In this work, a prey-predator model with both state-dependent impulsive harvesting and constant rate harvesting is investigated, where the replenishment rate of prey and the harvesting rate are linearly related with the selected threshold. By first using the successor function method and differential equation geometry theory, the existence, uniqueness and asymptotic stability of the order-1 periodic solution are discussed. And then numerical simulations with an example are given to illustrate the feasibility of the theorem-related results. Moreover, in order to increase the total profit, the optimization strategy is presented and the optimal threshold is obtained.

In this work, a prey-predator model with square root response function under a state-dependent impulse is proposed. Firstly, according to the differential equation geometry theory and the method of successor function, the existence, uniqueness and attractiveness of the order-1 periodic solution are analyzed. Then the stability of the order-1 periodic solution is discussed by the Poincaré criterion for impulsive differential equations. Finally, we show a specific example and carry out numerical simulations to verify the theoretical results.

In this paper, a mathematical model for microbial treatment in livestock and poultry sewage is proposed and analyzed. We consider periodic addition of microbial flocculants to treat microorganisms such as Escherichia coli in sewage. Different from the traditional models, a class of composite dynamics models composed of impulsive differential equations is established. Our aim is to study the relationship between substrate, microorganisms and flocculants in sewage systems as well as the treatment strategies of microorganisms. Precisely, we first show the process of mathematical modeling by using impulsive differential equations. Then by using the theory of impulsive differential equations, the dynamics of the model is investigated. Our results show that the system has a microorganismsextinction periodic solution which is globally asymptotically stable when a certain threshold value is less than one, and the system is permanent when a certain threshold value is greater than one. Furthermore, the control strategy for microorganisms treatment is discussed. Finally, some numerical simulations are carried out to illustrate the theoretical results.