  We study a two-dimensional process $(X, Y)$ arising as the unique nonnegative
solution to a pair of stochastic differential equations driven by independent
Brownian motions and compensated spectrally positive L\'evy random measures.
Both processes $X$ and $Y$ can be identified as continuous-state nonlinear
branching processes where the evolution of $Y$ is negatively affected by $X$.
Assuming that process $X$ extinguishes, i.e. it converges to $0$ but never
reaches $0$ in finite time, and process $Y$ converges to $0$, we identify
rather sharp conditions under which the process $Y$ exhibits, respectively, one
of the following behaviors: extinction with probability one, extinguishing with
probability one or both extinction and extinguishing occurring with strictly
positive probabilities.
