Numerical and Analytical Solutions of Volterra’s Population Model Malee Alexander Gabriela Rodriguez
Jan 19, 2016
Numerical and Analytical Solutions of Volterra’s
Population Model
Malee AlexanderGabriela Rodriguez
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
Volterra’s Model
a > 0 is the birth rate coefficientb > 0 is the crowding coefficientc > 0 is the toxicity coefficient
Nondimensionalization
For u(0)=u0 where k=c/ab
Variables are dimensionlessFewer parameters
t
dxxuuuudt
du
0
2 )(
Numerical Solution
Solve it in the form of a coupled system of differential equations
Substitute:yeu
uy
ln
Simplify:
Differentiate with respect to t to obtain a pure ordinary differential equation:
Substitute: and to get:
'yx uuy /''
Coupled Initial Value SystemSubstitute: and
and therefore:
So we have the coupled system:
'' yeu y yeu uxuyu ''
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
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:
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:
To obtain the fourth slope, useto produce a point on the lineso we get the pointTo obtain the fourth slope:
Take the average of the four slopes.
Slopes that come from the points with must be counted twice as heavily as the others:
Runge-Kutta SolutionTherefore, our general solution is:
Solution to coupled system of Volterra Model:
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
Phase Plane Analysis
ux
x
1
x(0)=
)1( 0u
u(0)= 0u
t
dxxuy0
)(
System:
Define in the original problem…
xuu
…to produce the following system
Our equation:
uy y (0) =0
)1( yu
dy
du u(0)= 0u
y
euyyu
)1()1()( 0
Phase portrait of with )(yu ,5.0
Methods
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*
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.