If is to eventually dominate will be driven to zero in finite time at a non zero value of (negative populations make no sense and the problem stops at such a point, or rather the rate of change of both populations are zero from that point on, or if we do not impose this restriction when one population becomes negative the other starts growing rather than contracting ... ).

That is there is a solution of:

which is the case iff:

(it is possible to get the explicit time dependence of the two populations by differentiating one of the equations again and then substituting from the other to get a second order linear constant coefficient homogeneous ODE which can be solved by the usual methods)

CB