Download 5-5-09Presentation

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
http://www.sialis.org/images/nesteggsphotoalbum/images/28CarolinaWren.jpg
BEWICK’S WREN
http://www.roysephotos.com/zzBewicksWren6.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?
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)
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
Aim 1: Single Species Model
HOUSE WREN
BEWICK’S WREN
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’s
wren
K
Two Species Model
House
wren
Bewick’s
wren
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
Do BBS data reflect populations?
(B/A) * R * D
• Convert to density
• Extrapolate for region
• Detection adjustments
Aim 2: Two Species Model
VS
HOUSE WREN
BEWICK’S WREN
Model Equations
Non-Dimensionalization
Final Equations
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 (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
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•
•
•
(0,0)
(0,1)
(1,0)
(n1 *,n2 *)
– n1 * = (1-alpha2/beta)/ (1-alpha1alpha2)
– n2 * = (1 – alpha1beta(1 – alpha2beta/(1alpha1alpha2)))
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
•
•
•
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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
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
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
(R  D)
A
• 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
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Zari Johnson
Meghan McCabe
Kelly Pippins
Mahati Sharma
Robert “Bobby” Sheehan
Gina Siddiqui
Laura Sloofman
Peter Ucciferro
Dr. John A. Pelesko
References
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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