Existence of periodic solution and persistence for a delayed predator-prey system with diffusion and impulse (Q2890557)

This paper deals with a prey-predator impulsive system of the form NEWLINENEWLINE\[NEWLINE\begin{aligned} x_{1}^{\prime}(t)&=x_{1}(t)(r_{1}(t)-a_{11}(t)x_{1}(t)-a_{12}(t)y(t))+D_{21}(t)x_{2}(t)-D_{12}(t)x_{1}(t),\quad t\neq t_{k},\\ x_{2}^\prime (t)&=x_{2}(t)(r_{2}(t)-a_{21}(t)x_{2}(t))+D_{12}(t)x_{1}(t)-D_{21}(t)x_{2}(t),\quad t\neq t_{k},\\ NEWLINEy^{\prime }(t)&=a_{31}(t)x_{1}(t-\tau )y(t-\tau)+r(t)y(t)-a_{32}(t)y^{2}(t),\quad t\neq t_{k},\\ \Delta x_{1}(t_{k})&=x_1 (t_{k}^{+})-x_{1}(t_{k}^{-})=b_{1k}x_{1}(t_{k}),\\ \Delta x_{2}(t_{k})&=x_2 (t_{k}^{+})-x_{2}(t_{k}^{-})=b_{2k}x_{2}(t_{k}),\\ \Delta y(t_{k})&=y(t_{k}^{+})-y(t_{k}^{-})=b_{3k}y(t_{k})\end{aligned}NEWLINE\]NEWLINE with the initial conditions NEWLINENEWLINE\[NEWLINEx_{i}(\theta )=\phi _{i}(\theta )\geq 0,~y(\theta )=\psi (\theta)\geq 0,~\theta \in (-\tau ,0],~\phi _{i}(0)>0,~\phi (0)>0,~i=1,2.NEWLINE\]NEWLINE By using Mawhin's continuation theorem, sufficient conditions for the existence of periodic solutions are obtained. Later, the boundedness and uniform permanence of the system is investigated. Moreover, an example and simulation are given to show the validity of main results.