PredatorPrey 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 
LotkaVolterra Equations



R = rabbits, F = foxes 

dR/dt = r_{1}R(1  R  a_{1}F) 

dF/dt = r_{2}F(1  F  a_{2}R) 
Types of Interactions
Equilibrium Solutions



dR/dt = r_{1}R(1  R  a_{1}F)
= 0 

dF/dt = r_{2}F(1  F  a_{2}R)
= 0 
Coexistence



With N species, there are 2^{N}
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. 
Alternate Spatial
LotkaVolterra Equations
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 
Typical Results
Dominant Species
Fluctuations in Cluster
Probability
Power Spectrum
of Cluster Probability
Sensitivity to Initial
Conditions
Results



Most species die out 

Coexistence is possible 

Densities can fluctuate chaotically 

Complex spatial patterns spontaneously
arise 
Romance
(Romeo and Juliet)



Let R = Romeo’s love for Juliet 

Let J = Juliet’s love for Romeo 

Assume R and J obey LotkaVolterra
Equations 

Ignore spatial effects 
Romantic Styles
Pairings  Stable Mutual
Love
Love Triangles



There are 46 variables 



Stable coexisting love is rare 



Chaotic solutions are possible 



But…none were found in LV model 



Other models do show chaos 


Summary



Nature is complex 





Simple models may suffice 
References



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



sprott@juno.physics.wisc.edu 