A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction - Liu Ling (Open Mathematics)
We investigate the two-species chemotaxis predator-prey system given by the following system: ut=Δu−χ∇⋅(u∇w)+u(λ1−μ1ur1−1+av),x∈Ω,t>0,vt=Δv+ξ∇⋅(v∇z)+v(λ2−μ2vr2−1−bu),x∈Ω,t>0,0=Δw−w+v,x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0,\left\{\begin{array}{ll}{u}_{t}=\Delta u-\chi \nabla \cdot \left(u\nabla w)+u\left({\lambda }_{1}-{\mu }_{1}{u}^{{r}_{1}-1}+av),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta v+\xi \nabla \cdot \left(v\nabla z)+v\left({\lambda }_{2}-{\mu }_{2}{v}^{{r}_{2}-1}-bu),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ 0=\Delta w-w+v,& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ 0=\Delta z-z+u,& x\in \Omega ,\hspace{0.33em}t\gt 0,\end{array}\right. in a bounded domain Ω⊂RN(N≥1)\Omega \subset {{\mathbb{R}}}^{N}\left(N\ge 1) with smooth boundary, where parameters χ,ξ,λi,μi>0\chi ,\xi ,{\lambda }_{i},{\mu }_{i}\gt 0, and ri>1(i=1,2){r}_{i}\gt 1\hspace{0.33em}\left(i=1,2). Under appropriate conditions, utilizing suitable a priori estimates, we demonstrate that if (N−2)+N<max(r1−1)(r2−1),4N2,(r1−1)2N,(r2−1)2N\frac{{\left(N-2)}_{+}}{N}\lt \max \left\{\phantom{\rule[-0.95em]{}{0ex}},\left({r}_{1}-1)\left({r}_{2}-1),\frac{4}{{N}^{2}},\left({r}_{1}-1)\frac{2}{N},\left({r}_{2}-1)\frac{2}{N}\right\}, then the system admits a unique, uniformly bounded global classical solution. This finding extends the results of several previous studies.