Narrow Results

We consider the dynamic behaviors of a mathematical chemostat model with state dependent impulsive perturbations. By using the Poincaré map and analogue of Poincaré's criterion, some conditions for the existence and stability of positive periodic solution are obtained. Moreover, we show that there is no periodic solution with order larger than or equal to three. Numerical simulation are carried out to illustrate the feasibility of our main results, thus implying that the presence of pulses makes the dynamic behavior more complex.

In a dual risk model, the premiums are considered as the costs and the claims are regarded as the profits. The surplus can be interpreted as the wealth of a venture capital, whose profits depend on research and development. In most of the existing literature of dual risk models, the profits follow the compound Poisson model and the cost is constant. In this paper, we develop a state-dependent dual risk model, in which the arrival rate of the profits and the costs depend on the current state of the wealth process. Ruin probabilities are obtained in closed-forms. Further properties and results will also be discussed.