The author deals with the following reaction-diffusion system \[ u_t-\Delta u= au- buv,\quad v_t-\Delta v= cu- duv- ev \] in a bounded and regular domain \(\Omega\) of \(\mathbb{R}^d\), with smooth initial conditions \(u_0,v_0\geq 0\), \(u_0\neq 0\) and homogeneous Dirichlet boundary conditions. The author is interested in under what choice of the parameters the system evolves towards a stationary solution. Here, the author provides a partial answer to this problem: he establishes under what choice of parameters there exists a nontrivial periodic solution and a (compact) global attractor.