Using mathematical models to simulate competition between House and Bewick’s Wrens

1. Using mathematical models to simulate competition between House and Bewick’s Wrens MATH 260 Speakers: Laura Sloofman, Gina Siddiqui, Zariel Johnson, Peter Ucciferro Advisor: Dr. John A. Pelesko

2. Distribution of House and Bewick’s Wrens HOUSE WREN BEWICK’S WREN http://www.roysephotos.com/zzBewicksWren6.jpg http://www.sialis.org/images/nesteggsphotoalbum/images/28CarolinaWren.jpg

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 http://www.suttoncenter.org/images/House-Wren-Carroll.jpg(wren) http://byteshuffler.com/rospo/blog/uploaded_images/NestEggs-729160.jpg (nest)

5. Egg Photo courtesy of The Nova Scotia Museum at http://museum.gov.ns.ca/mnh/nature/nsbirds/bns0276.htm

6. Data Supporting Nest Vandalism Bewick’s Wrens’ nests are failing due to Bewick’s Wrens Vandalized House Wren nests may Yield 30% or fewer offspring than intact nests

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. Major Model Assumption Interspecies competition with House Wrens is the only major contribution to the failing Bewick’s Wren population

12. Single Species Model House wren Bewick’swren

13. K

14. Two Species Model House wren Bewick’swren

15. So what is a competition coefficient? • α12 is the effect of species 2 on species 1 • α21 is the effect of species 1 on species 2 • Quantifies how much every additional organism of species 1 fills the niche of species 2 • If α> 0, competing species has limiting effect • If a > 1, the effect of competing species is greater than the effect of species on its own members

16. Do BBS data reflect populations? (B/A) * R * D • Convert to density • Extrapolate for region • Detection adjustments

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

18. Model Equations

20. Non-Dimensionalization

21. Final Equations

22. 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 Reproduction Rates This data was retrieved from The Birds of British Columbia - a reference work on 472 species of birds in the area.

23. Calculate carrying capacity for each species (or whatever Meghan has to put here) • Relate indiviual data and the logistic equation, growth rate

24. Linear Stability at Critical Points of the Model

25. 4 Critical Points • (0,0) • (0,1) • (1,0) • (n1 *,n2 *) • n1 * = (1-alpha2/beta)/ (1-alpha1alpha2) • n2 * = (1 – alpha1beta(1 – alpha2beta/(1- alpha1alpha2)))

26. 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

27. 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

28. Aim 3: Compare Models With Biological Data from BBS • Species interactions have mostly taken place where “northern” and “southern” regions of the U.S. came together

29. Types of BBS Regions • FWS Regions • Divides U.S. into large regions based on state boundaries • Physiographic Strata of the U.S. • Areas of similar geographic and vegetation features instead of state boundaries • Allow for examination of bird species in a small area that experiences a specific climate

30. Large Range Data from FWS Regions • 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 • Used as “control” data to estimate behavior without competition

31. 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

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

33. Overlap Data from Physiographic Strata Regions • Data taken from areas of species overlap shows general trend of decrease in Bewick’s population and increase in House population • Some data showed variance from this trend • Region 22 showed stable House populations and sharp decrease in Bewick’s • Region 33 showed stable Bewick’s populations while House increased • Possibly due to region-specific factors

34. Strata 15 – Lexington Plain(Tennessee area) Bewick’s Wren House Wren

35. Strata 19 – Ozark-Ouachita Plateau(Missouri area) Bewick’s Wren House Wren

36. Pending Questions • Will the competition between the birds lead to the extinction of one species or will they continue to coexist in the same regions? • Timing of departure from steady population varies between regions. What does this mean about validity of assumptions. • Can we use our model to estimate how much of the behavior of the populations is due to competition and not other factors? • How well does the information obtained from using the model match up with known values?

37. Do BBS data reflect populations? • Convert to density • Extrapolate for region • Detection adjustments

38. 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

39. 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.”

40. Example: House Wren data for region 87 – R.A. = 0.28

41. Contributors • Zari Johnson • Meghan McCabe • Kelly Pippins • Mahati Sharma • Robert “Bobby” Sheehan • Gina Siddiqui • Laura Sloofman • Peter Ucciferro • Dr. John A. Pelesko

42. References • Bewick’s map: http://www.mbr-pwrc.usgs.gov/bbs/htm03/trn2003/tr07190.htm • House map: http://www.mbr-pwrc.usgs.gov/bbs/htm03/trn2003/tr07210.htm • Region 2 Data: http://www.mbr-pwrc.usgs.gov/cgi-bin/atlasa99.pl?RE2&2&07 • Region 6 Data: http://www.mbr-pwrc.usgs.gov/cgi-bin/atlasa99.pl?RE6&2&07 • 15 Lexington Plain: http://www.mbr-pwrc.usgs.gov/cgi-bin/atlasa99.pl?S15&2&07 • 19 Ozark-Ouachita Plateau: http://www.mbr-pwrc.usgs.gov/cgi-bin/atlasa99.pl?S19&2&07 • Region 87 Intermountain Grasslands: http://www.mbr-pwrc.usgs.gov/cgi-bin/atlasa99.pl?S87&2&07 • Physiographic Strata Map: http://www.mbr-pwrc.usgs.gov/bbs/physio.html • FWS Region Map: http://www.fws.gov/irm/bpim/foiawhere.html