Numerical and Analytical Solutions of Volterra’s Population Model Malee Alexander Gabriela Rodriguez

Overview Volterra’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 coefficient b > 0 is the crowding coefficient c > 0 is the toxicity coefficient

Nondimensionalization For u(0)=u 0 where k=c/ab Variables are dimensionless Fewer parameters

Numerical Solution Solve it in the form of a coupled system of differential equations Substitute:

Simplify: Differentiate with respect to t to obtain a pure ordinary differential equation: Substitute: and to get:

Coupled Initial Value System Substitute: and and therefore: So we have the coupled system:

Solving using Runge-Kutta The 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 Process First 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 line along a line of slope to obtain a new number So the third slope is:

To obtain the fourth slope, use to produce a point on the line so we get the point To 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 Solution Therefore, our general solution is:

Solution to coupled system of Volterra Model:

Phase Plane Analysis Phase 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 x(0)= u(0)= System: Define in the original problem…

…to produce the following system Our equation: y (0) =0 u(0)=

Phase portrait of with

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 analysis 3)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, Thomson Brooks/Cole, Belmont, CA, ai_n ai_n TeBeest, Kevin. Numerical and Analytical Solutions of Volterra’s Population Model. Siam Review, Vol. 39, No. 3. (Sept 1997). Pp