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[Hong Qian <qian@rialto.amath.washington.edu>]

the reference:

Three-Dimensional Competitive Lotka--Volterra Systems with no 
Periodic Orbits

P. van den Driessche, M. L. Zeeman 

The following conjecture of M. L. Zeeman is proved. If three interacting
species modeled by a competitive Lotka--Volterra system can each resist 
invasion at carrying capacity, then there can be no coexistence of the 
species. Indeed, two of the species are driven to extinction. It is also 
proved that in the other extreme, if none of the species can resist 
invasion from either of the others, then there is stable coexistence of 
at least two of the species. In this case, if the system has a fixed 
point in the interior of the positive cone in R3, then that fixed point 
is globally asymptotically stable, representing stable coexistence of 
all three species. Otherwise, there is a globally asymptotically stable 
fixed point in one of the coordinate planes of R3, representing stable 
coexistence of two of the species. 

SIAM Journal on Applied Mathematics
Volume 58, Number 1
pp. 227-234