Permanence and extinction for a single-species system with jump-diffusion

In this study, we consider the effect of jump-diffusion random environmental perturbations on the permanence and extinction of a single-species dispersal periodic system in poor patchy environments with the possibility of species loss during their dispersion among patches. First, we prove that there is a unique global positive solution to the system with any initial positive value with probability 1, and we obtain the sufficient conditions that stochastically ensure the ultimate boundedness as well as the asymptotic polynomial growth of the population system. Next, we establish the sufficient conditions for the almost sure permanence in the mean, stochastic permanence, and extinction of the system. In particular, by constructing an appropriate integrating factor, we obtain the sufficient conditions for the almost sure weak permanence of the patch. The conditions obtained for permanence generalize the sufficient conditions established previously on the system without random environmental perturbations. Finally, we discuss the biological implications of the main results.