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better description of the structures, in the one-dimensional case, they [14] further considered
the global bifurcation of the non-constant steady states arising from the simple bifurcation (i.e.
the case d
j
= d
k
in [14]) by taking the “effective” diffusion rate d as bifurcation parameter.
This paper continues the analytic works of [14] with the goal of revealing a new or rich
solution structure and achieving a deeper understanding of the Turing patterns operating in
the system (1). We still view the effective diffusion rate d as the bifurcation parameter, and
maintain the basic hypothesis on the system parameters, i.e. condition (H) below, which results
in that no spatiotemporal pattern exists for the bifurcation parameter d. Thus, our main
purpose is only to study Turing structures, especially bifurcating from the double eigenvalue
(i.e. the case d
j
= d
k
) by using Lyapunov-Schmidt technique and singularity theory [15], and
further determine the stability and multiplicity of the bifurcating non-constant steady state
solutions, in particular the simple bifurcation solutions obtained in [14]. The results enrich and
perfect the earlier works of [14] in order to seeking a complete mathematical investigations of
the Turing patterns for the Lengyel-Epstein system (1), and are of practical significance for
the researches on pattern formation in complex reaction-diffusion systems. We believe that our
theoretical studies of spatially nonhomogeneous steady states are new advance, and until now
little or no results is known about the effect of diffusion on such double bifurcation structure.
1 Turing bifurcation when some d
j
= d
k
for j = k
By introducing the new variable d = c/b, δ = σb and α = a/5, the system (1) in the
one-dimensional interval Ω = (0, l) can be written as
∂u
∂t
=
∂
2
u
∂x
2
+ 5α − u −
4uv
1 + u
2
, x ∈ (0, l), t > 0,
∂v
∂t
= δ(d
∂
2
v
∂x
2
+ u −
uv
1 + u
2
), x ∈ (0, l), t > 0,
∂u
∂x
=
∂v
∂x
= 0, x = 0, l, t > 0,
u(x, 0) = u
0
(x) > 0, v(x, 0) = v
0
(x) > 0, x ∈ (0, l).
(2)
Clearly, system (1) and (2) both has a unique spatially homogeneous steady state (u
∗
, v
∗
) =
(α, 1 + α
2
). As in [13], we denote
f(u, v) = 5α − u −
4uv
1 + u
2
, g(u, v) = u −
uv
1 + u
2
,
f
0
:= f
u
(u
∗
, v
∗
) =
3α
2
− 5
1 + α
2
< 3, f
1
:= f
v
(u
∗
, v
∗
) = −
4α
1 + α
2
< 0,
g
0
:= g
u
(u
∗
, v
∗
) =
2α
2
1 + α
2
> 0, g
1
:= g
v
(u
∗
, v
∗
) = −
α
1 + α
2
< 0,
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