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