixed time steps (survival per week, year, decade)
Result of natural selection
Organisms exist to maximize lifetime reproductive success
Represent successful ways of allocating limited resources to carry out various functions of living organisms
Survival, growth, reproduction

13. Questions we can answer with age/stage structured population models:
How much harvest of a population can occur while still have less than an X% chance of extinction?
What life history stages should conservation (or eradication) efforts be focused on to achieve the biggest change in population size/growth rate?
How many populations of a species need be preserved to ensure reasonable protection from periodic local extinctions and infrequent catastrophic events?
Which life-history strategies are more or less vulnerable to exploitation and extinction?
Is it worth the effort to try and recover a particular population, or is it so likely to go extinct that limited resources should be invested elsewhere?
and hundreds more!

14. Single species population growth models

15. Matrix population models

16. Basic matrix construction1 2 3
a21
a32
a33
a13
aij transition prob. to row i from column j (per timestep)
Columns = j (from)
Rows = i (to)

17. Basic matrix construction1 2 3
a21
a32
a33
a13
If there were only one class, what would this look like? x
Matrix (A) x population vector (nt) = population vector (nt+1)
Underlying this model is an assumption of EXPONENTIAL growth

18. 3 x 3 age-structured matrix (also called Leslie matrix)
P=probability of surviving from one age to the next
F=fecundity of individuals at each age
In this case, there are two pre-reproductive years (maturity at age three)
*only need one subscript here because indivds must move ages each time step
See any inconsistency here?

19. 4 x 4 size-structured matrix (also called Lefkovitch matrix)
Pij=probability of growing from one size to the next or remaining the same size
(need subscripts to denote new possibilities)
F=fecundity of individuals at each size
In this case, there are three pre-reproductive sizes (maturity at age four).
**additional complexities like shrinking or moving more than one class back or forward is easy to incorporate

20. Matrix multiplication1 2 3
Semipalmated sandpiper
data from Hitchcock & Gatto-Trevor (1997)
Three stages (1,2,3+ yrs), with reproduction at each stage.
a21
a32
a33 1 2 3 x nt
per ha
Average matrix 1 2 3

21. Matrix multiplication1 2 3 x nt
How do we calculate the number present in each class next year (nt+1)? =

22. Matrix multiplicationnt
n(1)
n(2)
n(3)
n(4)
n(5)
n(6)
n(7)
n(8)

23. Matrix multiplicationnt
n(1)
n(2)
n(3)
n(4)
n(5)
n(6)
n(7)
n(8)
Is this population declining/increasing/stable?

24. Matrix multiplicationnt
n(1)
n(2)
n(3)
n(4)
n(5)
n(6)
n(7)
n(8)
= sum (n2) / sum(n1)
What’s the proportional change in the population from one time to the next?
= lambda (λ)
(0.639)

25. Matrix multiplication
Semipalmated sandpiper
data from Hitchcock & Gatto-Trevor (1997) nt
Average matrix
Is this population declining/increasing/stable?
Dominant eigenvalue (1)
*stable environment or average matrix*
Asymptototic measure of geometric, density-independent population growth rate
Is the population at a stable distribution of stages/ages?
Dominant eigenvector (w)

26. What to do with a deterministic matrix?
Fixed environment assumption is unrealistic.
BUT…
can evaluate the relative performance of different management/conservation options
can use the framework to conduct ‘thought experiments’ not possible in natural contexts
can ask whether the results of a short-term experiment/study affecting survival/reproduction could influence population dynamics

27. A short detour for an example

28. low UV-B
moderate UV-B
high UV-B
% survival
-UV
+UV
-UV
+UV
% survival
+UV
% survival
-UV
How does egg mortality change across a natural gradient of UV-B exposure? 28

29. 29

30. ?? UV 10,000 eggs 3,000 larvae 6,000 larvae 100 juveniles
dispersal UV ??
10,000 eggs
6,000 larvae
3,000 larvae
adults (4-10 years)
eggs (1-3 weeks)
Talk about variability in recruitment to adult population
“Population” in most cases includes many breeding sites across the landscape
larvae (8 weeks-3+ years)
100 juveniles 30

31. What to do with a deterministic matrix?
Fixed environment assumption is unrealistic.
BUT…
can evaluate the relative performance of different management/conservation options
can use the framework to conduct ‘thought experiments’ not possible in natural contexts
can ask whether the results of a short-term experiment/study affecting survival/reproduction could influence population dynamics
*can evaluate the relative sensitivity of  to different vital rates

32. Matrix element sensitivities
Semipalmated sandpiper
data from Hitchcock & Gatto-Trevor (1997) nt
Average matrix
Reproductive value
Sensitivity of 1 to element aij is Sij
Prop in stable age
constant

33. Matrix element sensitivities
Semipalmated sandpiper
data from Hitchcock & Gatto-Trevor (1997) nt
Average matrix
Sensitivity analysis of a deterministic matrix (by brute force):
Vary vital rates individually (small change vs. biologically realistic range?)
Re-calculate deterministic 1
Plot change in each rate versus change in 1

34. Matrix element sensitivities
Semipalmated sandpiper
data from Hitchcock & Gatto-Trevor (1997) nt
Average matrix
Sensitivity analysis of a deterministic matrix: X

35. Matrix element sensitivities
Semipalmated sandpiper
data from Hitchcock & Gatto-Trevor (1997) nt
Average matrix
Relative sensitivities (comparable across vital rates) called Elasticities
proportional change of vital rates compared to proportional change in lambda
Why does might this matter?