DRAFT 2 PRESENTATION

Dr. Pelesko

MATH 260

Distribution of House and Bewick’s Wrens

HOUSE WREN BEWICK’S WRENhttp://www.roysephotos.com/zzBewicksWren6.jpghttp://www.sialis.org/images/nesteggsphotoalbum/images/28CarolinaWren.jpg

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

http://byteshuffler.com/rospo/blog/uploaded_images/NestEggs-729160.jpg (nest)

http://www.suttoncenter.org/images/House-Wren-Carroll.jpg (wren)

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

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 WREN BEWICK’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

€

dN2

dt= (r2)(N2)

K2 −N2 −N1α 21

K2

Reproduction Rates

House Wrenr = .84

Of 36 nests 24 produced at least one

fledgling

Bewick’s Wrenr = .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)• (n1

*,n2 *)

– n1 * = (1-alpha2/beta)/ (1-alpha1alpha2)

– n2 * = (1 – alpha1beta(1 – alpha2beta/(1-

alpha1alpha2)))

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

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 Wren House Wren

Strata 31 –Till Plains - Midwest

Bewick’s Wren• No data for species

House Wren

Strata 92 – California Foothills – Southern California

Bewick’s Wren House 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?

€

B

A(R ×D)

• 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