The influence of dispersal on a predator–prey system with two habitats

• We investigate a predator–prey system on two coupled patches.
• We evaluate stability and bifurcations for many model classes.
• Losses due to emigration and adaptive dispersal can be stabilizing.
• Dispersal can induce pattern-forming bifurcations.
• Adaptive dispersal leads to antisynchronous oscillations.

Dispersal between different habitats influences the dynamics and stability of populations considerably. Furthermore, these effects depend on the local interactions of a population with other species. Here, we perform a general and comprehensive study of the simplest possible system that includes dispersal and local interactions, namely a 2-patch 2-species system. We evaluate the impact of dispersal on stability and on the occurrence of bifurcations, including pattern forming bifurcations that lead to spatial heterogeneity, in 19 different classes of models with the help of the generalized modelling approach. We find that dispersal often destabilizes equilibria, but it can stabilize them if it increases population losses. If dispersal is nonrandom, i.e. if emigration or immigration rates depend on population densities, the correlation of stability with dispersal rates is positive in part of the models. We also find that many systems show all four types of bifurcations and that antisynchronous oscillations occur mostly with nonrandom dispersal.