This paper deals with a food chain model with density dependence as follows: \(\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u+u(1-u)-b_1uv,& \quad x\in \Omega ,t>0,\\ v_t=\Delta (\gamma _1(u)v )+uv-b_2vw+\theta _1(v-v^2),& \quad x\in \Omega , t>0,\\ w_t=\Delta (\gamma _2(v)w )+vw+\theta _2(w-w^2), & \quad x\in \Omega ,t>0,\\ \end{array}\right. } \end{aligned}\) under homogeneous Neumann boundary conditions in a smooth-bounded domain \(\Omega \subset \mathbb {R}^2.\) For \(i=1,2\) , the parameters \(b_i\) and \(\theta _i\) are positive and the function \(\gamma _i\) satisfies \(\gamma _i\in C^3([0,\infty ))\) and \(\gamma _i(s)>0\) for all \(s\ge 0\) . It is proved that this system possesses globally bounded classical solutions. Moreover, by constructing Lyapunov functionals, the global stability of the semi-coexistence steady states and the coexistence steady state under suitable conditions on parameters are established.
