Global stability of three trophic levels predator–prey model with alarm-taxis

This paper is concerned with the three trophic levels predator–prey system with alarm-taxis \(\begin{aligned} \left\{ \begin{array}{lll} u_{t}=d_{1} \Delta u+u\left( 1-u-\frac{a v}{v+\rho }\right) , &{} x \in \Omega , &{} t>0, \\ v_{t}=d_{2} \Delta v+v\left( \frac{b u}{v+\rho }-\alpha -\frac{c w}{w+\sigma }\right) , &{} x \in \Omega , &{} t>0, \\ w_{t}=d_{3} \Delta w-\chi \nabla \cdot \left( w\nabla (uv)\right) +w\left( \frac{m v}{w+\sigma }-\beta \right) , &{} x \in \Omega , &{} t>0 \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{u}_t=d_1\mathrm{\Delta }u+u\left(1,-,u,-,\frac{\mathit{av}}{v+\rho }\right),\hfill & x\in \mathrm{\Omega },\hfill & t>0,\hfill \\ v_t=d_2\mathrm{\Delta }v+v\left(\frac{\mathit{bu}}{v+\rho },-,\alpha ,-,\frac{\mathit{cw}}{w+\sigma }\right),\hfill & x\in \mathrm{\Omega },\hfill & t>0,\hfill \\ w_t=d_3\mathrm{\Delta }w-\chi \mathrm{\nabla }·\left(w,\mathrm{\nabla },(,u,v,)\right)+w\left(\frac{\mathit{mv}}{w+\sigma },-,\beta \right),\hfill & x\in \mathrm{\Omega },\hfill & t>0\hfill \end{array}\right)\end{array}$$ under homogeneous Neumann boundary condition in smooth bounded domains \(\Omega \subset {\mathbb {R}}^n (n\ge 1)\) $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 1)$ . We prove that the system possesses a unique global bounded classical solution for all sufficiently smooth initial data. Moreover, we show the large time behavior of the solution with convergence rates and perform some numerical simulations to verify the analytic results.