Periodic solutions and stationary distribution of mutualism models in random environments

• The mutualism model perturbed by white noise admits a positive periodic solution.
• Existence, global attractivity of the boundary periodic solutions are discussed.
• The mutualism model perturbed by both white noise and color noise is discussed.

This paper is concerned with mutualism models in random environments. For the periodic mutualism model disturbed by white noise, using Has’minskii theory of periodic solution, we show that this model admits a nontrivial positive periodic solution. Then sufficient conditions for the existence and global attractivity of the boundary periodic solutions are established. For the mutualism model disturbed by both white noise and telephone noise, sufficient conditions for positive recurrence and the existence of ergodic stationary distribution of the solution are established. Finally, examples are introduced to illustrate the results developed.