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This paper deals with the two-species chemotaxis system with two chemicals \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\nabla\cdot(u\chi_1(v)\nabla v)+\mu_1 u(1-u-a_1w),\quad x'>under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} ( \begin{document}$ n\geq1 $\end{document} ), where the parameters \begin{document}$ d_1,d_2,d_3,d_4>0 $\end{document} , \begin{document}$ \mu_1,\mu_2>0 $\end{document} , \begin{document}$ a_1,a_2>0 $\end{document} and \begin{document}$ \alpha, \beta>0 $\end{document} . The chemotactic function \begin{document}$ \chi_i $\end{document} ( \begin{document}$ i = 1,2 $\end{document} ) and the signal production function \begin{document}$ f_i $\end{document} ( \begin{document}$ i = 1,2 $\end{document} ) are smooth. If \begin{document}$ n = 2 $\end{document} , it is shown that this system possesses a unique global bounded classical solution provided that \begin{document}$ |\chi'_i| $\end{document} ( \begin{document}$ i = 1,2 $\end{document} ) are bounded. If \begin{document}$ n\leq3 $\end{document} , this system possesses a unique global bounded classical solution provided that \begin{document}$ \mu_i $\end{document} ( \begin{document}$ i = 1,2 $\end{document} ) are sufficiently large. Specifically, we first obtain an explicit formula \begin{document}$ \mu_{i0}>0 $\end{document} such that this system has no blow-up whenever \begin{document}$ \mu_i>\mu_{i0} $\end{document} . Moreover, by constructing suitable energy functions, it is shown that: \begin{document}$ \bullet $\end{document} If \begin{document}$ a_1,a_2\in(0,1) $\end{document} and \begin{document}$ \mu_1 $\end{document} and \begin{document}$ \mu_2 $\end{document} are sufficiently large, then any global bounded solution exponentially converges to \begin{document}$\bigg(\frac{1-a_1}{1-a_1a_2},f_1(\frac{1-a_2}{1-a_1a_2})/\alpha,\frac{1-a_2}{1-a_1a_2},$\end{document} \begin{document}$ f_2(\frac{1-a_1}{1-a_1a_2})/\beta\bigg)$\end{document} as \begin{document}$ t\rightarrow\infty $\end{document} ; \begin{document}$ \bullet $\end{document} If \begin{document}$ a_1>1>a_2>0 $\end{document} and \begin{document}$ \mu_2 $\end{document} is sufficiently large, then any global bounded solution exponentially converges to \begin{document}$ (0,f_1(1)/\alpha,1,0) $\end{document} as \begin{document}$ t\rightarrow\infty $\end{document} ; \begin{document}$ \bullet $\end{document} If \begin{document}$ a_1 = 1>a_2>0 $\end{document} and \begin{document}$ \mu_2 $\end{document} is sufficiently large, then any global bounded solution algebraically converges to \begin{document}$ (0,f_1(1)/\alpha,1,0) $\end{document} as \begin{document}$ t\rightarrow\infty $\end{document} .

This paper deals with the following two-species chemotaxis system \begin{document}$\left\{ \begin{array}{*{35}{l}} \ \ {{u}_{t}}=\Delta u-{{\chi }_{1}}\nabla \cdot (u\nabla v)+{{\mu }_{1}}u(1-u-{{a}_{1}}w), & x\in \Omega ,t>0, & \\ \ \ {{v}_{t}}=\Delta v-v+h(w), & x\in \Omega ,t>0, & \\ \ \ {{w}_{t}}=\Delta w-{{\chi }_{2}}\nabla \cdot (w\nabla z)+{{\mu }_{2}}w(1-w-{{a}_{2}}u), & x\in \Omega ,t>0, & \\ \ \ {{z}_{t}}=\Delta z-z+h(u),& x\in \Omega ,t>0, & \\\end{array} \right.$ \end{document} under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$Ω\subset\mathbb{R}^{n}$\end{document} with smooth boundary. The parameters in the system are positive and the signal production function h is a prescribed C1-regular function. The main objectives of this paper are two-fold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium \begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document} may be globally attractive in the weak competition case (i.e., \begin{document}$0 ), while the constant stationary solution (0, h(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e., \begin{document}$a_{1}>1>a_{2}>0$\end{document} ). In the fully strong competition case (i.e. \begin{document}$a_{1}, a_{2}>1$\end{document} ), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous \begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document} . The matter which species ultimately wins out depends crucially on the starting advantage each species has.

In this work we consider the chemotaxis system with singular sensitivity and signal production in a two dimensional bounded domain. We present the global existence of weak solutions under appropriate regularity assumptions on the initial data. Our results generalize some well-known results in the literature.

In this paper we develop a new and convenient technique, with fractional Gagliardo-Nirenberg type inequalities inter alia involved, to treat the quasilinear fully parabolic chemotaxis system with indirect signal production: \begin{document}$ u_t = \nabla\cdot(D(u)\nabla u-S(u)\nabla v) $\end{document} , \begin{document}$ \tau_1v_t = \Delta v-a_1v+b_1w $\end{document} , \begin{document}$ \tau_2w_t = \Delta w-a_2w+b_2u $\end{document} , under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset\Bbb{R}^{n} $\end{document} ( \begin{document}$ n\geq 1 $\end{document} ), where \begin{document}$ \tau_i,a_i,b_i>0 $\end{document} ( \begin{document}$ i = 1,2 $\end{document} ) are constants, and the diffusivity \begin{document}$ D $\end{document} and the density-dependent sensitivity \begin{document}$ S $\end{document} satisfy \begin{document}$ D(s)\geq a_0(s+1)^{-\alpha} $\end{document} and \begin{document}$ 0\leq S(s)\leq b_0(s+1)^{\beta} $\end{document} for all \begin{document}$ s\geq 0 $\end{document} with \begin{document}$ a_0,b_0>0 $\end{document} and \begin{document}$ \alpha,\beta\in\Bbb R $\end{document} . It is proved that if \begin{document}$ \alpha+\beta and \begin{document}$ n = 1 $\end{document} , or \begin{document}$ \alpha+\beta with \begin{document}$ n\geq 2 $\end{document} , for any properly regular initial data, this problem has a globally bounded and classical solution. Furthermore, consider the quasilinear attraction-repulsion chemotaxis model: \begin{document}$ u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla z)+\xi\nabla\cdot(u\nabla w) $\end{document} , \begin{document}$ z_t = \Delta z-\rho z+\mu u $\end{document} , \begin{document}$ w_t = \Delta w-\delta w+\gamma u $\end{document} , where \begin{document}$ \chi,\mu,\xi,\gamma,\rho,\delta>0 $\end{document} , and the diffusivity \begin{document}$ D $\end{document} fulfills \begin{document}$ D(s)\geq c_0(s+1)^{M-1} $\end{document} for any \begin{document}$ s\geq 0 $\end{document} with \begin{document}$ c_0>0 $\end{document} and \begin{document}$ M\in\Bbb R $\end{document} . As a corollary of the aforementioned assertion, it is shown that when the repulsion cancels the attraction (i.e. \begin{document}$ \chi\mu = \xi\gamma $\end{document} ), the solution is globally bounded if \begin{document}$ M>-1 $\end{document} and \begin{document}$ n = 1 $\end{document} , or \begin{document}$ M>2-4/n $\end{document} with \begin{document}$ n\geq 2 $\end{document} . This seems to be the first result for this quasilinear fully parabolic problem that genuinely concerns the contribution of repulsion.

In this paper we consider the following chemotaxis-growth system with nonlinear signal production and logistic source \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi\nabla\cdot( u\nabla v)+\lambda u-\mu u^{\alpha}, \quad x'>with homogeneous Neumann boundary conditions in the ball \begin{document}$ \Omega = B_R(0)\subset \mathbb{R}^n $\end{document} for \begin{document}$ n\geq 1 $\end{document} and \begin{document}$ R>0 $\end{document} , where \begin{document}$ \chi, \lambda, \mu>0 $\end{document} , \begin{document}$ \alpha>1 $\end{document} , and \begin{document}$ f $\end{document} is an appropriate regular function satisfying \begin{document}$ f(u)\geq ku^{\kappa} $\end{document} for all \begin{document}$ u\geq1, \kappa>0 $\end{document} with some constants \begin{document}$ k>0 $\end{document} . If the number \begin{document}$ \kappa $\end{document} and \begin{document}$ \alpha $\end{document} satisfy \begin{document}$ \begin{equation*} \kappa+1>\alpha\left(\frac{2}{n}+1\right), \end{equation*} $\end{document} then there exists appropriate initial data such that the corresponding solution \begin{document}$ (u, v) $\end{document} of the system blow up in finite time. This result extends the blow-up result of the chemotaxis model without logistic cell kinetics in [ 45 ]. Apparently, for the case \begin{document}$ \kappa = 1 $\end{document} , this provides a rigorous detection for blow-up of solution in spaces-dimensions three and four.