Permanence for General Nonautonomous Impulsive Population Systems of Functional Differential Equations and Its Applications

Abstract

In this paper, we investigate general impulsive nonautonomous population dynamical systems of functional differential equations. By utilizing the method of multiple Liapunov-like functionals to construct the permanence region, a general criterion on the permanence for the system is established. Furthermore, as applications of this general criterion, a class of impulsive nonautonomous n-species Lotka-Volterra competitive systems with delays and a class of impulsive nonautonomous 3-species Lotka-Volterra food chain systems with delays are discussed. Some new and useful sufficient conditions on the permanence for these systems are established.

The periodic solutions for general periodic impulsive population systems of functional differential equations and its applications

“…Recently, a few studies on permanence of general nonautonomous impulsive systems (1) and (2) and their special cases were made in [1,5,20,21,2,6,7,23,24,12,[25][26][27][28]30,31,35,17,36,18,4]. Therefore, applying Theorem 1 and Corollary 1 of this paper we obviously can obtain that if systems are ω-periodic, then there is at least one positive ω-periodic solution under the permanent conditions.…”

“…Choose an integer p ≥ 0 such that t 2 ∈ (t 1 + pω, t 1 + (p + 1)ω], then from (33) and (36) it follows that…”

“…For more details see Theorem 3.1 in [19]. At the same time, we also see that the dynamical properties for many special cases of systems (1) and (2) have been extensively studied, for example [1,5,20,21,2,3,22,6,7,[23][24][25][26][27][28][29][30][31][32][33]8,[34][35][36] and the references cited therein. The main study subjects are the permanence, persistence and extinction of species, the local and global asymptotic stability of systems, the existence and uniqueness of positive periodic solution and almost periodic solution, and the bifurcation and dynamical complexity, etc.…”

Coexistence for an Almost Periodic Predator-Prey Model with Intermittent Predation Driven by Discontinuous Prey Dispersal

“…Currently, theories of impulsive differential equations [16] have been introduced into population dynamics. A large number of models have been described by impulsive diffusion (see [14,[17][18][19][20][21][22][23][24]) during the past couple of decades.…”

Permanence and global stability for nonautonomous N-species Lotka–Volterra competitive system with impulses and infinite delays

“…Many important and interesting population dynamical systems have been extensively studied, see [5][6][7][8]10,[15][16][17]22,31]. In [6], the authors investigated the following nonautonomous N-species Lotka-Volterra competitive system with impulsive effects u i (t) = u i (t) a i (t) − n l=1 b il (t)u j (t) , t = t k , u i t + k = (1 + p ik )u i (t k ), i = 1, 2, .…”