On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property

In this paper, we study some properties of the solutions of a stochastic Lotka–Volterra predator-prey model, namely, the boundedness in the mean of numerical solutions, the strong convergence for this kind of solutions, and the turnpike property of solutions of an optimal control problem in a population modelled by a Lotka–Volterra system with stochastic environmental fluctuations. Even though there are numerous results in the deterministic case, there are few results for the behavior of numerical solutions in a population dynamic with random fluctuations. First, we show, using the Euler–Maruyama scheme, that the boundedness of numerical solutions and the convergence of the scheme are preserved in the stochastic case. Second, we analyze a property of the long-term behavior of a Lotka–Volterra system with stochastic environmental fluctuations known as turnpike property. In optimal control theory, the optimal solutions dwell mostly in the neighborhood of a balanced equilibrium path, corresponding to the optimal steady-state solution. Our study shows, by means of the Stochastic Maximum Principle, that this turnpike property is preserved, when the noise in the system is small. Numerical simulations are implemented to support our results.