Analysis On Dynamics Of Some Stochastic Population Systems With Lévy Jumps

In recent years, the dynamical behaviors of stochastic population systems perturbedby white noise have been extensively studied, and many important results have beenreported. However, population systems may suffer some sudden environmental shockswhich can cause jumps in population size. Therefore, it is necessary to study the dynam-ics of stochastic population systems perturbed by jumping noise. This paper will focuson the dynamics of stochastic population systems driven by L′evy process and reveal howthe jumping noise affects the population systems. The main results are as follows:1. The survival analysis and stability of stochastic Gilpin-Ayala model with L′evyjumps. By stochastic analysis, this paper studies the strong persistence in the mean, ex-tinction, stochastic permanence and exponential stability, and obtains the critical numberbetween strong persistence in the mean and extinction. Results show that white noiseis unfavorable for the survival of the species, and if the coe?cient of jumping noise ispositive, then it is favorable for the persistence of the species while if the coe?cient ofjumping noise is negative then it is unfavorable.2. The dynamics of a stochastic Gilpin-Ayala mutualism system with L′evy jump-s. This paper establishes the su?cient conditions for the uniqueness and existence ofthe global positive solution, stochastic ultimate boundedness, strong persistence in themean, stochastic permanence and non-persistence. Some asymptotic properties are alsostudied. Results show that when the intensity of the white noise is large, the system is non-persistent, when the intensity of the white noise is su?ciently small, the system will keepthe persistence; the jumping noise can make non-persistent system strongly persistent inthe mean, it can also make persistent system non-persistent.3. The dynamics of a stochastic non-autonomous Gilpin-Ayala competition systemwith L′evy jumps. This paper investigates the existence and uniqueness of the globalpositive solution, stochastic ultimate boundedness, pathwise estimation, and establishessu?cient conditions for stochastic permanence and extinction. Results show that whitenoise on the competition coe?cients has little effect on the survival of the species, and thejumping noise on the growth rate plays an important role in the persistence and extinctionof the species.4. The dynamics of a stochastic Holling II one-predator two-completing prey systemwith L′evy jumps. On the basis of existence and uniqueness of the global positive solution,su?cient conditions for persistence and extinction of each species are obtained. Resultsshow that it is advantageous to the system when the coe?cient of the jumping noise ispositive, and it is disadvantageous when the coe?cients of the jumping noise is negative.5. A stochastic Holling II predator-prey model under Markovian switching withL′evy jumps. This paper establishes su?cient conditions for the strong persistence in themean and extinction of each species, and also obtains the critical number between theextinction and strong persistence in the mean of the prey species if there is no predatorspecies. Results show that telephone noise and jumping noise have different effects on thesurvival of the population.