An n-Dimensional Extension of the Volterra-Lotka Model Ariel Krasik-Geiger Jon (Ari) Miller Math 314-Differential Equations 12/3/2008

Modeling Food Chain Behavior(1) Key Elements of Volterra-Lotka: 1. Each species exhibits one prey relation 2. Each species exhibits one predator relation 3. Each species has an attrition (death) rate ww.juzprint.com/blog/wp-content/uploads/2008/07/captd5ec864c8f2c454191fc0006a58c2867china_obese_clinic_xhg109.jpg Example of Volterra-Lotka: Chinese > noodle Conceptually, we began with the Volterra-Lotka model as inspiration for food chain behavior.

x Rabbits x Foxes Modeling Food Chain Behavior(2) x paper x scissors x rock xnxn x n-1 x1x1 x2x2 xixi x3x3 Note that the Triangle Inequality still holds, and the system has a Hauzzenstraβe factor of Log(13i) Volterra-Lotka Model # of species 2 3 n Species behave like a game of Rock-Paper-Scissors. (RPS 3 ). The RPS n system. No relations exist outside of successive ones, as indicated by this monkey and dog.

The RPS n Model System of rate equationsInitial condition vector Model Assumptions: The initial populations of all species are non-negative. No interaction between non-successive species within any n-gon. Parameter conditions as indicated above.

RPS n Notation Parameter Matrix Initial Condition Vector

The Volterra-Lotka Case (RPS 2 )

Results(1) Equilibrium Solutions

Results(2) Periodic Solutions Non-Periodic/Equilibrium Solutions

Procedure Guessing → Educated Guessing → Generalizations –steady state solutions –periodic solutions –solutions which tend toward infinity –solutions with finite limiting behavior

Result 1:,, and Equilibrium Solutions(1)

Result 2: n even,,, and Result 3:, and Proof strategy of these results is similar to proof of Result 1. 1.Determine rate equations with given parameter matrix. 2.Evaluate rate equations at given initial conditions. 3.Show each species rate equations to be zero. Equilibrium Solutions (2)

Observations –Periodic solutions curves “grow” out of the straight line solutions. –In RPS 3, same amplitudes –In RPS 4, similar amplitudes Periodic Solutions Result 4:,, and Periodic Solutions

Similar Proof. –Show that every species’ rate equation is identical. Observations –Non-Periodic/Equilibrium Solutions occur the most. –Reason for not concentrating on these. –Unrealistic behavior. Result 5:,,, and Non-Periodic/Equilibrium Solutions

Interesting Examples RPS n can exhibit systems which have elements of periodicity, as well as overall increasing or decreasing Parameter and IC sensitivity

Equilibrium Solution Results

4. 5. Periodic Solution Results Non-Periodic/Equilibrium Solutions Results

Conclusion Qualitatively observe bifurcation values Why focus on these specialized cases? Much more work to be done –Food ladder –Other model variations

References Blanchard, P., Devaney, R. and Hall, G. Differential Equations. 3ed. Boston, USA: Thomson-Brooks/Cole, pp , 482. Chauvet, E., Paullet, J., Previte, J. and Walls, Z. A Lotka-Volterra Three-Species Food Chain. Mathematics Magazine, 75(4): , October Mathematica 6. Computer software. Wolfram Research Inc., 2008; 32-bit Windows, v