Strong convergence on a stochastic controlled Lotka-Volterra 3-species model with L´evy jumps

Abstract

In this paper we study two properties of the numerical solutions of a controlled stochastic Lotka-Volterra one-predator-two-prey model, namely the boundedness in the mean of the numerical solutions and the strong convergence of these solutions. We also establish and solve, by means of the Stochastic Maximum Principle, the corresponding optimal control problem in a population modeled by a Lotka-Volterra system with two types of stochastic environmental fluctuations: white noise and L´evy jumps. Our study shows, assuming standard linear growth and Lipschitz conditions on the drift and diffusion coefficients, that the boundedness of the numerical solutions and the strong convergence of the scheme are preserved in this stochastic model.