In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “ ” and a chemical stimulus “ ” in a bounded and regular domain of . The equation for is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as for . The chemical substance distribution satisfies the elliptic equation The evolution of is also determined by a logistic type growth term . The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for and any .