Long-term behaviour of ODEs

Hi,

If I have an autonomous locally Lipschitz system of n coupled (non-
linear) ODEs, for which all (unique) solutions are bounded to remain
in some compact subset of \real^n (and thus all solutions exist for
all t >= t_0), am I guaranteed that as t -> \infty, solutions will
converge to some equilibrium solution? If so, where can I find this
result? I have a feeling I read such a result (or something similar) a
while back but can't remember where.

If not, is there usual way to prove that all solutions to an ODE
system do exhibit this behaviour? That they always do is clear
numerically but I've no idea how to go about proving it (if the above
result does not hold). An example might be:

da/dt = -a / (a + b) + c
db/dt = -b / (a + b) + a / (a + b)
dc/dt = -c + b / (a + b)

with initial conditions a_0, b_0, c_0 > 0.

The class of ODEs in question model a fixed population, i.e. satisfy
x'_1(t) + ... + x'_n(t) = 0 for all times t >= t_0 in case this helps.
Similar ODEs I presume arise in biological/chemical context, but I
can't find any useful results.

Thanks,

Richard.
.

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