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An n -Dimensional Extension of the Volterra-Lotka Model

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). Conceptually, we began with the Volterra-Lotka model as inspiration for food chain behavior.

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An n -Dimensional Extension of the Volterra-Lotka Model

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

  2. Modeling Food Chain Behavior(1) Conceptually, we began with the Volterra-Lotka model as inspiration for food chain behavior. 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

  3. xRabbits x1 x2 xFoxes xn x3 xn-1 xi xrock xpaper xscissors Modeling Food Chain Behavior(2) # of species 2 3 n Volterra-Lotka Model Species behave like a game of Rock-Paper-Scissors. (RPS3). The RPSn system. No relations exist outside of successive ones, as indicated by this monkey and dog. Note that the Triangle Inequality still holds, and the system has a Hauzzenstraβe factor of Log(13i)

  4. The RPSn Model Initial condition vector System of rate equations • 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.

  5. RPSn Notation Parameter Matrix Initial Condition Vector

  6. The Volterra-Lotka Case (RPS2)

  7. Results(1) Equilibrium Solutions 1. 2. 3.

  8. Results(2) Periodic Solutions 4. 5. Non-Periodic/Equilibrium Solutions http://cache.daylife.com/imageserve/02PI7QC7rbh0G/610x.jpg

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

  10. Equilibrium Solutions(1) Result 1: , , and http://ghostleg.com/blog/wp-content/uploads/2008/08/christin.jpg

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

  12. Periodic Solutions Result 4: , , and Periodic Solutions • Observations • Periodic solutions curves “grow” out of the straight line solutions. • In RPS3, same amplitudes • In RPS4, similar amplitudes

  13. Result 5: , , , and Non-Periodic/Equilibrium 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.

  14. Interesting Examples • RPSn can exhibit systems which have elements of periodicity, as well as overall increasing or decreasing • Parameter and IC sensitivity http://8vsb.files.wordpress.com/2008/03/rock-paper-scissors.png

  15. Equilibrium Solution Results 1. 2. 3.

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

  17. Conclusion • Qualitatively observe bifurcation values • Why focus on these specialized cases? • Much more work to be done • Food ladder • Other model variations http://thefuntimesguide.com/images/blogs/rockpaperscissors.jpg

  18. References • Blanchard, P., Devaney, R. and Hall, G. Differential Equations. 3ed. • Boston, USA: Thomson-Brooks/Cole, 2006. pp. 11-13, 482. • Chauvet, E., Paullet, J., Previte, J. and Walls, Z. A Lotka-Volterra Three-Species Food Chain. Mathematics Magazine, • 75(4):243-255, October 2002. • Mathematica 6. Computer software. Wolfram Research Inc., 2008; 32-bit Windows, v. 6.0.2.1. http://boilingpotusa.files.wordpress.com/2006/11/snakefrog.JPG

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