Mars Astrobiology: Recent Status and Progress - Planetary Exploration and Science: Recent Results and Advances - page 206

The LLV model was used to study such following problems: a biological system
composed of three competing species, which gives rise to a stationary pattern of
vortices and strings, clusters, in this bio-system (Tainaka 1989 ); the evolution of
a system of N interacting biological species mimicking the dynamics of cyclic
food (nutrients) chain, which gives rise to a spatial organization, clusters, in such
system (Frachebourg et al. 1996 ; Saunders and Bazin 1974 ); and the dynamics of
open reactive systems capable of giving rise to oscillatory behavior, which displays
a result that in low dimensional supports, the system prefers specific oscillation
frequencies, which gives rise to a spatial clustering mechanism (Provata et al. 1999 ).
Using mean field analysis and Monte Carlo simulations, it is found that spon-
taneous local clustering on lattice and homogeneous initial distribution turn into
clustered structures and reactions among and between molecular and biological
species occur within their interfaces adopting a fractal structure, i.e., besides
clustering, there is fractality within the clusters' domains and borders (Tsekouras
and Provata 2002 ); at the reaction limited regime, on square lattice, in the cyclic
LLV model, there is the spontaneous development of dynamical patterns, in
the form of consecutive stripes or rings, which give rise to clustering, and the
borderlines between consecutive stripes are fractal (Provata and Tsekouras 2003 ); it
is numerically shown that the LLV model, when realized on a square lattice support,
gives rise to a finite production of the above mentioned non-extensive q-entropy S q
(Tsekouras and Provata 2004 ), and this evidence of non-extensivity is consistent
with the spontaneous emergence of local domains of identical particles (which can
be anything, from molecules to microorganisms, for instance), e.g., clustering, with
fractal boundaries and competing interactions at long range (Tsallis and Gell-Mann
2004 ).
Schematically, the Turing mechanism has the following form:
8
<

@ u
@t Dr

2 u C u v u Ǜ

: @v

@t D d r

2 v C LJ u v

LJ
LJ Ǜ

u ss D LJ Ǜ I v ss D

s 1 C

p LJd 1 ! / c D LJ p LJd C

d
2

4LJ

.Ǜ LJ/ 2

Ǜd
2d .Ǜ LJ/

k 2

D

Based on the Jacobian matrix, there is also the activator-inhibitor and substrate-
depletion Turing model. This model is called activator-inhibitor if the matrix is
C
C

and substrate-depletion if the matrix is ˙˙ .