We construct a Lyapunov function for tridiagonal competitive-cooperative systems. The same function is a Lyapunov function for Kolmogorov tridiagonal systems, which are defined on a closed positive orthant in R n . We show that all bounded orbits converge to the set of equilibria. Moreover, we show that there can be no heteroclinic cycles on the boundary of the first orthant, extending the results of H. I. Freedman and H. L. Smith [Differential Equations Dynam. Systems, 3 (1995), pp. 367--382].
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Fiedler et al. (1999) studied this question.
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