Predator-Prey%20Dynamics%20for%20Rabbits,%20Trees,%20

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Predator-Prey Dynamics for Rabbits, Trees,
Romance

• J. C. Sprott
• Department of Physics
• University of Wisconsin - Madison
• Presented to
• International Conference on Complex Systems
• in Nashua, NH
• on May 10, 2002

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Lotka-Volterra Equations

• R rabbits, F foxes
• dR/dt r1R(1 - R - a1F)
• dF/dt r2F(1 - F - a2R)

r and a can be or -
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Types of Interactions
dR/dt r1R(1 - R - a1F) dF/dt r2F(1 - F - a2R)

a2r2
Prey- Predator
Competition
-

a1r1
Predator- Prey
Cooperation
-
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Equilibrium Solutions

• dR/dt r1R(1 - R - a1F) 0
• dF/dt r2F(1 - F - a2R) 0

Equilibria

• R 0, F 0
• R 0, F 1
• R 1, F 0
• R (1 - a1) / (1 - a1a2), F (1 - a2) / (1 -
  a1a2)

F
R
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Coexistence

• With N species, there are 2N equilibria, only one
  of which represents coexistence.
• Coexistence is unlikely unless the species
  compete only weakly with one another.
• Diversity in nature may result from having so
  many species from which to choose.
• There may be coexisting niches into which
  organisms evolve.
• Species may segregate spatially.

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Alternate Spatial Lotka-Volterra Equations

• Let Si(x,y) be density of the ith species
  (rabbits, trees, seeds, )
• dSi / dt riSi(1 - Si - SaijSj)

j?i
where
S Sx-1,y Sx,y-1 Sx1,y Sx,y1 aSx,y
2-D grid
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Features of the Model

• Purely deterministic
• (no randomness)
• Purely endogenous
• (no external effects)
• Purely homogeneous
• (every cell is equivalent)
• Purely egalitarian
• (all species obey same equation)
• Continuous time

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Typical Results
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Dominant Species
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Fluctuations in Cluster Probability
Cluster probability
Time
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Power Spectrumof Cluster Probability
Power
Frequency
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Sensitivity to Initial Conditions
Error in Biomass
Time
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Results

• Most species die out
• Co-existence is possible
• Densities can fluctuate chaotically
• Complex spatial patterns spontaneously arise

One implies the other
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Romance(Romeo and Juliet)

• Let R Romeos love for Juliet
• Let J Juliets love for Romeo
• Assume R and J obey Lotka-Volterra Equations
• Ignore spatial effects

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Romantic Styles
dR/dt rR(1 - R - aJ)

a
Narcissistic nerd
Cautious lover
-

r
Eager beaver
Hermit
-
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Pairings - Stable Mutual Love
Cautious Lover
Eager Beaver
Narcissistic Nerd
Hermit
Narcissistic Nerd
46
67
5
0
Eager Beaver
67
39
0
0
Cautious Lover
5
0
0
0
Hermit
0
0
0
0
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Love Triangles

• There are 4-6 variables
• Stable co-existing love is rare
• Chaotic solutions are possible
• Butnone were found in LV model
• Other models do show chaos

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Summary

• Nature is complex
• Simple models may suffice

but
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References

• http//sprott.physics.wisc.edu/
  lectures/iccs2002/ (This talk)
• sprott_at_juno.physics.wisc.edu