In this paper, we study the global existence and boundedness of a Neumann initial-boundary value problem for the following three-species spatial food chain model with nonlinear diffusion mechanisms and prey-taxis \(\begin{aligned} \left\{ \begin{aligned}&u_t=d_1\Delta u+u(1-u)-b_1uv,\\&v_t=\nabla \cdot (D_2(v)\nabla v)-\nabla \cdot (\xi (v)\nabla u)+uv-b_2vw-\theta _1v,\\&w_t=\nabla \cdot (D_3(w)\nabla w)-\nabla \cdot (\chi (w)\nabla v)+vw-\theta _2w, \end{aligned} \right. \end{aligned}\) where \(D_2(v)=(v+1)^m\,\) , \(D_3(w)=(w+1)^k\,\) , \(\xi (v)=v(1+v)^{\alpha -1}\,\) , \(\chi (w)=w(1+w)^{\beta -1}\) . The model can be regarded as an extension of the three-species food chain model (Jin et al., 2022). We show that when \(m\) , \(\alpha \) , \(k\) and \(\beta \) satisfy either of the following conditions (i). \(m\ge 1\) , \(\alpha \le 1\) , \(k>\max \left\{ \frac{n}{4},1\right\} \) , \(\beta \le 1\) ;
(ii). \(\alpha \ge 0\) , \(\alpha -m<\frac{2}{n}\) , \(k>\max \left\{ \frac{n}{4},1\right\} \) , \(\beta \le 1 \) ,
the problem has a global classical solution in a \(n\) -dimensional bounded domain. Moreover, it is proved the global existence of weak solutions in the sense of Definition 1.1 provided \(m\ge 1\) , \(\alpha \le 1\,\) , \(k>1\,\) , \(\beta \le 1\) or \(\alpha \ge 0\,\) , \(\alpha -m<\frac{2}{n},k>1\) , \(\beta \le 1\) . On the other hand, when \(m=k=0\,\) , this paper also proves the global existence of classical solutions in a \(n\) -dimensional bounded domain for \(\alpha ,\beta \in (-\infty ,\frac{2}{n})\) .
