Positive piecewise continuous quasi-periodic solutions to logistic impulsive differential equations

Abstract

ABSTRACT To prove the existence of piecewise continuous solutions to a logistic quasi-periodic differential system with impulses (whose coefficients have rationally independent periods), this system is divided into a differential equation and a difference equation. The quasi-periodicity of a function is proved by showing that this function is the uniform limit of a series of trigonometric polynomials with a finite total number of frequencies. The asymptotically stable quasi-periodic positive and piecewise continuous solution is proved to exist and to be unique. Quasi-periodic variation of the environment leads to a quasi-periodic growth of the population size in the sense that the rationally independent frequencies of the system are also frequencies of the quasi-periodic solution. The positive solutions have a repeated behavior similar to that of the quasi-periodic solution for a sufficiently long time due to asymptotical stability. The separation of the continuous-discrete system into a differential equation and a difference equation is a method of proving the existence of a quasi-periodic solution with perturbed coefficients of the impulsive system.