On a parabolic-elliptic Keller–Segel system with nonlinear signal production and nonlocal growth term

and nonlocal growth term\[\begin{cases}u_t = \Delta u - \chi \nabla \cdot (u^m \nabla v) + u \Biggl( a_0 - a_1 u^\alpha + a_2 \displaystyle \int_\Omega u^\sigma dx \Biggr) & (x, t) \in \Omega \times (0,\infty) \; , \\0=\Delta - v + u^\gamma , & (x, t) \in \Omega \times (0,\infty) \; , \\\end{cases}\]under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset \mathbb{R}^n (n \geq 1)$, where $\chi \in \mathbb{R}, m, \gamma \geq 1$ and $a_0, a_1, a_2, \alpha \gt 0$. • When $\chi \gt 0$, the solution of the above system is global and uniformly bounded, if the parameters satisfy certain suitable assumptions. • When $\chi \gt 0$, the system possesses a globally bounded classical solution, provided that $a_1 \gt a_2 \lvert \Omega \rvert$. These results indicate that the repulsive mechanism plays a crucial role in ensuring the global boundedness of solutions. In addition, the paper derives the large time behavior of globally bounded solutions for the chemo-attractive or chemo-repulsive system by constructing energy functionals.