Positive periodic solutions of an n-species ecological system with infinite delay

The existence of positive periodic solution of n-species ecological system with infinite delay x ˙ i ( t ) = h i ( t , x t ) b i ( t , x t ) - a i ( t , x t ) x i ( t ) - ∫ - ∞ t G i ( t , s , x 1 ( s ) , … , x n ( s ) ) d s , i = 1 , … , n is studied by the authors. Under the suitable conditions, the main results are obtained by using of the method developed by Wang [On the existence of periodic solutions of functional differential equations, Chinese Ann. Math. 10A(3) (1989) 366–372 (in Chinese)] and the essential Theorems 2.1–2.3 of Sawano [Exponential asymptotic stability for functional differential equations with infinite retardations, Tohoku Math. J. 31 (1979) 363–382]. Our results differ from the ones given by Wang [Positive periodic solutions of an n -species ecological system, Acta Math. Appl. Sinica 17(1) (1994) 1–8 (in Chinese)].