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We study a two-dimensional process $(X, Y)$ arising as the unique nonnegativesolution to a pair of stochastic differential equations driven by independentBrownian motions and compensated spectrally positive L\'evy random measures.Both processes $X$ and $Y$ can be identified as continuous-state nonlinearbranching processes where the evolution of $Y$ is negatively affected by $X$.Assuming that process $X$ extinguishes, i.e. it converges to $0$ but neverreaches $0$ in finite time, and process $Y$ converges to $0$, we identifyrather sharp conditions under which the process $Y$ exhibits, respectively, oneof the following behaviors: extinction with probability one, extinguishing withprobability one or both extinction and extinguishing occurring with strictlypositive probabilities.

On the extinction-extinguishing dichotomy for a stochastic Lotka-Volterra type population dynamical system