Impulsive stabilization and stability analysis for Gilpin–Ayala competition model involved in harmful species via LMI approach and variational methods

Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle

“…Remark 1. (A1) expands greatly the allowable range of parameters θ i , compared with the previous related literature (see, e.g., in [6,7]). For example, the harsh condition "0 < θ i < 1 withθ i being an even number, andθ i being an odd number" is deleted.…”

“…It is well known that Gilpin-Ayala competition model (GACM) has been hotly discussed (see in [1][2][3][4][5][6][7]) due to its importance in simulating two or more competing biological populations in nature. As diffusion is an essential characteristic of most biological populations, Ling Bai and Ke Wang began to investigate the global stability of reaction-diffusion Gilpin-Ayala ecosystem under Neumann zero boundary value in 2005 (see in [8]), and obtained good results.…”

“…However, many animal populations are at the edge of the biosphere, where the population density is usually zero, which is not reflected by Neumann zero boundary value. Thus, the Dirichlet zero boundary value was considered in recent literature [6,7]. In recent years, linear impulsive control and nonlinear impulsive control technology are widely used in ordinary differential dynamical systems and infinite dimensional dynamical systems [9][10][11][12][13][14][15][16].…”

“…Therefore, the linear impulsive control is considered in this paper. Note that impulse control is employed to make the GACM stable globally in [6,7], but this paper involves the impulsive disturbance, which is not limited to impulsive control. Minimax principle will be employed to derive the existence of multiple stationary solutions, which improve the method of Mountain Pass Lemma in [6].…”

“…Note that impulse control is employed to make the GACM stable globally in [6,7], but this paper involves the impulsive disturbance, which is not limited to impulsive control. Minimax principle will be employed to derive the existence of multiple stationary solutions, which improve the method of Mountain Pass Lemma in [6]. On the other hand, impulsive disturbance is considered in this paper, not just impulse control.…”