m ranges over the Hm and we could allow different kinds
of ‘noise’ dBti , having particular forms of quadratic variation
that may, in fact, represent a projection of environmental fac-
tors under something like a rate distortion manifold (Wallace
and Fullilove, 2008).
As usual for such systems, there will be multiple quasi-
stable points within a given system’s Hm, representing a class
of generalized resilience modes accessible via punctuation.
Second, however, may well be analogs to fragmentation
when the system exceeds the critical values of Kc according
to the approach of Wallace and Wallace (2008). That is, the
K -parameter structure will represent full-scale fragmentation
of the entire structure, and not just punctuation within it.
We thus infer two classes of punctuation possible for this
kind of structure.
There are other possible patterns:
[1] Setting equation (19) equal to zero and solving for sta-
tionary points again gives attractor states since the noise
terms preclude unstable equilibria.
[2] This system may converge to limit cycle or ‘strange at-
tractor’ behaviors in which the system seems to chase its tail
endlessly, e.g., the cycle of climate-driven phenotype changes
in persistent temperate region plants.
[3] What is converged to in both cases is not a simple state
or limit cycle of states. Rather it is an equivalence class, or
set of them, of highly dynamic information sources coupled
by mutual interaction through crosstalk. Thus ‘stability’ in
this extended model represents particular patterns of ongoing
dynamics rather than some identifiable ‘state’, although such
dynamics may be indexed by a ‘stable’ set of phenotypes.
Here we become enmeshed in a highly recursive phenomeno-
logical stochastic differential equations, but at a deeper level
than the stochastic reaction model of Zhu et al. (2007), and in
a dynamic rather than static manner: the objects of this dy-
namical system are equivalence classes of information sources
and their crosstalk, rather than simple ‘states’ of a dynamical
or reactive chemical system.
Although we are studying local, reciprocally interacting,
developmental phenomena of gene expression within an or-
ganism, it is interesting to compare our results to those of
Dieckmann and Law (1996), who use population-level evolu-
tionary game dynamics to obtain a first order canonical equa-
tion describing large scale coevolutionary process. They find
a relation of the form
dsi/dt = Ki(s)∂Wi(s0i, s)|s0i=s
(20)
The si, with i = 1, ..., N denote adaptive trait values in a
community comprising N species. The Wi (s0i , s) are measures
of fitness of individuals with trait values s0i in the environment
determined by the resident trait values s, and the Ki (s) are
non-negative coefficients, possibly distinct for each species,
that scale the rate of evolutionary change. Adaptive dynamics
of this kind have frequently been postulated, based either on
the notion of a hill-climbing process on an adaptive landscape
or some other sort of plausibility argument.
When this equation is set equal to zero, so there is no time
dependence, one obtains what are characterized as ‘evolution-
ary singularities’, i.e., stationary points.
Dieckmann and Law contend that their formal derivation
of this equation satisfies four critical requirements:
[1] The evolutionary process needs to be considered in a
coevolutionary context.
[2] A proper mathematical theory of evolution should be
dynamical.
[3] The coevolutionary dynamics ought to be underpinned
by a microscopic theory.
[4] The evolutionary process has important stochastic ele-
ments.
Equation (19) is analogous, although we study a different
phenomenon, local gene expression rather than population
level evolutionary change.
Champagnat et al. (2006), in their coevolutionary pop-
ulation dynamics work, derive a higher order canonical ap-
proximation extending equation (20) that is very much closer
equation to (19). Further, Champagnat et al. (2006) use
a large deviations argument to analyze dynamical coevolu-
tionary paths, not merely evolutionary singularities. They
contend that in general, the issue of evolutionary dynamics
drifting away from tra jectories predicted by the canonical
equation can be investigated by considering the asymptotic
of the probability of ‘rare events’ for the sample paths of the
diffusion. By ‘rare events’ they mean diffusion paths drifting
far away from the canonical equation.
The probability of such rare events is governed by a large
deviation principle: when a critical parameter (designated )
goes to zero, the probability that the sample path of the diffu-
sion is close to a given rare path φ decreases exponentially to 0
with rate I (φ), where the ‘rate function’ I can be expressed in
terms of the parameters of the diffusion. This result, in their
view, can be used to study long-time behavior of the diffu-
sion process when there are multiple attractive evolutionary
singularities.
Under proper conditions the most likely path followed by
the diffusion when exiting a basin of attraction is the one
minimizing the rate function I over all the appropriate tra-
jectories. The time needed to exit the basin is of the order
exp(H/) where H is a quasi-potential representing the min-
imum of the rate function I over all possible tra jectories.
An essential fact of large deviations theory is that the rate
function I which Champagnat et al. (2006) invoke can al-
most always be expressed as a kind of entropy, that is, in the
form I = - j Pj log(Pj ) for some probability distribution.
This result goes under a number of names; Sanov’s Theorem,
Cramer’s Theorem, the Gartner-Ellis Theorem, the Shannon-
McMillan Theorem, and so forth (Dembo and Zeitouni, 1998).
In combination with the cognitive paradigm for gene expres-
sion, this will suggest a mechanism for environmental effects
in developmental process. We will argue below that the fluc-
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