Exotic Bifurcations in Three Connected Populations with Allee Effect

We consider three connected populations with strong Allee effect, and give a complete classification of the steady state structure of the system with respect to the Allee threshold and the dispersal rate, describing the bifurcations at each critical point where the number of steady states change. One may expect that by increasing the dispersal rate between the patches, the system would become more well-mixed, hence simpler. However, we show that it is not always the case, and the number of steady states may (temporarily) go up by increasing the dispersal rate. Besides sequences of pitchfork and saddle-node bifurcations, we find triple-transcritical bifurcations and also a sun-ray shaped bifurcation where twelve steady states meet at a single point then disappear. The major tool of our investigations is a novel algorithm that decomposes the parameter space with respect to the number of steady states and finds the bifurcation values using cylindrical algebraic decomposition with respect to the discriminant variety of the polynomial system.

Automated summary

The study investigates the steady state structure of three connected populations with strong Allee effect, showing that increasing the dispersal rate between patches does not always simplify the system and may temporarily increase the number of steady states. The research also identifies various bifurcation patterns, including triple-transcritical bifurcations and a sun-ray shaped bifurcation where twelve steady states meet at a single point and then disappear. The main tool used in the investigation is a novel algorithm that decomposes the parameter space and finds bifurcation values using cylindrical algebraic decomposition.