1. DRAFT 2 PRESENTATION
Dr. Pelesko
MATH 260

2. Distribution of House and Bewick’s Wrens
HOUSE WREN
BEWICK’S WREN

3. Biological Problem
House-Wren and Bewick’s Wren competition relatively new (within the last 10 years)
Didn’t share territory until recently (Kennedy et. al., 2007)
How will this new interaction affect the population dynamics of both species?

4. X
Bewick’s Wren Nest
(wren)
(nest)

5. Captures overlaps of niches
CONCEPT: cohabitation of same exact habitat is impossible
Nest issue is not only variable but it is what cued scientists to explore this behavior (also easiest to measure)
Egg Photo courtesy of The Nova Scotia Museum at

6. Data Supporting Nest Vandalism

7. Summary
We want to analyze the consequences of the cohabitation of the House Wren and Bewick’s Wren on their populations
Will this result in fewer Bewick’s Wrens?
Will this result in more House Wrens?

8. Mathematical Problem
How can build a mathematical model of the population dynamics of the Bewick’s Wren and the House Wren?

9. Specific Aims
Aim 1: Examine single-species population model for both Bewick’s Wren and House Wren Aim 2: Create two species model of competition between Bewick’s Wren and House Wren Aim 3: Compare Models with biological data from BBS

10. Aim 1: Single Species Model
HOUSE WREN
BEWICK’S WREN

11. Model Assumptions
Interspecies competition with House Wrens is the only major contribution to the failing Bewick’s Wren population
Move to end gina’s slid

12. Single Species Model (Gina)

13. Aim 2: Two Species Model VS
HOUSE WREN
BEWICK’S WREN

14. Model Equations
Aim 2!

15. Non-Dimensionalization

16. Final Equations

17. So what is a competition coefficient?
Quantifies how every additional organism of species 1 fills the niche of species 2
Aim 2

18. Reproduction Rates House Wren r = .84
Of 36 nests 24 produced at least one fledgling
Bewick’s Wren
r = .67
Of 535 nests 449 produced at least one fledgling
Aim 3
This data was retrieved from The Birds of British Columbia - a reference work on 472 species of birds in the area.

19. Calculate carrying capacity for each species
Relate indiviual data and the logistic equation, growth rate
Aim 3

20. Linear Stability at Critical Points of the Model
Aim 2 ending, pete

21. 4 Critical Points (0,0) (0,1) (1,0) (n1 *,n2 *)
n1 * = (1-alpha2/beta)/ (1-alpha1alpha2)
n2 * = (1 – alpha1beta(1 – alpha2beta/(1- alpha1alpha2)))

22. Linear Stability We notice that similar to a scalar ODE
dx/dt = Ax ,x(0) = x0 where denotes vector
Has solution
x(t) = x0 exp(At), where A is the Jacobian matrix

23. Decomposing A By writing A = SDS-1 Exp(At) = exp[(SDS-1)t]
then taylor expanding the following
sum{ (SDS-1 t)n / n! } from 0…inf
we can see that the eigenvalues of A determine the behavior of the solution.
If Eig(A(criticalpt)) = both neg. then the point is stable
If Eig(A(criticalpt)) = both pos. then the point is unstable
If Eig(A(criticalpt)) = pos/ neg. then it is a saddle point

24. Tedious details of Analysis
This needs to be typed in latex
Show all A matrices evaluated at each critical point
Eigenvalues of each matrix A
Phase plane behavior determined by above. A couple plots for different cases of alphas, betas, etc. would be nice

25. Aim 3: Compare Models with biological data from BBS
BBS has separated data by several classes, including Fish & Wildlife Service Regions

26. Species interactions have mostly taken place where “northern” and “southern” regions of the U.S. came together
Breeze through this one

27. Physiographic Strata of the U.S.
Areas of similar geographic and vegetation features
Developed by modifying vegetation and soil distribution maps
Allow for examination of bird species in a small area that experiences a specific climate
Ignores state boundaries, concentrates on geographical boundaries
Aim 3

28. Large Range Data
Data from wider geographical regions allowed us to evaluate the behavior of each species' population somewhat individually
This data from larger areas, reflected less of the effect of interaction with the other species

29. Region 2: Southern Midwest U.S.
Bewick's wren and House wren populations stable throughout BBS data collection
Average Bewick's population much lower than that of House wren

30. Region 6: Northern Midwest U.S.
Bewick's wren population: slowly increasing
House wren population: slowly increasing until early 1990's before stabilizing

31. Region 3: Northern Central U.S.
Bewick's wren population: decreasing
House wren population: slowly increasing

32. Region 4: Southern Central U.S.
Bewick's wren population: decreasing rapidly until around 1980 and then stable
House wren population: increasing rapidly throughout survey
Bewick's wren population: decreasing rapidly until around 1980 and then stable
House wren population: increasing rapidly throughout survey

33. Wren Population Patterns
Bewick’s Wren populations seem largest in the southwest
Strongest areas with no House Wrens are in southern Texas, in Strata 53, 54, 56
House Wren populations seem largest in the northern US
Strongest areas with no Bewick’s Wrens are in the north and midwest, in Strata 31,32,40
Overlap between the two is most prevalent in southwestern California, in Strata 92,94, and 83
NOTE: Check data with data from overlap regions – do we repeat anything?
Aim 3

34. Strata 54 – Rolling Red Plains- Texas
Bewick’s Wren
House Wren
Do we need these next 3 slides?

35. Strata 31 –Till Plains - Midwest
Bewick’s Wren
House Wren
No data for species

36. Strata 92 – California Foothills – Southern California
Bewick’s Wren
House Wren

37. Pending Questions
Do both the birds coexist (did you mean can they coexist for infinitely large t given their competitive nature)?
There is no data given in the BBS, where the two birds over lap (?). Looking at all the data , it seems that the House wrens exist at the central and east where as the Bewicks wren at the west. There is no data that shows their existence together. The possible problem that House Wrens and Bewicks Wrens compete might be true as there are certain states where the, population changes inversely. While the Bewicks Increase the House Wren decreases.
FIX

38. Do BBS data reflect populations?
Convert to density
Extrapolate for region
Detection adjustments
NEED PICTURE/IMAGE FOR BOTTOM – IS THIS IN THE CORRECT SPOT?
Aim 3

39. Interpreting Data From BBS Graphs
The vertical axis of population graphs from the BBS website was labeled “count”.
Clearly, this was not the raw number of birds counted because there were often data points that appeared to show fractional birds being observed
Aim 3

40. Vertical Axis: Relative Abundance
The vertical axis of these graphs is not the raw number of birds of a given species counted
BBS has calculated the relative abundance (R.A.) for each species and region – the number of birds per route
According to BBS, “[…] an approximate measure of how many birds are seen on a route in the region.”
Aim 3

41. Example: House Wren data for region 87 – R.A. = 0.28
WE NEED A CONCLUSIONS/REFERENCES SLIDE Maybe instead of putting hyperlinks in pictures, we should use a numbering system and reference everything in a references slide (which we don’t need to show)