  We give a theoretical framework of stochastic non-canonical Hamiltonian
systems as well as their modified symplectic structure which is named
stochastic K-symplectic structure. The framework can be applied to the study of
the Lotka--Volterra model perturbed by external noises. In terms of internal
properties of the stochastic Lotka--Volterra model, we propose different kinds
of stochastic K-symplectic integrators which could inherit the positivity of
the solution. The K-symplectic conditions are also obtained to ensure that the
proposed schemes admit the same geometric structure as the original system.
Besides, the first-order condition of the proposed schemes in $L^1(\Omega)$
sense are given based on the uniform boundedness of both the exact solution and
the numerical one. Several numerical examples are illustrated to verify above
properties of proposed schemes compared with non-K-symplectic ones.
