1. NATIONAL CENTER FOR CASE STUDY TEACHING IN SCIENCE
A Presentation to Accompany the Case Study:
Life Tables, Darwin’s Finches, and the Dynamics of Small Island Populations
A Case Study Adaptation of Grant & Grant 1992
Whitley R Lehto, Shannon M Murphy, Mayra C Vidal, and Robin M Tinghitella University of Denver, Denver, CO

2. Describing and Measuring Populations
Different ways we measure and estimate populations:
Their size (mark and recapture)
Their relative abundance
Their distribution (dispersion)
Can describe the dynamics of populations with demography
Different measured or calculated quantities tell us how populations change and grow:
Age Structure
Mortality
Life Expectancy
Reproductive Rate

3. Survivorship Curves
Graphical representation of the number of individuals surviving to each age for a group (within a population or species).
Survivorship Curve for Dall Mountain Sheep
Data from Murie 1944

4. Life Tables Standardized presentation of age structure data
Cohort life table: tracks a group of individuals born at the same time until the last individual dies
Static life table: sample of population and ages of individuals at one time point
A life table for Dall sheep at Denali National Park, Alaska
(data from Murie, 1944)
Age in Years (X)
Number Alive at Start of Age (Nx)
Proportion Surviving as Fraction of Newborn (Lx)
Number Dying in Age Interval (Dx)
608
1.000
121 1
487
0.801 7 2
480
0.789 8 3
472
0.776 4
465
0.765 18 5
447
0.735 28 6
419
0.689 29
390
0.641 42
348
0.572 80 9
268
0.441
114 10
154
0.253 95 11 59
0.097 55 12
0.0066 13
0.003 14
0.000

5. Life Tables Warm-up
Life tables can provide information on survivorship, deaths, and mortality rate.
How do we calculate these?
Find a partner.
Fill in the table on the LIFE TABLES WARM-UP using the provided formulas and their descriptions.

6. Isolated Populations Often small and experience unique ecology
Island populations
How they persist despite genetic drift and variable environments is a long-standing question in ecology and evolutionary biology

7. Case Study: Background
Galapagos Island Finches
Grant and Grant 1992
Galapagos Islands
Geospiza scandens (Common cactus-finch)
Daphne Major
Geospiza fortis (Medium ground-finch)

8. Finch Life Tables: Cohorts of Different Species
Complete Table 3 in Part II of the case study handout.
Values from life table warm-up were taken from the finch life tables; those values are filled in for you in Table 2.
MAKE SURE YOU GOT THE SAME ANSWERS!
Complete the Questions 1–4.
#4: Remember all graph components: don’t forget the “year” labels and arrows
G. scandens
G. fortis

9. Survivorship in a Changing Environment: Same-Species Cohorts Differing in Year
View the graph in Part III of the handout.
Complete Questions 1–4.

10. Net Reproductive Rate = R0
Reproduction: Same-Species Cohorts Differing in Year
Net Reproductive Rate = R0
The expected number of (female) offspring produced by an average (female) individual over its lifetime
If R0>1, population is increasing
If R0=1, population is stable
If R0<1, population is decreasing

11. Reproduction: Same-Species Cohorts Differing in Year
Complete Part IV of the case study handout and answer the questions.
Note: The Grants tracked the reproduction of a subset of the cohorts they were studying by tracking the fledglings each year.

12. lx=Proportion Surviving to x (Nx/N0)
Life Tables Warm-up
Age Class
(x)
Number Alive
(Nx) lx dx qx 82
 1.00 1 42 2 25 3 17
lx=Proportion Surviving to x (Nx/N0)
the proportion of individuals surviving to each age class with respect to the initial number of individuals

13. lx=Proportion Surviving to x (Nx/N0)
Life Tables Warm-up
Age Class
(x)
Number Alive
(Nx) lx dx qx 82
 1.00 1 42
 0.512 2 25
 0.305 3 17
0.207 
N0/N0 = 82/82 = 1.00
N1/N0 = 42/82 = 0.512
N2/N0 = 25/82 = 0.305
lx=Proportion Surviving to x (Nx/N0)

14. the proportion of individuals dying between age classes x and x+1
Life Tables Warm-up
Age Class
(x)
Number Alive
(Nx) lx dx qx 82
 1.00
 0.488 1 42
 0.512 2 25
 0.305 3 17
0.207 
dx=Proportion Dying between x and x+1 (lx-lx+1)
the proportion of individuals dying between age classes x and x+1

15. Life Tables Warm-up dx=Proportion Dying between x and x+1 (lx-lx+1)
Age Class
(x)
Number Alive
(Nx) lx dx qx 82
 1.00
 0.488 1 42
 0.512
0.207  2 25
 0.305
 0.098 3 17 … 
l0-l1 = = 0.488
l1-l2 = = 0.207
l2-l3 = = 0.098
Question to pose to students: Why can’t you calculate the last one of these? That was shaded in on your warm up for a reason.
Answer: Need lx for x=4 to calculate it.
dx=Proportion Dying between x and x+1 (lx-lx+1)

16. qx=Mortality Rate (dx/lx)
Life Tables Warm-up
Age Class
(x)
Number Alive
(Nx) lx dx qx 82
 1.00
 0.488
0.488 1 42
 0.512
0.207  2 25
 0.305
 0.098 3 17 … 
qx=Mortality Rate (dx/lx)
the proportion dying in a given age class as compared to the proportion surviving to that age class

17. qx=Mortality Rate (dx/lx)
Life Tables Warm-up
Age Class
(x)
Number Alive
(Nx) lx dx qx 82
 1.00
 0.488
0.488 1 42
 0.512
0.207 
0.405 2 25
 0.305
 0.098
0.302 3 17 …  …
d0/l0 = 0.488/1.00 = 0.488
d1/l1 = 0.207/0.512 = 0.405
d2/l2 = 0.098/0.305 = 0.302
qx=Mortality Rate (dx/lx)

18. Reproduction: Same-Species Cohorts Differing in Year
G. scandens 1978 cohort females
Age Class
(x)
Number Alive
(Nx) lx mx
lxmx 90
1.000
0.000 1 39
0.433
0.051 2 33
0.367
0.667 3 30
0.333
1.500
N0/N0 = 90/90 = 1.00
N1/N0 = 39/90 = 0.433
N2/N0 = 33/90 = 0.367
lx=Proportion Surviving to x (Nx/N0)

19. Reproduction: Same-Species Cohorts Differing in Year
G. scandens 1978 cohort females
Age Class
(x)
Number Alive
(Nx) lx mx
lxmx 90
1.000
0.000 1 39
0.433
0.051
0.022 2 33
0.367
0.667
0.245 3 30
0.333
1.500 ?
l0m0 = 1.000*0.000 = 0.000
l1m1 = 0.433*0.051 = 0.022
l2m2 = 0.367*0.667 = 0.245

20. Reproduction: Same-Species Cohorts Differing in Year
Σ(lx*mx*x)
Generation Time
Tc =
Σ(lx*mx)
G. scandens 1978 cohort females
l0m0 *0= 0.000*0= 0.000
Age Class
(x)
Number Alive
(Nx) lx mx
lxmx 90
1.000
0.000 1 39
0.433
0.051
0.022 2 33
0.367
0.667
0.245 3 30
0.333
1.500
l1m1*1 = 0.022*1 = 0.022
l2m2 *2= 0.245*2 = 0.490 +
l12m12 *12= . . .
Numerator of Tc

21. Image Credits
Slide 1-
Left: Image of Charles Darwin taken in Photo credit: Herbert Rose Barraud ( ).
Right: Image of Peter and Rosemary Grant conducting research on Daphne Major in the Galapagos Islands. Photo credit: K.T. Grant. Original caption: “Peter and Rosemary Grant on Daphne Major, capturing and measuring finches. Their work documented the evolution of finches in the genus Geospiza in real time.” Appeared in New York Times article “In Darwin’s Footsteps” by Jonathan Weiner, published online
Slide 3-Dall sheep.
Slide 4-Dall sheep.
Slide 6-Image of different habitat islands (as described in Wilson and Bossert 1971).
Slide 7
Map of Galapagos Islands (Ecuador, South America). (Image on the left.) Photo credit:
Common Cactus Finch Geospiza scandens Santa Cruz Island, Galapagos Islands. (Upper image on the right.) Photo credit:
Medium Ground Finch Geospiza fortis Santa Cruz, Galapagos. (Lower image on the right.) Photo credit:
Slide 8-Images of G. fortis and G. scandens from Grant, PR and Grant, BR Conspecific versus heterospecific gene exchange between populations of Darwin's finches. Phil. Trans. R. Soc. B, 365: