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This paper deals with the two-species chemotaxis-competition models\begin{align*}  \begin{cases}  u_t = d_1 \Delta u  - \chi_1 \nabla \cdot (u \nabla w)  + \mu_1 u (1- u^{\kappa_1-1} - a_1 v^{\lambda_1-1}),  &\quad x \in \Omega,\ t>0,\\ %  v_t = d_2 \Delta v  - \chi_2 \nabla \cdot (v \nabla w)  + \mu_2 v (1- a_2 u^{\lambda_2-1} - v^{\kappa_2-1}),  &\quad x \in \Omega,\ t>0,\\ %  0 = d_3 \Delta w + \alpha u + \beta v - h(u,v,w),  &\quad x \in \Omega,\ t>0,  \end{cases}  \end{align*} where $\Omega \subset \mathbb{R}^n$ $(n\ge2)$ is a boundeddomain with smooth boundary, and $h=\gamma w$ or$h=\frac{1}{|\Omega|}\int_\Omega(\alpha u+ \beta v)\,dx$. In the case that$\kappa_1=\lambda_1=\kappa_2=\lambda_2=2$ and $h=\gamma w$, it is known thatsmallness conditions for the chemotacic effects lead to boundedness ofsolutions (Math.\ Methods Appl.\ Sci.; 2018; 41; 234--249). However, the casethat the chemotactic effects are large seems not to have been studied yet;therefore it remains to consider the question whether the solution is boundedalso in the case that the chemotactic effects are large. The purpose of thispaper is to give a negative answer to this question.

Can chemotactic effects lead to blow-up or not in two-species chemotaxis-competition models?