= Solution
The nonnegative quadrant is invariant because both right-hand sides have the corresponding population as a factor. There $u_2'\leq\rho\gamma u_2$. The differential inequality, or the <Gronwall inequality>, gives
$$
0\leq u_2(\tau)\leq u_2(0)e^{\rho\gamma\tau}\longrightarrow0\qquad(\gamma<0).
$$
Biologically $\gamma<0$ means $\delta>r_2$: per-capita removal exceeds even the grey squirrels' maximum uncrowded growth rate. Competition can only strengthen the decline.
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