TY  - GEN
PB  - Springer
TI  - Carrying simplicies of competitive maps
CY  - Cham, Switzerland
N1  - This version is the author accepted manuscript. For information on re-use, please refer to the publisher?s terms and conditions.
ID  - discovery10078384
Y1  - 2019/07//
UR  - https://doi.org/10.1007/978-3-030-20016-9
AV  - public
T3  - Springer Proceedings in Mathematics & Statistics
A1  - Baigent, S
N2  - 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.
ER  -