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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
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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
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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?
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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)
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Egg Photo courtesy of The Nova Scotia Museum at http://museum.gov.ns.ca/mnh/nature/nsbirds/bns0276.htm
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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
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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?
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Mathematical Problem
• How can build a mathematical model of the population dynamics of the Bewick’s Wren and the House Wren?
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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
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HOUSE WREN BEWICK’S WREN
Aim 1: Single Species Model
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Major Model Assumption
Interspecies competition with House Wrens is the only major
contribution to the failing Bewick’s Wren population
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Single Species Model
House wren
Bewick’swren
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K
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Two Species Model
House wren
Bewick’swren
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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
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Do BBS data reflect populations?
(B/A) * R * D
• Convert to density• Extrapolate for region• Detection adjustments
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HOUSE WREN
VS
BEWICK’S WREN
Aim 2: Two Species Model
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Model Equations
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Non-Dimensionalization
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Final Equations
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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.
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Calculate carrying capacity for each species (or whatever Meghan has to put here)
• Relate indiviual data and the logistic equation, growth rate
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Linear Stability at Critical Points of the Model
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4 Critical Points
• (0,0)• (0,1)• (1,0)• (n1
*,n2 *)
– n1 * = (1-alpha2/beta)/ (1-alpha1alpha2)
– n2 * = (1 – alpha1beta(1 – alpha2beta/(1-
alpha1alpha2)))
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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
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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
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• 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
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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
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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
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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
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Region 6: Northern Midwest U.S.• Bewick's wren population: slowly increasing• House wren population: slowly increasing until early
1990's before stabilizing
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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
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Strata 15 – Lexington Plain(Tennessee area)
Bewick’s Wren House Wren
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Strata 19 – Ozark-Ouachita Plateau(Missouri area)
Bewick’s Wren House Wren
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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?
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Do BBS data reflect populations?
B
A(RD)
• Convert to density• Extrapolate for region• Detection adjustments
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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
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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.”
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Example: House Wren data for region 87 – R.A. = 0.28
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Contributors• Zari Johnson• Meghan McCabe• Kelly Pippins• Mahati Sharma• Robert “Bobby” Sheehan• Gina Siddiqui• Laura Sloofman• Peter Ucciferro• Dr. John A. Pelesko
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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