




J. C. Sprott 

Department of Physics 

University of Wisconsin  Madison 



Presented to 

International Conference on Complex Systems 

in Nashua, NH 

on May 10, 2002 




R = rabbits, F = foxes 

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

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





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

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




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. 





Purely deterministic 

(no randomness) 

Purely endogenous 

(no external effects) 

Purely homogeneous 

(every cell is equivalent) 

Purely egalitarian 

(all species obey same equation) 

Continuous time 









Most species die out 

Coexistence is possible 

Densities can fluctuate chaotically 

Complex spatial patterns spontaneously arise 




Let R = Romeo’s love for Juliet 

Let J = Juliet’s love for Romeo 

Assume R and J obey LotkaVolterra Equations 

Ignore spatial effects 






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 






Nature is complex 





Simple models may suffice 




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



sprott@juno.physics.wisc.edu 
