Carrying Simplices for Competitive Maps

Abstract

The carrying simplex is a finite-dimensional, attracting Lipschitz invariant manifold that is commonly found in both continuous and discrete-time competition models from Ecology. It can be studied using the graph transform and cone conditions often applied to study attractors in continuous-time finite and infinitedimensional models from applied mathematics, including chemical reaction networks and reaction diffusion equations. Here we show that the carrying simplex can also be studied from the point of view of the graph transform and cone conditions. However, unlike many of the models mentioned above, we do not use-at least directly-a gap condition that is often used to establish existence of a globally and exponentially attracting manifold. Instead we use contraction of phase volume to 'suck' hypersurfaces together uniformly, and ultimately onto the carrying simplex. We give a proof of the existence of the carrying simplex for a class of competitive maps, viewed here as also normally monotone maps. The result is not new, but is carried out in the framework of the graph transform to indicate how the carrying simplex relates to other well-known classes of invariant manifolds. We also discuss the relation between hypersurfaces with positive normals, unordered hypersurfaces and also the type of maps that preserve these types of hypersurfaces. Finally we review several examples from models in Ecology where the carrying simplex is known to exist.

Invariant manifolds of Competitive Selection–Recombination dynamics

“…That is, are there methods that do not require an application of Liouville's theorem, and therefore do not require the inequality (7.10) in addition to M ≥ 0 (6.30)? Consider, for example, the treatment of carrying simplices which are codimension-one invariant manifolds of competitive population models, where global attraction usually requires only mild additional conditions beyond competitiveness (see, for example, [27,28,29,30]). In the continuous time case, in his seminal paper on carrying simplices [14], Hirsch merely adds to competition (that the per-capita growth function has all nonpositive entries) the stronger condition that at any nonzero equilibrium the per-capita growth function has all negative entries) (although as stated in [28], the proof is not complete and we are not aware of a published correction).…”