Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop

This paper deals with the two-species chemotaxis-competition system with loop $$\begin{aligned} \left\{ \begin{array}{llll} \partial _{t} u_{1}=d_1\Delta u_{1}-\chi _{11}\nabla \cdot (u_{1}\nabla v_{1}) -\chi _{12}\nabla \cdot (u_{1}\nabla v_{2}) +\mu _{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ \partial _{t} u_{2}=d_2\Delta u_{2}-\chi _{21}\nabla \cdot (u_{2}\nabla v_{1}) -\chi _{22}\nabla \cdot (u_{2}\nabla v_{2}) +\mu _{2}u_{2}(1-u_{2}-a_{2}u_{1}),\\ \partial _t v_1=d_3\Delta v_{1}-\lambda _{1} v_{1}+\alpha _{11}u_{1}+\alpha _{12}u_{2},\\ \partial _t v_2=d_4\Delta v_{2}-\lambda _{2} v_{2}+\alpha _{21}u_{1}+\alpha _{22}u_{2}, \\ \end{array} \right. \end{aligned}$$ subject to homogeneous Neumann boundary conditions in a smooth bounded domain $$\Omega \subset {\mathbb {R}}^{3}$$ , where $$\chi _{ij}>0$$ , $$\mu _{i}>0$$ , $$a_i>0$$ , $$\alpha _{ij}>0$$ , $$\lambda _{i}>0$$ , $$d_k>0$$ $$(i, j=1, 2, k=1, 2, 3, 4)$$ . Our main purpose is to extend the global boundedness result to the 3D setting. To address this issue, based on a new coupled function, by selecting sufficiently large $$\mu _1$$ and $$\mu _2$$ , we construct a Gronwall type inequality which directly renders the uniform boundedness of solutions.