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.