using mathematical models to simulate competition between house and bewick’s wrens

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

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

Bewick’s Wrens’ nests are failing due to Bewick’s Wrens

Vandalized House Wren nests may Yield 30% or fewer offspring than

intact nests

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

Major Model Assumption

Interspecies competition with House Wrens is the only major

contribution to the failing Bewick’s Wren population

Single Species Model

House wren

Bewick’swren

K

Two Species Model

House wren

Bewick’swren

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 > 1, the effect of competing species is

greater than the effect of species on its own members

Do BBS data reflect populations?

(B/A) * R * D

• Convert to density• Extrapolate for region• Detection adjustments

HOUSE WREN

VS

BEWICK’S WREN

Aim 2: Two Species Model

Model Equations

Non-Dimensionalization

Final Equations

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 (or whatever Meghan has to put here)

• 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

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

Aim 3: Compare Models With Biological Data from BBS

Types of BBS Regions

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

FWS Regions• Divides U.S. into large regions based on

state boundaries

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

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

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

Strata 15 – Lexington Plain(Tennessee area)

Bewick’s Wren House Wren

Strata 19 – Ozark-Ouachita Plateau(Missouri area)

Bewick’s Wren House Wren

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?

Do BBS data reflect populations?

B

A(RD)

• 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

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

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