Turnpike property in a particular class of stochastic Lotka-Volterra systems

We study a controlled stochastic Lotka-Volterra model for a one-predator-two-prey model with white noise. We consider an optimal control problem for this model and the stability of its solutions. We discuss a type of stability for state trajectories, optimal control and adjoint trajectories, called the Turnpike property. We assume linear growth and Lipschitz conditions in the drift and diffusion terms, prove the Turnpike property, applying the Hamiltonian formalism and the maximum principle to this stochastic control problem, and express the optimal control in terms of state and adjoint variables. Finally, we illustrate our results with an example in which we numerically solve the stochastic differential equation systems of the model.