Boundedness of Solutions to a Fully Parabolic Indirect Pursuit–Evasion Predator–Prey System with Density-Dependent Diffusion in \({{\mathbb{R}}}^2\)

This paper deals with a fully parabolic indirect pursuit–evasion predator–prey system with density-dependent diffusion \(u_{t}=\Delta (\psi _1(w)u)+u(\lambda -u+\alpha v), v_{t}=\Delta (\psi _2(z) v)+v(\mu -v-\beta u), w_{t}=\Delta w -w+v, z_{t}=\Delta z-z+u\) $u_t=\mathrm{\Delta }({\psi }_1(w)u)+u(\lambda -u+\alpha v),v_t=\mathrm{\Delta }({\psi }_2(z)v)+v(\mu -v-\beta u),w_t=\mathrm{\Delta }w-w+v,z_t=\mathrm{\Delta }z-z+u$ under a smooth bounded domain \(\Omega \subset {\mathbb{R}}^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with homogeneous Neumann boundary conditions, where the parameters \(\lambda , \mu , \alpha\) $\lambda ,\mu ,\alpha$ and \(\beta\) $\beta$ are assumed to be positive. Through the establishment of appropriate conditions for the density-dependent diffusion functions \(\psi _1(w)\) ${\psi }_1(w)$ and \(\psi _2(z),\) ${\psi }_2(z),$ it is revealed that a unique classical solution exists for the corresponding initial-boundary problem, which remains uniformly bounded over time.