Dynamics of a stochastic predator–prey system in a polluted environment with pulse toxicant input and impulsive perturbations

Stochastic Periodic Solution and Persistence of a Nonautonomous Impulsive System with Nonlinear Self-Interaction

“…For ecological system, some discrete effects often appear at some short time interval, such as periodic spraying pesticides, harvesting, and stocking, which affect the growth of species and are often modeled by impulsive parameters. In last decades, many impulsive systems have been proposed and many good results have been reported, see, e.g., [10][11][12][13][14][15] and references cited therein. For example, Tan et al [15] investigated the existence of solution, stochastic permanence of the following impulsive model: dx 1 (t) � x 1 (t) r 1 (t) − a 11 (t)x 1 (t) − a 12 (t)x 2 (t) − b 1 (t)x 2 1 (t) + σ 1 (t)x 1 (t)dω 1 (t), t ≠ t k , dx 2 (t) � x 2 (t) r 2 (t) − a 21 (t)x 1 (t) − a 22 (t)x 2 (t) − b 2 (t)x 2 2 (t) + σ 2 (t)x 2 (t)dω 2 (t), t ≠ t k ,…”

“…Similarly, for stochastic system, it is very interesting to study the existence of stochastic periodic solution (periodic Markovian process). On the other hand, the extinction and permanence in the mean and stochastic persistence in probability are all very important dynamical behaviors (see [12,13,23,24]), but all these are not investigated in [15]. Hence, it is necessary for us to further explore these dynamical behaviors of (2).…”

Dynamical Analysis of a Stochastic Predator-Prey System with Lévy Noise and Impulsive Toxicant Input

“…Zhang and Tan [11] considered a stochastic predator-prey system in a polluted environment with impulsive toxicant input and impulsive perturbations.…”

The dynamic behaviors of a new impulsive predator prey model with impulsive control at different fixed moments

“…Meng DOI: 10.14736/kyb-2018-3-0522 et al proposed a new SEIRS epidemic disease model with two profitless delays and nonlinear incidence and analyzed the dynamic behaviors of the model under pulse vaccination [29]. Jin et al proposed a pulse vaccination SIR model with periodic infection rate, and studied the stability of the infection-free periodic solution and the existence of the positive periodic solution [38]. Gao et al formulated an SEIRS epidemic model with time delays and pulse vaccination, and obtained the exact infection-free periodic solution of the impulsive epidemic system and proved its global attractability and persistence [24].…”