Numerical branch continuation and long-time dynamics in a one-dimensional diffusive Rosenzweig–MacArthur system with Holling type-II response

We study the one-dimensional diffusive Rosenzweig–MacArthur predator–prey system with Holling type-II functional response under homogeneous Neumann boundary conditions. In order to examine its long-time dynamics, we employ a Strang splitting scheme that combines exact diffusion in cosine space with a positivity-preserving reaction update. Guided by the bifurcation theory of Yi, Wei, and Shi, we also use numerical continuation to compute spatially nonhomogeneous periodic and steady-state branches. For selected periodic branches, Floquet analysis and direct time stepping identify stable nonhomogeneous periodic orbits and suggest possible bistability with the homogeneous periodic orbit. For steady states, continuation reveals a regular connecting geometry for most computed branches, together with an exceptional folded branch consisting of two components. Overall, the computations uncover nontrivial branch organization and point to further phenomena, such as mode coupling and Hopf–steady-state interaction.