Remarks on the number of persistent states to a fragmented predator–prey model system

• Heterogeneous predator–prey model system posed on a fragmented spatial domain, resident vs. traveling predators.
• Anthropization of the environment, well-fed vs. starving predators.
• Lotka–Volterra vs. Holling type II functional responses to predation.
• Single vs. multiple persistent predator–prey stationary states.

We consider a predator–prey model system for spatially distributed species over patches. Each predator species has a unique preferred patch (shelter and reproduction site) and travel for chasing prey. Its individuals are split into resident from the preferred patch and travelers. Further there is at most one resident predator species per patch. Depending on the availability of local anthropized resources not related to local prey on the preferred patch, one distinguishes between well-fed and starving predators. We assume prey species do not disperse at the predator scale.In this study we are interested in the number of persistent stationary states for the resulting ordinary differential equations model system. There exists at most one persistent predator–prey stationary state when there is exactly one starving resident predators per patch provided all functional responses to predation are Lotka–Volterra like or when a single starving resident predators is available. Else multiple persistent predator–prey stationary state are likely to exist. A specific emphasis is put on toy-model systems with 2 or 3 patches. Slow–fast dynamical methodology is also used for locally asymptotically stable purposes.Numerical experiments suggest that several scalings may govern the dynamics at stabilization.