DRAFT 2 PRESENTATION Dr. Pelesko MATH 260

Distribution of House and Bewick’s Wrens HOUSE WRENBEWICK’S WREN

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?

Bewick’s Wren Nest X (nest) (wren)

Egg Photo courtesy of The Nova Scotia Museum at

Data Supporting Nest Vandalism

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?

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

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

HOUSE WRENBEWICK’S WREN Aim 1: Single Species Model

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

Single Species Model (Gina)

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

Model Equations

Non-Dimensionalization

Final Equations

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

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

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

Linear Stability at Critical Points of the Model

4 Critical Points (0,0) (0,1) (1,0) (n 1 *,n 2 * ) – n 1 * = (1-alpha2/beta)/ (1-alpha1alpha2) – n 2 * = (1 – alpha1beta(1 – alpha2beta/(1- alpha1alpha2)))

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

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

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

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

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

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

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

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

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

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

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

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

Strata 54 – Rolling Red Plains- Texas Bewick’s WrenHouse Wren

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

Strata 92 – California Foothills – Southern California Bewick’s WrenHouse Wren

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.

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

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

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.”

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