Predator-Prey Dynamics for Rabbits, Trees,

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Predator-Prey Dynamics for Rabbits, Trees,
Romance

• J. C. Sprott
• Department of Physics
• University of Wisconsin - Madison
• Presented at the
• Swiss Federal Research Institute (WSL)
• in Birmendsdorf, Switzerland
• on April 29, 2002

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Collaborators

• Janine Bolliger
• Swiss Federal Research Institute
• Warren Porter
• University of Wisconsin
• George Rowlands
• University of Warwick (UK)

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Rabbit Dynamics

• Let R of rabbits
• dR/dt bR - dR

rR
r b - d
Birth rate
Death rate

• r gt 0 growth
• r 0 equilibrium
• r lt 0 extinction

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Exponential Growth

• dR/dt rR
• Solution R R0ert

R
r gt 0
r 0
rabbits
r lt 0
t
time
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Logistic Differential Equation

• dR/dt rR(1 - R)

1
R
r gt 0
rabbits
t
0
time
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Effect of Predators

• Let F of foxes
• dR/dt rR(1 - R - aF)

Intraspecies competition
Interspecies competition
But The foxes have their own dynamics...
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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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Stable Focus(Predator-Prey)
r1(1 - a1) lt -r2(1 - a2)
r1 1 r2 -1 a1 2 a2 1.9
r1 1 r2 -1 a1 2 a2 2.1
F
F
R
R
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Stable Saddle-Node(Competition)
a1 lt 1, a2 lt 1
r1 1 r2 1 a1 1.1 a2 1.1
r1 1 r2 1 a1 .9 a2 .9
Node
Saddle point
F
F
Principle of Competitive Exclusion
R
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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Reaction-Diffusion Model

• Let Si(x,y) be density of the ith species
  (rabbits, trees, seeds, )
• dSi / dt riSi(1 - Si - SaijSj) Di?2Si

j?i
reaction
diffusion
where
?2Si Sx-1,y Sx,y-1 Sx1,y Sx,y1 - 4Sx,y
2-D grid
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Typical Results
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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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Parameters of the Model
Growth rates
Interaction matrix
1 r2 r3 r4 r5 r6
1 a12 a13 a14 a15 a16 a21 1 a23 a24 a25 a26 a31
a32 1 a34 a35 a36 a41 a42 a43 1 a45 a46 a51
a52 a53 a54 1 a56 a61 a62 a63 a64 a65 1
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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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Typical Results
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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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Fluctuations in Total Biomass
Time Derivative of biomass
Time
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Power Spectrumof Total Biomass
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/predprey/ (This talk)
• sprott_at_juno.physics.wisc.edu