Positive periodic solutions for an impulsive predator-prey system with delay and diffusion (Q2910574)

The authors study the following predator-prey system with prey dispersal and impulsive effect: NEWLINE\[NEWLINE\begin{cases} \dot x_1(t)=&x_1(t)(b_1(t)-d_1(t)x_1(t))-\frac{c(t)x_1(t)y^2(t)}{x_1^2(t)+m^2y^2(t)}+D_1(t)(x_2(t)-x_1(t)),\\ \dot x_2(t)=&x_2(t)(b_2(t)-d_2(t)x_2(t))+D_2(t)(x_1(t)-x_2(t)),\\ \dot y(t)=&y(t)(\frac{e(t)x_1(t-\tau(t))y(t-\tau(t))}{x_1^2(t-\tau(t))+m^2 y^2(t-\tau(t))}-d_3(t)),\;t\neq t_n,\;n\in Z^+,\\ x_i(t_n^+)=&(1+h_{in})x_i(t_n), \;i=1,2, y(t_n^+)=(1+h_{3n})y(t_n),\;t=t_n,\;n\in Z^+, \end{cases}\tag{1}NEWLINE\]NEWLINE where \(x_i(t)\) represents the density of the prey population at patch \(i\) \((i=1,2)\), \(y(t)\) represents the the density of the predator population at time \(t\), \(b_i(t)\) represents the intrinsic growth rate of the prey at patch \(i\), \(D_i(t)\) is the dispersal rate of the prey species, \(c(t)\) and \(e(t)\) are the capturing rate and the conversion rate of the predator, respectively. The functions above are continuous positive \(T\)-periodic; \(h_{in} (i=1,2,3)\) are constants and there is a positive integer \(q\) such that \(h_{in+q}=h_{in} (i=1,2,3),\;t_{n+q}=t_n+T.\) By using the continuation theorem in the coincidence degree theory, sufficient conditions are obtained for the existence of positive periodic solutions to system (1).