Dynamical analysis of a five-dimensioned chemostat model with impulsive diffusion and pulse input environmental toxicant

Dynamics of periodic switched systems describing a single population system with impulses and hibernation

“…Following the basic ideas and structure of the mathematical modeling of biological populations [1,3,4,10,11], the population dynamics is carried out under the following assumptions.…”

“…Presence of toxicant in the environments decreases the growth rate of species and its carrying capacity. In recent years, some investigations have been carried out to study the effect of toxicant on a single species population [8][9][10][11], and a lot of scholars have adopted mathematical modeling approach to study the influence of environmental pollution on the surviving of biological population [12,13]. Most of the previous work assumed that input of toxicant was continuous.…”

Dynamics of a stage-structured single population system with winter hibernation and impulsive effect in polluted environment

“…Impulsive equations are found in almost every domain of applied science and have been studied in many investigations [10,11,13,14,[16][17][18][20][21][22], they generally describe phenomena which are subject to steep or instantaneous changes. Especially, Jiao et al [10] suggested releasing pesticides is combined with transmitting infective pests into an SI model.…”

Existence and persistence of positive solution for a stochastic turbidostat model

“…They also pointed out that a small enough stochastic disturbance could cause the microorganism to die out even if the microorganism could be persistent in the deterministic model. More mathematical models about microorganism cultivation with constant dilution rate and perturbed phenomena could be found in [17][18][19][20][21][22][23][24]. The study of stochastic population models has been a focus of some scholars in recent years (see [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]).…”