Existence of periodic solution for a harvested system with impulses at variable times

In this Letter, a mathematical model for the impulsive exploitation of a single species is proposed and analyzed. An impulsive harvested population model at variable times is investigated, and the sufficient and necessary conditions for the existence of periodic solution are established. We find that the system has a unique nontrivial periodic solution if and only if the initial value and the impulsive times satisfy some conditions specifically, which assure that density of the single species is a periodic surge. A numerical example is presented to illustrate the efficiency of the proposed results.