Geoscience Reference
In-Depth Information
and their nutrients within a unit volume inside a homogeneously composed soil
in the ground. The bacteria exhibit long-range interactions via signal molecules.
We can think that after some enough time, the bacteria will be arranged in
clusters. And this actually occurs. This can be mathematically well verified via
the theoretical analysis of Turing structures (Murray
1989
), which are stationary,
spatially periodic morphological patterns (or clusters), with fractal boundaries
behavior, resulting from the interplay between pure diffusion and nonlinear reaction
kinetics mechanisms at nonequilibrium.
Interestingly, the first experimental observation of Turing patterns occurred
almost 40 years after Turing's work (Turing
1952
), in a chemical diffusion-reaction
system apparatus (Castets et al.
1990
; Ouyang and Swinney
1991
).
There is a model which describes the dynamics of predator-prey systems, which
is the Lotka-Volterra model (LV) (Murray
1989
; Lotka
1920
; Volterra
1926
,
1936
).
And there is a description that it is a minimal complexity model, with mean field
conservative dynamics which can be directly implemented on lattice and involves
only two reactive species
X
1
and
X
2
(adsorbed on a lattice support) and the empty
sites of the support S.
This description is named lattice Lotka-Volterra model (LLV) (Frachebourg et al.
1996
; Provata et al.
1999
). All reactive steps are bimolecular, and the reaction occurs
via hard-core interactions. Schematically, the LLV model has the following form:
k
s
X
1
C
X
2
!
2X
2
k
1
X
2
C
S
!
2S
k
2
S
C
X
1
!
2X
1
In the LLV model, the kinetic rate equations are:
dx
1
dt
D
x
1
.-k
s
x
2
C
k
2
s/
dx
2
dt
D
x
2
.k
s
x
1
-k
1
s/
ds
dt
D
s.-k
2
x
1
C
k
1
x
2
/; with the entropies k:
And the associated non-extensive
q
-entropy in the LLV model is:
X
M
X
M
p
i
.t/=q - 1; S
1
D
-
S
q
D
1-
p
i
.t/ ln p
i
.t/
i
D
1
i
D
1
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