"Random is a non-linear dynamical system"? No!
jim mcnamara said:
FWIW - there are papers that claim random is really a non-linear dynamical system.
I assume jim just choose his words badly, but I urgently recommend that anyone who actually believes this should immediately read:
J. D. Murray,
Mathematical Biology, 2nd Ed., Springer, 1993. Note the many examples of
nonlinear dynamical systems which model highly
structured geometric phenomena such as coat patterns.
E. Atlee Jackson,
Perspectives on Nonlinear Dynamics, two volumes, Cambridge University Press, 1994. A wonderful picture book, full of excellent information, which provides a superb portrait of modern dynamical systems and which even includes some biological examples.
About dispersion patterns: the "scattered" versus "random" distinction is terribly crude by mathematical standards, but there is a mathematically valid intuition underlying this distinction.
When we speak of "choosing a position at random", we always must have in mind--- to use the language of mathematics--- some
probability measure. When we have in mind a geometric setting, such as a
metric space, we usually will want to choose a measure which is "compatible" with the
topology of our space, a so-called
Borel probability measure. In this case we probably have in mind something like a uniform measure induced from Lebesgue measure on a rectangle. If so, roughly speaking, "independently and randomly choosing many positions" using such a measure will result in a pattern in which some of the "random positions" happen to be quite close to each other. But if we are using some other means to generate our positions, the positions may tend to keep some minimal distance apart from each other (for example in a model of birds sitting on a telephone wire).
In a nonmathematical example which will nonetheless probably be familiar to most biologists: "pseudorandom number generators" are algorithms which attempt to mimic the behavior of "random and independent choice of positions" on the unit interval \left[ 0,1\right]. A common problem with naive algorithms is that statistical tests reveal that an algorithm produces positions which are "too scattered" to mimic the behavior of the Borel probability measure induced by "normalizing" Lebesgue measure on the unit interval.
