An n-Dimensional Extension of the Volterra-Lotka Model Ariel Krasik-Geiger Jon (Ari) Miller Math...
-
Upload
imogene-hubbard -
Category
Documents
-
view
216 -
download
0
Transcript of An n-Dimensional Extension of the Volterra-Lotka Model Ariel Krasik-Geiger Jon (Ari) Miller Math...
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
ww
.juzprint.com/blog/w
p-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.
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
http://cache.daylife.com/im
ageserve/02PI7Q
C7rbh0G
/610x.jpg
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
/blog/wp-content/uploads/2008/08/christin.jpg
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
http://8vsb.files.wordpress.com/2008/03/rock-paper-scissors.png
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
http://thefuntimesguide.com/images/blogs/rockpaperscissors.jpg
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