Unidirectional Migration of Populations with Allee Effect

In this note we consider two populations living on identical patches, connected by unidirectional migration, and subject to strong Allee effect. We show that by increasing the migration rate, there are more bifurcation sequences than previous works showed. In particular, the number of steady states can change from 9 (small migration) to 3 (large migration) at a single bifurcation point, or via a sequence of bifurcations with the system having 9, 7, 5, 3 steady states or 9, 7, 9, 3 steady states, depending on the Allee threshold. This is in contrast with the case of bidirectional migration, where the number of steady states always goes through the same bifurcation sequence of 9, 5, 3 steady states as we increase the migration rate, regardless of the value of the Allee threshold. These results have practical implications as well in spatial ecology.