Dynamics of a new stage-structured population model with transient and nontransient impulsive effects in a polluted environment

In this paper, we consider a new stage-structured population model with transient and nontransient impulsive effects in a polluted environment. By using the theories of impulsive differential equations, we obtain the globally asymptotically stable condition of a population-extinction solution; we also present the permanent condition for the investigated system. The results indicate that the nontransient and transient impulsive harvesting rate play important roles in system permanence. Finally, numerical analyses are carried out to illustrate the results. Our results provide effective methods for biological resource management in a polluted environment.

Dynamics Analysis of a Stochastic Hybrid Logistic Model with Delay and Two-Pulse Perturbations

In this paper, we propose and discuss a stochastic logistic model with delay, Markovian switching, Lévy jump, and two-pulse perturbations. First, sufficient criteria for extinction, nonpersistence in the mean, weak persistence, persistence in the mean, and stochastic permanence of the solution are gained. Then, we investigate the lower (upper) growth rate of the solutions. At last, we make use of Matlab to illustrate the main results and give an explanation of biological implications: the large stochastic disturbances are disadvantageous for the persistence of the population; excessive impulsive harvesting or toxin input can lead to extinction of the population.

Bifurcation Analysis and Application for Impulsive Systems with Delayed Impulses

In this article, we present a systematic approach to bifurcation analysis of impulsive systems with autonomous or periodic right-hand sides that may exhibit delayed impulse terms. Methods include Lyapunov–Schmidt reduction and center manifold reduction. Both methods are presented abstractly in the context of the stroboscopic map associated to a given impulsive system, and are illustrated by way of two in-depth examples: the analysis of a SIR model of disease transmission with seasonality and unevenly distributed moments of treatment, and a scalar logistic differential equation with a delayed census impulsive harvesting effort. It is proven that in some special cases, the logistic equation can exhibit a codimension two bifurcation at a 1:1 resonance point.