Periodic solution for stochastic non-autonomous multispecies Lotka–Volterra mutualism type ecosystem

• We give sufficient conditions for the existence of positive periodic solution.
• We give sufficient conditions for the global attractiveness of the periodic solution.
• We prove that all species will be extinct if the intensity of noise is large enough.
• The results are illustrated by computer simulations.

The paper characterizes some qualitative dynamic properties of a stochastic non-autonomous multi-species mutualism model, with continuous periodic parameters. Using Khasminskii theory of stability with suitable Lyapunov functions, and M-Matrices, sufficient conditions are established to guarantee existence of positive periodic solutions to the system. We also provide conditions for the global attractiveness of the latter, or extinction of all species for sufficiently high volatility levels. Results are finally supported by numerical computations.