Convexity of invariant manifolds in competitive population models
摘要：Many competitive population models have attracting invariant manifolds of codimension-one. These manifolds are Lipschitz and they are unordered in the sense that no two points on the manifold can be ordered component-wise. In some cases it is also possible to show that the manifolds are convex or concave. I will give a graph-transform proof of the existence of an attracting invariant manifold in 2-species continuous-time and discrete-time competitive population models. Moreover, I will show that some of these invariant manifolds are convex or concave. Finally, I will discuss how the convexity/concavity of the manifold can inform the asymptotic stability of equilibrium points, and how all these ideas translate into 3-species models.