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Axiomatic Panbiogeography

offers an application of incidence geometry to historical biogeography by defining collection localities as points, tracks as lines and generalized tracks as planes.
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Incidence Geometry
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Quaternion Algebraic Geom
Primate Vicariances
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Panbiogeographic Chaos

Biological Disproof of the Poincare’-Bendixson Theorem

Can Panbiogeography prove that even apart from singularities - -  that differential equations of phylogenetic interation yield to more complicated orbits than a closed circuit?  Does biogeography not provide evidence that chaos is recoverable in the plane of lat/long distribution locations?   Yes.

Abraham has extended Thom’s metabolic modeling environment to one of a coupleable dynamic bifurcation scheme being as a directed graph with bifurcation and serial coupling control verticies linked.  Viewing biogeographical records with verticies as species and location overlaps as edges these data provide a planar structure from which properties of labyrinths might be conceived, synthesized,  and directed.

Armed with the notions of tracks and vicariance via nodes and masses it is possible to demonstrate a counter example, where non-closed orbits are possible.  Panbiogeography demonstrates a chaos of geography  provided by the ongoing reproductive interation of demes.  This was suggested by Wright as a possible evolutionary dynamic but no one had yet shown the specific context-dependent constraints which are dissected below.

It may be argued that this is not truly two dimensional  (since there are Earth rotations and revolutions) but as a measure of general deme structures of metacommunity stability (and hence measure of future potenialglobal climate change effects etc) this is a disproof and shows that biology can contribute to changes in the way pure math is thought.  This provides an orthogenesis on top of Thom’s morphogenesis and demonstrates that Abraham macrons (lower dimensional versions of infinite dimensional attractors)  are visible in panbiogeography with genetics providing the fundamental algebra that the disproof displayed (it also indicates that Robinson miscognized the genetic relation among algebras,  trees and shapes). Biological systems are more complicated than purely physical ones.  

Hubbard and West thought that even sans singularity that 2-D labyrithns still could not disprove a consequence of the Poincare'-Bendixson Theorem.  They offered a "tadpole" labyrinth that contained a limit set by restricting the flow lines to infintely narrowing areas but said it yet did not permit chaos, odd behavior associated with  infinitely many periodic orbits and or invariant Cantor sets becuase there was no way to "coherently" direct the diminishing "tails" of the accumulating and narrowing path traversals.  Panbiogeographic generalization suggests how it might be done.

The classification of macrons is thus ripe for improvement here. It should now be possible to construct a classification table of  how  cycles and cycles and cycles and chaos interact in the panbiogeographic macrons plotted in latitudes and longitudes.

Furthermore this envisioning of panbiogeography as an applied discipline of dynamical systems theory uniting vicariance and bifurcation through the generalized track that does not need a center of origin as source, shows how bankrupt and out dated is the form of argument recently exemplified by De Queiroz (see reviews by Heads and Flowers). Evidence ( such as Vintana and Caecilian distributions) is not needed (as was used on this page) as pure logic itself suffices..  How far the apparent randomness of chaos explains the chance of selection remains to be seen.