Global exponential stability of positive periodic solution of the n-species impulsive Gilpin–Ayala competition model with discrete and distributed time delays

Figures & data

Table

$u_i\left(t\right)$	the population density	$i=1,2,\dots ,n$
$r_i\left(t\right)>0$	the natural growth rates	$i=1,2,\dots ,n$
$a_{ii}\left(t\right),b_{ii}\left(t\right),c_{ii}\left(t\right)>0$	the intraspecific competition rates	$i=1,2,\dots ,n$
$a_{ij}\left(t\right),b_{ij}\left(t\right),c_{ij}\left(t\right)>0$	the interspecific competition rates	$i,j=1,2,\dots ,n,i\ne j$
${\tau }_{ij}\left(t\right)>0$	the discrete time-delay	$i,j=1,2,\dots ,n$
$K_{ij}\left(s\right)>0$	the infinite distributed delay	$i,j=1,2,\dots ,n$
$\{t_k{\}}_{k=1}^{\mathrm{\infty }}$	the pulse sequence	$t_1<\dots <t_k<\dots ,t_k\to \mathrm{\infty }$
$u_i\left(t_k^{+}\right)$	the right limit of $u_i\left(t\right)$	$i=1,2,\dots ,n,k\in \mathbb{N}$
$u_i\left(t_k^{-}\right)=u_i\left(t_k\right)$	the left limit of $u_i\left(t\right)$	$i=1,2,\dots ,n,k\in \mathbb{N}$
${\alpha }_{ij},{\beta }_{ij},{\gamma }_{ij}>0$	the nonlinear measures	$i,j=1,2,\dots ,n$
$I_{ik}>-1$	the constants	$i=1,2,\dots ,n,k\in {\mathbb{N}}^{+}$