Existence and global attractivity of a periodic solution for a prey–predator model with sex-structure

Abstract

In this paper, we investigate the following periodic sex-structure model: x 1 ′ ( t ) = x 1 ( t ) b 1 ( t ) x 2 ( t ) x 1 ( t ) - d 1 ( t ) - k ( t ) x 1 ( t ) - k ( t ) x 2 ( t ) - c 1 ( t ) x 3 ( t ) , x 2 ′ ( t ) = x 2 ( t ) β ( t ) - k ( t ) x 1 ( t ) - k ( t ) x 2 ( t ) - c 1 ( t ) x 3 ( t ) , x 3 ′ ( t ) = x 3 ( t ) - a ( t ) - b ( t ) x 3 ( t ) + c 2 ( t ) x 1 ( t ) + c 2 ( t ) x 2 ( t ) . The sufficient and realistic conditions are obtained for the existence of positive periodic solution of above system. Further, by constructing a V functional and using the result of the existence of positive periodic solutions, the global attractivity of a positive periodic solution for above system is obtained.