Global generalized solutions to a three species predator-prey model with prey-taxis

Abstract

<p style='text-indent:20px;'>In this paper, we study the following three species predator-prey model with prey-taxis:</p><p style='text-indent:20px;'><disp-formula> <label>*</label> <tex-math id="E1"> \begin{document}$ \left\{ \begin{array}{lll} u_t = d_1\Delta u+\chi_1\nabla\cdot(u\nabla v)+r_1u(1-u-kv-b_1w), &\quad x\in \Omega, t>0, \\ v_t = d_2\Delta v+r_2v(1-hu-v-b_2w), &\quad x\in \Omega, t>0, \\ w_t = d_3\Delta w-\chi_2\nabla\cdot(w\nabla u)-\chi_3\nabla\cdot(w\nabla v)\\ \ \ \ \ \ \ \ +r_3w(-1+au+av-w), &\quad x\in \Omega, t>0. \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove that if (1.7) and (1.6) hold, the model (<inline-formula><tex-math id="M1">\begin{document}$ \ast $\end{document}</tex-math></inline-formula>) admits at least one global generalized solution in any dimension.</p>