The authors consider the parabolic system
where , , , bounded with smooth boundary, together with boundary conditions on , in . The main result states that this problem has a nonnegative -periodic solution if and are -periodic in , is quasipositive ( if , ) and growth conditions , for with , ( with a special provision for ) hold.
The proof uses a fixed point argument for the Poincaré map . It carries over to Robin type boundary condition; in the Neumann case a more stringent growth condition on is required.