Transcript The Canadian Lynx vs. the Snowshoe Hare: Predator

The Canadian Lynx vs. the
Snowshoe Hare: The
Predator-Prey
Relationship and the
Lotka-Volterra Model
By: Ryan Winters and Cameron
Kerst
Photo Courtesy of
http://taggart.glg.msu.edu/bs110/lynx1.gif
An Overview
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The Canadian Lynx population fluctuates based upon the
Snowshoe Hare population
Share a common habitat in the Boreal forests of Canada
All data comes from records of the Hudson Bay fur
company
Hare and Lynx Populations
90
80
70
Population
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60
Hare Population
Lynx Population
50
40
30
20
10
0
1895
1900
1905
1910
Year
1915
1920
1925
Background of the LotkaVolterra Model
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Developed Simultaneously by
Alfred J. Lotka and Vito
Volterra
Volterra, an Italian professor of
math, developed the model
while trying to explain his son’s
observations of fish predators.
Lotka, a chemist, demographer
ecologist and mathematician,
addressed the model in his
book Elements of Physical
Biology.
Explaining the Model
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dH/dt is Malthusian, depends :
aH(t)
extension of the basic Verhulst
(logistic) Model
Outputs rate at which the
respective population in
changing at time t
a=intrinsic rate of Hare
population increase (births)
b=predation rate coefficient
c=reproduction rate of
predators per 1 prey eaten
e=predator mortality rate
dH= aH(t)-bH(t)L(t)
dt
dL= cH(t)L(t)-eL(t)
dt
Applying the Data
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We chose values for our
coefficients that best fit
our population data graph
Also initial conditions
were taken under
consideration in order to
most accurately depict
our original data
This yielded these rate
equations
H’=0.7R(t)-1.25R(t)L(t)
L’=R(t)L(t)-L(t)
a=0.7
b=1.25
c=1
e=1
IVP and the Model
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After finding our rate
equations we then
formed an IVP with an
initial conditions and rate
equations
We used our coefficients
that we found and used
Euler’s Method to
compare our model with
the actual data
I.C.=
H(1900)=3*
L(1900)=.4*
*Population in thousands
R.E.=
H’= aH(t)-bH(t)L(t)
L’= cH(t)L(t)-eL(t)
Works Consulted
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Works Consulted:
Lotka, Alfred J. Elements of Physical Biology.
Mahaffy, Joseph M. “Lotka-Volterra Models.” San Diego State University: 2000.
http://wwwrohan.sdsu.edu/~jmahaffy/courses/f00/math122/lectures/qual_de2/qualde2.ht
ml
McKelvey, Steve. “Lotka-Volterra Two Species Model.”
<http://www.stolaf.edu/people/mckelvey/envision.dir/lotka-volt.html >
Sharov, Alexei. “Lotka-Volterra Model.” 01/12/1996. <
http://www.gypsymoth.ento.vt.edu/~sharov/PopEcol/lec10/lotka.html >
“Vito Voltera.” School of Mathematics and Statistics, University of St. Andrews,
Scotland. December 1996. < http://www-groups.dcs.stand.ac.uk/~history/Mathematicians/Volterra.html >
“Alfred J. Lotka.” Wikipedia.
<http://www.stolaf.edu/people/mckelvey/envision.dir/lotka-volt.html >
12/01/2005
The End
Thank You!!!