An An nn-Dimensional Extension -Dimensional Extension of the Volterra-Lotka Modelof the Volterra-Lotka Model

Ariel Krasik-GeigerAriel Krasik-GeigerJon (Ari) MillerJon (Ari) Miller

Math 314-Differential EquationsMath 314-Differential Equations12/3/200812/3/2008

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

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Example of Volterra-Lotka:

Chinese > noodle

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

xRabbits xFoxes

Modeling Food Chain Behavior(2)Modeling Food Chain Behavior(2)

xpaperxscissors

xrock

xn

xn-1

x1 x2

xi

x3

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. (RPS3).

The RPSn system.No relations exist outside of successive ones, as indicated by this monkey and dog.

The RPSn ModelSystem of rate equations

n

i

x

x

x

x

tX

0

0

0

0

0

2

1

)(

111

11

232122222

12111111

nnnnnnnn

iiiiiiiii

n

xxxxxdt

dx

xxxxxdt

dx

xxxxxdt

dx

xxxxxdt

dx

dt

Xd

ParameterPrey 0

ParameterPredator 0

ParameterDecay 0

i

i

i

Initial 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.

RPSn Notation

Parameter Matrix

n

i

x

x

x

x

RPS

nnn

iii

n

0

0

0

0

222

111

2

1

,

Initial Condition Vector

The Volterra-Lotka Case (RPS2)

0 5 10 15t

0 .5

1 .0

1 .5

2 .0x xi t v. t

0 5 1 0 1 5t

0 .5

1 .0

1 .5

2 .0x xi t v. t

1

1,

011

1012RPS

1

75.1,

011

1012RPS

Results(1)

1.

2.

3.

Equilibrium Solutions

Results(2)

4.

5.

Periodic Solutions

Non-Periodic/Equilibrium Solutions

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Procedure

Guessing → Educated Guessing → Generalizations

– steady state solutions– periodic solutions– solutions which tend toward infinity– solutions with finite limiting behavior

Result 1: , , and 0i ii

c

c

tX

)( 0

Equilibrium Solutions(1)http://ghostleg.com

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Result 2: n even, , , and 0i ii

b

c

b

c

tX

)( 0

Result 3: , and iii

1

1

)( 0 tX

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 RPS3, same amplitudes– In RPS4, similar amplitudes

Periodic Solutions

Result 4: , , and 0i ii ci

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 bi ci di

a

a

tX

)( 0

Non-Periodic/Equilibrium Solutions

Interesting Examples

• RPSn can exhibit systems which have elements of periodicity, as well as overall increasing or decreasing

• Parameter and IC sensitivity

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Equilibrium Solution Results1.

2.

3.

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

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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.

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