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Numerical and Analytical Solutions of Volterra’s
Population Model
Malee AlexanderGabriela Rodriguez
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OverviewVolterra’s equation models
population growth of a species in a closed system
We will present two ways of solving this equation:◦Numerically: as a coupled system of
two first-order initial value problems◦Analytically: phase plane analysis
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Volterra’s Model
a > 0 is the birth rate coefficientb > 0 is the crowding coefficientc > 0 is the toxicity coefficient
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Nondimensionalization
For u(0)=u0 where k=c/ab
Variables are dimensionlessFewer parameters
t
dxxuuuudt
du
0
2 )(
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Numerical Solution
Solve it in the form of a coupled system of differential equations
Substitute:yeu
uy
ln
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Simplify:
Differentiate with respect to t to obtain a pure ordinary differential equation:
Substitute: and to get:
'yx uuy /''
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Coupled Initial Value SystemSubstitute: and
and therefore:
So we have the coupled system:
'' yeu y yeu uxuyu ''
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Solving using Runge-KuttaThe Runge-Kutta method
considers a weighted average of slopes in order to solve the equation
More accurate than Euler’s method
Need 4 slopes given by a function f( t , y) that defines the differential equation
Slopes denoted:Also need several intermediate
variables
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Runge-Kutta ProcessFirst slope: Second slope: need to go halfway
along t-axis to to produce a point where then use the function to determine second slope:
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Follow same steps again but with new slope to obtain third slope:
So, go from to the linealong a line of slopeto obtain a new number
So the third slope is:
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To obtain the fourth slope, useto produce a point on the lineso we get the pointTo obtain the fourth slope:
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Take the average of the four slopes.
Slopes that come from the points with must be counted twice as heavily as the others:
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Runge-Kutta SolutionTherefore, our general solution is:
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Solution to coupled system of Volterra Model:
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Phase Plane AnalysisPhase lines of similar to first order
differential equations. Phase planes
◦ Have points for each ordered pair of the population for each dependent variable
◦ Are not explicitly shown at a specific time. ◦ A solution taken as t evolves.
Plot many solutions in a phase plane simultaneously = phase portrait
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Phase Plane Analysis
ux
x
1
x(0)=
)1( 0u
u(0)= 0u
t
dxxuy0
)(
System:
Define in the original problem…
xuu
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…to produce the following system
Our equation:
uy y (0) =0
)1( yu
dy
du u(0)= 0u
y
euyyu
)1()1()( 0
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Phase portrait of with )(yu ,5.0
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Methods
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Conclusion
Nondimensionalization of our solution
numerically solve and analyze the Volterra model.
1)solved numerically the equation in a first-order coupled system, 2)applied phase plane analysis3)Obtain results:
*The population approaches zero for any values of the parameters: birth rate, competition coefficient, and toxicity coefficient*
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Bibliography R. L. Burden and J.D. Faires,
Numerical Analysis, 5th ed., Prindle, Weber & Schmidt, Boston, MA, 1993.
Thomson Brooks/Cole, Belmont, CA, 2006.
http://findarticles.com/p/articles/mi_7109/is_/ai_n28552371
TeBeest, Kevin. Numerical and Analytical Solutions of Volterra’s Population Model. Siam Review, Vol. 39, No. 3. (Sept 1997). Pp. 484-493.