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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
Composite Construction
Quaternion Algebraic Geom
Primate Vicariances
Individual Track Construc
Generalized Tracks
Main Massings
Track Analysis and MetaCo
Martitrack Panbiogeograph
Replies to Criticism
Multimodel Selection
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Track Analysis beyond Pan

Track Analysis as a Proxy for Community Connectance

Applied Panbiogeography (beyond Croizat) began largerly with a focus on endemism and the issue of track construction in general.  Since the newer concept of the node has been  constructed and used it becomes possible to focus more on different generalized tracks rather than on simple aggregation of individual tracks in situ.  This has made it possible to consider a further application of the panbiogeographic track as a measure of metacommunity complexity.  Interestingly the spatial content of the Croizat track can provide spatial information that informs the relation of stability and diversity within metacommunities.  In general it is possible to "age" tracks such that older tracks may express different relationships of diversity and stability for a given level (number of nodes etc) connectivity (albeit "weak") in metacommunity persistance.  Once multiple aged tracks are plotted the geometry of such would have been thought of as the places dispersal (with respect to those species's lineages in the given community) will likely continue through (assuming the metacommunity is not the entire globe). Panbiogeography can suggest that the theoretical concept of qualitative stability based on a community matrix may need be modified since the existence of multiple aged tracks across a region provide many weak links that for a given species' population growth are effectively zero in ecological time but vary over geological time with an effect in evolutionary time.  Development of this application where mainland - island biogeography follows a different trajectory than predictable with island biogeography further solidifies the need for funding in panbiogeography even beyond that proposed for comparative biogeography.

Oscillatory Population Theory with Quaternions : Social Selection Application, Stability, Diversity, Complexity

The theory is applied to social selection and yields a 3D rotation form for each oscillation and  weak other species’ linekages.  The ability of the these solid rotations to combine places a relation between stability and diversity that depends on complexity such as sucssesional state.  The combinations will only be stable when the individual rotations still have most of their ability to rotate.  If the community forms an ecosystem functionality some of the individual flexibility may be diminished for the same return interms of offspring.

The oscilation between two species population numbers is modeled by a common 4D quaternion rotation able to accomplish the belt trick between the two such that the combined phase relations of the two species changes form 2pie rotations that equivalent to 0pie rotation.  The left and right coordination implies the rs correlated.  This might mathematically be accomplished  by exchanging the sin and cos amongst the two .  Biologically it appears that Fisher reproductive value variations have similar variances between the two such that stable age structure forcings match?  The parameter space that maps o pie to 4pie gives the 2 species system this advantage when placed in community with others not so synced.

This might be translated into a high order vector space but the4D quaternion representation is accomplished by splitting the vector space3.  Each species must independently evolve the same stableling age structured variations.

This could happen with social selection where each species has its own social selection infrastructure raising variations but follow a generally same physiological pleasure gradient (dopanmine etc).

Each oscillation thus possesses a soild rotation figure that associates a community and makes it stable when it is of  aform that can keep rotations going while moving through other couple oscilations.  Diversity and stability thus have an intricate relationship on this theory and depending on the complexity possible (the ecosystem , succession, particular large scale effects) sometimes less diversity is more stable sometimes more diversity is more stable.

In an end one may be able to determine track widths and geometry from the quaternion solids concatenated from analysis of local community belt tricked osciliations weakly linked to others in the track.  This would mean there two ways to find tracks (analysis and synthesis). Ozark sourced metacommunities indicate many sink  links to many taxa some seemingly viarcariating symmetrically and remaining formally from 100s of millions of years to date.  One might begin to construct a metacommunity that varies is space made up of different Parastacids, Mcirohylids and Chelids even though likely only very weakly linked.

Now with two generalized tracks of different ages the notion of congruence comes into play and is demonstrable by the absolute value of the baseline ( I assume that the tracks so synthesized do not involve life off Earth).

 This is an enitrely unique constructablility within  panbiogeography that uses an axiomatic panbiogeographic concept.  If there is no life off Earth involved and the total track to baseline relations can be relied on, down tot the angle (shape of nodes known) then geographic angle between generalized tracks is a constant part of life and Earth that evolves toghether and is as close to a law of nature that contingent natural history can provide.  It does not need to depend on the origin of life since it may be falsfied by a higher dimensional representation given other kinds of coded life off Earth ancestrally connected.