This paper deals with a two-species chemotaxis-consumption system involving nonlinear diffusion and chemotaxis $\left\{\begin{array}{cc}{u}_t=\Delta \left({\gamma }_1\left(w\right)u\right)+{\mu }_{1}u(1-u-a_{1}v),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=\Delta \left({\gamma }_2\left(w\right)v\right)+{\mu }_{2}v(1-a_{2}u-v),\hfill & x\in \Omega ,t>0,\hfill \\ w_t=\Delta w-(\alpha u+\beta v)w,\hfill & x\in \Omega ,t>0,\hfill \\ \hfill \end{array}\right.$in an arbitrary smooth bounded domain $\Omega \subset {\mathbb{R}}^n(n=2,3)$ under homogeneous Neumann boundary conditions, where ${\mu }_i,a_i,\alpha ,\beta$ are positive constants and the motility functions ${\gamma }_i\left(w\right)\in C^3\left([0,\infty )\right)$, ${\gamma }_i\left(w\right)>0$, ${\gamma }_i^{\prime }\left(w\right)<0$ for all $w\ge 0$, ${lim}_{w\to \infty }{\gamma }_i\left(w\right)$=0 and ${lim}_{w\to \infty }\frac{{\gamma }_i^{\prime }\left(w\right)}{{\gamma }_i\left(w\right)}$ exist for $i=1,2$. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in 2-D and in 3-D for ${\mu }_1,{\mu }_2$ being sufficiently large. Furthermore, in a spatially three-dimensional setting, the paper also proceeds to establish asymptotic stabilization of solutions to the above system, and the following properties hold:

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when $a_1,a_2\in (0,1)$, the global bounded classical solution $(u,v,w)$ exponentially converges to $(\frac{1-a_1}{1-a_1{a}_2},\frac{1-a_2}{1-a_1{a}_2},0)$ as $t\to \infty$;

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when $a_1>1>a_2$ , the global bounded classical solution $(u,v,w)$ exponentially converges to $(0,1,0)$ as $t\to \infty$;

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when $a_1=1>a_2$, the global bounded classical solution $(u,v,w)$ polynomially converges to $(0,1,0)$ as $t\to \infty$.

本文研究了一种涉及非线性扩散和趋化性的两种种群趋化性消耗系统。 $\left\{\begin{array}{cc}{ü}_{Ť}=\Delta （{\gamma }_{\mathrm{1个}}（w）ü）+{\mu }_{\mathrm{1个}}ü（\mathrm{1个}-ü-{\mathrm{一种}}_{\mathrm{1个}}v），\hfill & X\in \Omega ，Ť>0，\hfill \\ v_{Ť}=\Delta （{\gamma }_2（w）v）+{\mu }_{2}v（\mathrm{1个}-{\mathrm{一种}}_2ü-v），\hfill & X\in \Omega ，Ť>0，\hfill \\ w_{Ť}=\Delta w-（\alpha ü+\beta v）w，\hfill & X\in \Omega ，Ť>0，\hfill \\ \hfill \end{array}\right.$在任意光滑有界域中 $\Omega \subset {\mathbb{[R}}^{ñ}（ñ=2，3）$ 在齐次Neumann边界条件下， ${\mu }_{\mathrm{一世}}，{\mathrm{一种}}_{\mathrm{一世}}，\alpha ，\beta$ 是正常数和运动功能 ${\gamma }_{\mathrm{一世}}（w）\in C^3（[0，\infty ））$， ${\gamma }_{\mathrm{一世}}（w）>0$， ${\gamma }_{\mathrm{一世}}^{\prime }（w）<0$ 对所有人 $w\ge 0$， ${林}_{w\to \infty }{\gamma }_{\mathrm{一世}}（w）$= 0且 ${林}_{w\to \infty }\frac{{\gamma }_{\mathrm{一世}}^{\prime }（w）}{{\gamma }_{\mathrm{一世}}（w）}$ 存在于 $\mathrm{一世}=\mathrm{1个}，2$。证明了相应的初-边值问题在2-D和3-D中具有唯一的全局有界经典解${\mu }_{\mathrm{1个}}，{\mu }_2$足够大。此外，在空间三维设置中，本文还着手建立上述系统的解的渐近稳定，并且具有以下性质：

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什么时候 ${\mathrm{一种}}_{\mathrm{1个}}，{\mathrm{一种}}_2\in （0，\mathrm{1个}）$，全局有界经典解 $（ü，v，w）$ 指数地收敛到 $（\frac{\mathrm{1个}-{\mathrm{一种}}_{\mathrm{1个}}}{\mathrm{1个}-{\mathrm{一种}}_{\mathrm{1个}}{\mathrm{一种}}_2}，\frac{\mathrm{1个}-{\mathrm{一种}}_2}{\mathrm{1个}-{\mathrm{一种}}_{\mathrm{1个}}{\mathrm{一种}}_2}，0）$ 如 $Ť\to \infty$;

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什么时候 ${\mathrm{一种}}_{\mathrm{1个}}>\mathrm{1个}>{\mathrm{一种}}_2$ ，全局有界经典解 $（ü，v，w）$ 指数地收敛到 $（0，\mathrm{1个}，0）$ 如 $Ť\to \infty$;

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什么时候 ${\mathrm{一种}}_{\mathrm{1个}}=\mathrm{1个}>{\mathrm{一种}}_2$，全局有界经典解 $（ü，v，w）$ 多项式收敛到 $（0，\mathrm{1个}，0）$ 如 $Ť\to \infty$。