Permanence and global attractivity of a periodic predator-prey system with mutual interference and impulses (Q430290)

The main purpose of this paper is to study the permanence and global attractivity of the following \(T\)-periodic prey-predator system with impulses: NEWLINE\[NEWLINE \begin{aligned} x'(t)&=x(t)[r_1(t)-b_1(t)x(t)-c(t)y^m(t)], \;t\neq \tau_k, \\ y'(t)&=y(t)[-r_2(t)+\beta c(t)x(t)y^{m-1}(t)-b_2(t)y(t)], \;t\neq \tau_k, \\ x(\tau_k^+)&=(1+h_k)x(\tau_k), \;t=\tau_k, \\ y(\tau_k^+)&=(1+g_k)y(\tau_k), \;t=\tau_k. \end{aligned}NEWLINE\]NEWLINE Here, \(m\in (0,1)\) is the mutual interference constant between the predator and the prey, \(r_i\), \(b_i\), \(i=1,2\), and \(c\) are continuous periodic functions on \([0,+\infty)\) with a common period \(T>0\), and \(\beta, b_i(t), c(t)>0\), the \(\tau_k\) satisfy \(0=\tau_0<\tau_1<\tau_2<\dots\) with \(\tau_k\rightarrow +\infty\) as \(k\rightarrow \infty\), and that there is a positive integer \(q\) such that \(h_{k+q}=h_k\), \(g_{k+q}=g_k\), \(\tau_{k+q}=\tau_k+T\).NEWLINENEWLINEBy constructing a suitable Lyapunov function and using the comparison theorem for impulsive differential equations, and by imposing additional technical conditions, the authors are able to show that any positive solution of the above system is permanent and globally attractive.