On a macrophage and tumor cell chemotaxis system with both paracrine and autocrine loops

<p style='text-indent:20px;'>In this paper, we consider a homogeneous Neumann initial-boundary value problem (IBVP) for the following two-species and two-stimuli chemotaxis model with both paracrine and autocrine loops:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \label{IBVP} \left\{ \begin{aligned} &amp;u_t = \nabla\cdot(D_1(u)\nabla u-S_1(u)\nabla v), &amp;\qquad x\in\Omega, \, t&gt;0, \\ &amp; \tau_1 v_t = \Delta v- v+w, &amp;\qquad x\in\Omega, \, t&gt;0, \\ &amp;w_t = \nabla\cdot(D_2(w)\nabla w-S_2(w)\nabla z-S_3(w)\nabla v), &amp;\qquad x\in\Omega, \, t&gt;0, \\ &amp; \tau_2 z_t = \Delta z- z+ u, &amp;\qquad x\in\Omega, \, t&gt;0, \end{aligned} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ u(t, x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ w(t, x) $\end{document}</tex-math></inline-formula> denote the density of macrophages and tumor cells at time <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula> and location <inline-formula><tex-math id="M4">\begin{document}$ x\in \Omega, $\end{document}</tex-math></inline-formula> respectively, <inline-formula><tex-math id="M5">\begin{document}$ v(t, x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ z(t, x) $\end{document}</tex-math></inline-formula> represent the concentration of colony stimulating factor 1 (CSF-1) secreted by the tumor cells and epidermal growth factor (EGF) secreted by macrophages at time <inline-formula><tex-math id="M7">\begin{document}$ t $\end{document}</tex-math></inline-formula> and location <inline-formula><tex-math id="M8">\begin{document}$ x\in \Omega, $\end{document}</tex-math></inline-formula> respectively. <inline-formula><tex-math id="M9">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> is a bounded region with smooth boundary, <inline-formula><tex-math id="M10">\begin{document}$ \tau_i\ge 0 \; (i = 1, 2) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ D_i(s)\ge d_i(s+1)^{m_i-1} $\end{document}</tex-math></inline-formula> with parameters <inline-formula><tex-math id="M12">\begin{document}$ m_i\ge 1 \; (i = 1, 2) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ S_j(s)\lesssim (s+1)^{q_j} $\end{document}</tex-math></inline-formula> with parameters <inline-formula><tex-math id="M14">\begin{document}$ q_j&gt;0 \;(j = 1, 2, 3) $\end{document}</tex-math></inline-formula>. For the case without autocrine loop (i.e., <inline-formula><tex-math id="M15">\begin{document}$ S_3(w) = 0 $\end{document}</tex-math></inline-formula>), it is shown that when <inline-formula><tex-math id="M16">\begin{document}$ q_j\le 1 \; (j = 1, 2) $\end{document}</tex-math></inline-formula>, if one of <inline-formula><tex-math id="M17">\begin{document}$ q_j $\end{document}</tex-math></inline-formula> is smaller than one or one of <inline-formula><tex-math id="M18">\begin{document}$ m_i $\end{document}</tex-math></inline-formula> is larger than one, then the IBVP has a global classical solution which is uniformly bounded. Moreover, when <inline-formula><tex-math id="M19">\begin{document}$ m_1 = m_2 = q_1 = q_2 = 1 $\end{document}</tex-math></inline-formula>, an inequality involving the product <inline-formula><tex-math id="M20">\begin{document}$ d_1d_2 $\end{document}</tex-math></inline-formula> and the product of the two species' initial mass is obtained which guarantees the existence of global bounded classical solutions. More specifically, it allows one of <inline-formula><tex-math id="M21">\begin{document}$ d_i $\end{document}</tex-math></inline-formula> to be small or one of the species initial mass to be large. For the case with autocrine loop (i.e <inline-formula><tex-math id="M22">\begin{document}$ S_3(w)\ne 0 $\end{document}</tex-math></inline-formula>), similar results hold only if <inline-formula><tex-math id="M23">\begin{document}$ q_3&lt;1 $\end{document}</tex-math></inline-formula>. If <inline-formula><tex-math id="M24">\begin{document}$ q_3 = 1 $\end{document}</tex-math></inline-formula>, solutions to the IBVP exist globally only when <inline-formula><tex-math id="M25">\begin{document}$ d_2 $\end{document}</tex-math></inline-formula> is suitably large or the mass of species <inline-formula><tex-math id="M26">\begin{document}$ w $\end{document}</tex-math></inline-formula> is suitably small.</p>