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

Abstract: In this paper we study a two-dimensional process $(X, Y)$ arising as nonnegative solution to stochastic differential equations both 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 dies out in finite time, and process $Y$ converges to $0$,  we identify rather sharp conditions under which the process $Y$ exhibits, respectively, the following behaviors: extinction with probability one,  or extinguishing  with probability one, or both extinction and extinguishing occurring  with strictly positive probabilities.