Transcript 4-26-09 260 Presentation Outline 1x

DRAFT 2 PRESENTATION
Dr. Pelesko
MATH 260
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
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
Model Assumptions
Interspecies competition with House Wrens is the only
major contribution to the failing Bewick’s Wren
population
Single Species Model (Gina)
Aim 2: Two Species Model
VS
HOUSE WREN
BEWICK’S WREN
Model Equations
Non-Dimensionalization
Final Equations
So what is a competition coefficient?
dN 2
K 2  N 2  N1 21
 (r2 )(N 2 )
dt
K2
• 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)
(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
•
•
•
•
•
•
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 PlainsTexas
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
(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