Positive periodic solutions for three species Lotka-Volterra mixed ecosystems with periodic stocking

By using a new method, a set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions for three-species Lotka-Volterra mixed systems with periodic stocking:x′1(t) =x1 (t) (b1 (t) - a11 (t)x1 (t) - a12 (t)x2 (t) - a13 (t)x3 (t)) + S1 (t)x′2(t) = x2 (t) (- b2 (t) + a21 (t)x1 (t) - a22 (t)x2 (t) - a23 (t)x3 (t)) + S2 (t)x′3(t) = x3 (t)(- b3 (t) + a31 (t)x1 (t) - a32 (t)x2 (t) - a33 (t)x3 (t)) + S3 (t)where bi ( t ), aij (t) ( i,j= 1,2,3 ) are positive continuous T- periodic functions, Si (t) (i = 1,2,3) are nonnegative continuous T-periodic functions.