Existence and global attractivity of periodic solution for enterprise clusters based on ecology theory with impulse

Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls

“…The sufficient conditions which ensure the existence and stability of unique globally attractive periodic solution of the system without time delays are established. The obtained results are completely new and complement the published works of [25][26][27][28][29]46].…”

“…Considering that the change of environment, the output of enterprises A and B usually change rapidly, Xu and Shao [29] considered the existence and global attractivity of periodic solution for the following enterprise clusters model with impulse and varying coefficients…”

“…Applying coincidence degree theory, Xu and Shao [29] obtained a set of sufficient criteria to ensure the existence of at least a positive periodic solution, and by constructing a Lyapunov functional, they established a sufficient condition to guarantee the uniqueness and global attractivity of the positive periodic solution for (6).…”

New results on competition and cooperation model of two enterprises with multiple delays and feedback controls

“…In many cases, the environment often varies and the output of two enterprises is subjected to rapid change at certain instants in time. Xu and Shao [4] established the following two-enterprise interaction model with impulses:…”

“…are the impulses at moments t k and t 1 < t 2 < · · · is a strictly increasing sequence such that lim k→+∞ t k = +∞ and q is a positive integer. Using continuation theorem and constructing an appropriate Lyapunov functional, Xu and Shao [4] discussed the periodic solution and global attractivity of (1.4). Considering the different effect of time delay on the two-enterprise interaction, Liao et al [5] investigated the stability and Hopf bifurcation of the following two-enterprise interaction model with delays:…”