Date/Time Date(s) - 27/03/20252:30 pm - 3:30 pm
Speaker: Bingyang Cui
Location: Hamilton Hall, Room 410 & Zoom
Title: Global dynamics of a generalized Beverton-Holt model for two competing species
Abstract: This presentation investigates the global dynamics of a generalized Beverton–Holt model for two competing species. The model is first nondimensionalized to reduce the number of parameters. The well-posedness of the model is established by proving that every forward orbit eventually enters and remains in a compact region.
Some important properties of the map T are then established. In particular, T is shown to be order preserving with respect to the competitive order (i.e. the partial order induced by the cone K = {(u, v) ? R2 : u ? 0, v ? 0}), and is globally injective in the first quadrant. Moreover, we prove that any rectangular region in the first quadrant is an order interval that is also ?K -convex. These properties allow us to apply a result by Smith (1998) to conclude that every forward orbit of T must converge to a fixed point.
In the general case where the two species are not equally competitive, we find necessary and sufficient conditions for the existence of boundary fixed points, and we derive a sufficient condition that guarantees the uniqueness of an interior fixed point. Moreover, we prove sufficient conditions that guarantee the global stability of the boundary fixed points. In the symmetric case, where both species are equally competitive, the system can have either one or three interior fixed points. We find necessary and sufficient conditions for the existence of both a unique interior fixed point and three distinct interior fixed points, and we prove the global stability of the unique interior fixed point under certain conditions. Finally, a codimension-1 bifurcation analysis shows that interior fixed points enter the first quadrant through transcritical bifurcation and leave it by either saddle-node or pitchfork bifurcation.