Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 948480, 13 pages
doi:10.1155/2012/948480
Research Article
Dynamic Properties of a Forest Fire Model
Na Min, Hongyang Zhang, and Zhilin Qu
Department of Mathematics, Northeast Forestry University, Harbin 150040, China
Correspondence should be addressed to Zhilin Qu, zhynefu@163.com
Received 11 June 2012; Revised 7 September 2012; Accepted 17 September 2012
Academic Editor: Bashir Ahmad
Copyright q 2012 Na Min et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The reaction-diffusion equations have been widely used in physics, chemistry, and other areas.
Forest fire can also be described by such equations. We here propose a fighting forest fire model.
By using the normal form approach theory and center manifold theory, we analyze the stability of
the trivial solution and Hopf bifurcation of this model. Finally, we give the numerical simulations
to illustrate the effectiveness of our results.
1. Introduction
The forest fire is an important issue in the world. It has brought us huge losses. It not only
burns our forests but also destroys the local ecological environment. Many factors lead to
forest fires. Several authors have studied them in depth 1–6. Some important organizations,
especially the USDA Forest Service, have also researched them in their themes 7.
Reaction-diffusion equations have been applied in forest fire model for several
years. Some authors analyzed the dynamical behavior of the fire front propagations using
hyperbolic reaction-diffusion equations 8. Lots of articles related to percolation theory 9
and self-organized criticality 10 are trying to provide a different dynamical model for the
spread of the fire. In this paper, the model describes the condition that people are putting out
the fire when the fire is spreading. We analyze dynamic properties of the reaction-diffusion
equations. Kolmogorov et al. proposed the famous KPP model 11 in the 1930s. From then
on, it had been applied in various fields including forest fire:
ut d1 uxx u fu,
x ∈ R, t ≥ 0,
1.1
where u ux, t can be seen as the area of the burned forest. uxx is a diffusion term of u
in space, and d1 is the diffusion coefficient. fu is a nonlinear function. The equation can
2
Abstract and Applied Analysis
describe the speed of fire spreading. Zeldovich et al. gave the famous theory of combustion
and explosions 12. We can get inspiration from it:
The people will go to put out the fire as soon as they realize the forest fire. We can use
a reaction-diffusion equation to describe it.
vt d2 vxx − cv gv.
1.2
In this equation, v vx, t is the area where the fire has been put out. vxx is a diffusion term
of v in space, and d2 is the diffusion coefficient. c is the resurgence probability of v. gv is a
nonlinear function which represents the ability of people to put out the fire.
Now, let us consider the two reaction-diffusion equations together. As we know, u and
v influence each other. Thus, f and g must be functions of u, v. We define gu, v by referring
to the combustion model 13:
gu, v uv
,
bu 1
b > 0.
1.3
Since gu, v has opposite effect on the fire area or u, we can also define fu, v by
taking into account KPP model 8:
fu, v −au2 −
uv
.
bu 1
1.4
Then we get a new model:
ut − d1 uxx u − au2 −
vt − d2 vxx −cv uv
,
bu 1
uv
,
bu 1
ux 0, t vx 0, t 0,
ux lπ, t vx lπ, t 0,
ux, 0 ≥ 0,
x ∈ 0, lπ.
vx, 0 ≥ 0,
1.5
Define
H L2 0, π × L2 0, π f
: f, g ∈ L2 0, π ,
g
1.6
and an inner product is given by
2 π
2 π
f1
f
, 2
f1 , f2 L2 g1 , g2 L2 f1 f2 dx g1 g2 dx,
g1
g2
π 0
π 0
1.7
Abstract and Applied Analysis
3
where f1 , g1 T ∈ H, f2 , g2 T ∈ H. From the standpoint of biology, we are only interested in the
dynamics of model 1.5 in the region:
1.8
R2 {u, v | u > 0, v > 0}.
2. Stability Analysis
Firstly, we consider the location stability 14, 15 and the number of the equilibria of model
1.5 in R2 . We can also study autowave solutions 16 of the model. The interior equilibrium
point is a root of the following equation:
u − au2 −
−cv uv
0,
bu 1
2.1
uv
0.
bu 1
It is obvious that 2.1 has an only real solution Y0 u0 , v0 , where
u0 bc
,
1 − bc
v0 ab
1
− u0 1 u0 ,
a
2.2
and b < 1/ca 1.
Now, we analyze the asymptotic stability of u0 , v0 by Lyapunov function.
Lemma 2.1. For the model 1.5,
1 if a ≥ 1, u0 , v0 is global asymptotic stability.
2 if a < 1 and 1 − a/c ≤ b ≤ 1/ac c, u0 , v0 also has global asymptotic stability.
Proof. Defining
ωu, v lπ u
0
u0
1
r/br 1 − c
dr dx r/r 1
b
lπ v
0
v0
s − v0
ds dx,
s
2.3
we can get
∂ω 1
∂t
b
lπ
hu − hu0 pu − pu0 dx Y t,
2.4
0
where
u
,
pu 1 − auu 1,
u1
lπ h u 2
d2 v0 lπ vx2
Y t −d1 c
ux dx dx.
2
2
b
0 h u
0 v
hu 2.5
4
Abstract and Applied Analysis
In what follows, we split it into two cases to prove. If a ≥ 1, for all u > 0, p u < 0, since
h u 1/u 12 > 0, we can get
hu − hu0 pu − pu0 ≤ 0.
2.6
If a < 1 and 1 − a/c ≤ b ≤ 1/ac c equal to v0 ≤ b, we can still get 2.6. That is to say
2.7
wt u, v < 0.
We prove the conclusion.
Because of the conclusion of Lemma 2.1, we always assume a < 1 and 0 < b < 1 −
a/2ac − c. Introducing perturbations u∗ u − u0 , v∗ v − v0 , and replace u∗ , v∗ with
u, v, for which model 1.5 yields
ut − d1 uxx u u0 − au u0 2 −
vt − d2 vxx −cv v0 u u0 v v0 ,
bu u0 1
u u0 v v0 ,
bu u0 1
ux 0, t vx 0, t 0,
ux lπ, t vx lπ, t 0,
ux, 0 ≥ 0,
x ∈ 0, lπ.
vx, 0 ≥ 0,
2.8
Now we can get the linearized system of parametric model 2.8 at 0, 0,
ut
Lb u ,
Δ
vt
v
2.9
where
⎛
⎜
⎜
Δ
⎝
0
∂2
d2 2
∂x
⎞
∂2
d1 2 ⎟
∂x ⎟,
⎠
0
⎞
u0 1 − a − au0 −c
⎟
⎜
1 u0
⎟
⎜
⎟.
Lb ⎜
⎟
⎜
1 − au0
⎝
0⎠
1 u0
⎛
2.10
are as follows:
The eigenvalues of Δ
n2
n2
−d2 2 , −d1 2
l
l
∞
,
2.11
n0
and the corresponding eigenvectors as follows:
∞
2
βn1 , βnk
,
n0
2.12
Abstract and Applied Analysis
5
where
⎞
0
⎟
⎜
βn1 ⎝ n ⎠,
cos x
l
⎛
⎛ n ⎞
cos x
⎜
l ⎟
βn2 ⎝
⎠.
0
2.13
Define for all y ∈ H
⎛ 1⎞
βk
YkT ⎝ 2 ⎠,
y
βk
k1
n
⎞
⎛
y, βk1 ⎠.
Yk ⎝
y, βk2 2.14
Lb, if and only if the equation
It is easy to get, λ ∈ Δ
⎛
⎛
⎞⎞
⎛ ⎞
k2
n
0 ⎟⎟ βk1
⎜
⎜−d1 l2
T⎜
⎟⎟⎝ ⎠ 0
Yk ⎝Eλ − Lb − ⎜
⎝
k2 ⎠⎠ βk2
k1
0
−d2 2
l
2.15
⎛
⎞
n2
0 ⎟
⎜−d1 l2
⎟ 0.
Eλ − Lb − ⎜
⎝
n2 ⎠
0
−d2 2 l
2.16
λ2 − Tn bλ Dn b 0,
2.17
bc
d1 d2 n2
−
,
Tn b bc 1 − a − 2a
1 − bc
l2
2abc n2 d1 d2 n4
Dn b c1 − bc − abc − d2 bc 1 − a −
.
1 − bc l2
l4
2.18
is held.
We obtain
Rewrite it as
where
From 2.18, when 1 − a/ac c < b < 1 − a/c is held, we can get Tn b < 0, Dn b > 0. So
the system’s eigenvalues have negative real part, and u0 , v0 has local asymptotic stability.
Then, we can conclude that the system has Hopf bifurcation 14 in b ∈ 0, 1 − a/ac c.
Define
B b0 | Tn b0 0, Dn b0 > 0, Tj b0 /
0, Dj b0 /
0, ∀j /
n .
2.19
6
Abstract and Applied Analysis
Lb, where
b0 ∈ B ∪ 0, 1 − a/ac c, αb ± iωb are characteristic roots of Δ
αb 1
d 1 d2 2
n
Dn b − α2 b,
,
ωb
Ab −
2
l2
bc
Ab bc 1 − a − 2a
.
1 − bc
2.20
Now we compute transversality condition:
×
⎧
⎪
⎪
⎪
> 0,
⎪
⎨
⎪
⎪
⎪
⎪
⎩< 0,
2
1
4b0 c
b0 c
−1−
−
a
1 − b0 c
1 − b0 c
%
2a
1
1−
,
0 < b0 <
c
1a
%
1
2a
1−a
1−
< b0 <
.
c
1a
ac c
1
α b0 a1 − b0 c2
2
2.21
Now we consider A0 Ab0B 0 and Ab is positive in 0, b0B . So we can get the
maximum value of Ab defined as Ab∗ :
Define
&
ln n
d 1 d2
,
Ab∗ ⎛&
n ∈ N, Ab∗ a⎝
⎞2
√
1
1 − 2⎠ ,
a
2.22
B
B
for all l ∈ ln , ln1 , 0 ≤ j ≤ n, bj,−
and bj,
are two roots of the equation
Ab d 1 d2 2
j .
l2
2.23
It is easy to get
B
B
< · · · < bn,−
0 < b1,−
⎛
⎞
&
2a ⎠
1⎝
B
B
1−
< bn,
<
< · · · < b1,
< b0B .
c
1a
2.24
B
B
Then we give the condition of Dn bj,±
/
0 especially Dn bj,±
> 0.
As we know
Dn b ≥ ac − d2 Ab∗ n2
n4
d
d
.
1
2
l2
l4
2.25
Abstract and Applied Analysis
7
Then Dn b > 0 is held if and only if
d1 d2 > 0,
2.26
d2 Ab∗ 2 − 4d1 d2 ac > 0.
Theorem 2.2. Assume that d1 , d2 , c > 0, 0 < a < 1, and the equation is held:
d2 A2 b∗ − 4d1 ac > 0,
2.27
where
&
ln n
d 1 d2
,
Ab∗ ⎛&
n ∈ N, Ab a⎝
∗
⎞2
√
1
1 − 2⎠ ,
a
2.28
B
or b b0B ; there are Hopf bifurcations at the real solution of
then for all l ∈ ln , ln1 , existing b bj,±
model 1.5.
Furthermore
⎛
B
B
< · · · < bn,−
<
0 < b1,−
1⎝
1−
c
⎞
&
2a ⎠
B
B
< bn,
< · · · < b1,
< b0B .
1a
2.29
3. Hopf Bifurcation
In the above section, we have already obtained the conditions which ensure that model 2.8
undergoes the Hopf bifurcation at the critical values b0 or bj,± j 1, · · · . In the following
part, we will study the direction and stability of the Hopf bifurcation based on the normal
form approach theory and center manifold theory introduced by Hassard at al. 14.
Firstly, by the transformation u∗ u − u0 , v∗ v − v0 , and replacing u∗ , v∗ with u, v,
the parametric system 1.5 is equivalent to the following functional differential equation
FDE system:
∂U Δ Lb0 Fb0 , U,
∂t
3.1
where
T
U u, v ,
Fb0 , U f − fu u − fv v
g − gu u − g v v
.
3.2
8
Abstract and Applied Analysis
The adjoint operator of Ln b is defined as
⎛
⎞
∂2
fv b0 ⎜d1 2 fu b0 ⎟
∂x
⎟.
L∗ b0 ⎜
2
⎝
⎠
∂
gu b0 d2 2 gv b0 ∂x
3.3
u, Lb0 v L∗ b0 u, v.
3.4
It is easy to get
From the discussions in Section 2, define q∗ a∗n , bn∗ T cosn/lx. We have
L∗ b0 q∗ −iωq∗ ,
q∗ , q 1,
q∗ , q 0.
3.5
Decompose X as X X c ⊕ X s , where X c {zq zq | z ∈ C} and X s {u ∈ x | q∗ , u 0}.
For all u, v ∈ X, existing z ∈ C and ω ω1 , ω2 ∈ X s , we can obtain
ω1
u
.
zq zq ω2
v
3.6
Rewrite 3.1 as
ż iω0 z q∗ , F0∗ ,
3.7
ω̇ Lb0 ω Hz, z, ω,
where
(
'
Hz, z, ω F0∗ − q∗ , F0∗ q − q∗ , F0∗ q.
F0∗ zq zq ω,
3.8
Using the same notations as in 11,
F0∗ U 1
1
QU, U CU, U, U OU4 ,
2
6
3.9
where U u, v and Q, C are symmetrical multilinear functions. We can compute
⎛
A1n
⎞
n
Q q, q ⎝ 2 ⎠cos2 x,
An
l
⎛
Bn1
⎞
n
Q q, q ⎝ 2 ⎠cos2 x,
Bn
l
⎛
C q, q, q ⎝
Cn1
Cn2
⎞
⎠cos3 n x,
l
3.10
Abstract and Applied Analysis
9
where
A1n fuu a2n 2fuv an bn fvv bn2 ,
A2n guu a2n 2guv an bn gvv bn2 ,
Bn1 fuu |an |2 fuv an bn an bn fvv |bn |2 ,
Bn2 guu |an |2 guv an bn an bn gvv |bn |2 ,
3.11
Cn1 fuuu |an |2 an fuuv 2|an |2 bn a2n bn fuvv 2bn2 an bn2 an fvvv |bn |2 bn ,
Cn2 guuu |an |2 an guuv 2|an |2 bn a2n bn guvv 2bn2 an bn2 an gvvv |bn |2 bn .
Define
Hz, z, ω 1
1
H20 z2 H11 zz H02 z2 o|z||ω|,
2
2
3.12
where
'
(
H20 Q q, q − q∗ , Q q, q q − q∗ , Q q, q q,
H11
3.13
'
(
Q q, q − q∗ , Q q, q q − q∗ , Q q, q q.
On the center manifold, we have
ω
1
1
ω20 z2 ω11 zz ω02 z2 o |z|3 .
2
2
3.14
We can obtain
w20 2iω0 I − Ln b0 −1 H20 ,
ω11 Ln b0 −1 H11 .
3.15
Comparing 3.9 and 3.13, we can get
H20
⎧
⎛ ⎞
⎛ ⎞
⎪
A1n 1
A1n
⎪
n
1
⎪
2
2n
⎝
⎠
⎠
⎝
⎪
cos
x
−
cos
x
,
Q
q,
q
−
⎪
⎪
⎨
l
2
2
2
A2n
A2n
⎛ ⎞
⎛ 1⎞
⎪
A
a0
⎪
a
⎪⎝ 0 ⎠
0
⎪
⎪
− q∗ , Q q, q ⎝ ⎠,
− q∗ , Q q, q
⎪
⎩ A2
b0
b
0
0
n ∈ N∗ ,
3.16
n 0.
10
Abstract and Applied Analysis
Similarly
ω11
⎛ ⎞
⎧
⎪
Cn1 ⎪
n
1
⎪
⎪− Lbo −1 ⎝ ⎠ cos2 1 ,
n ∈ N∗ ,
⎪
⎪
2
2
2
⎪
⎨
Cn
⎛ ⎞
⎛ ⎞⎤
⎞
⎛
⎡
⎪
⎪
( a0
'
a10
c01
⎪
⎪
−1
⎪−Lb ⎣⎝ ⎠ − q∗ , Q q, q ⎝ ⎠ − q∗ , Q q, q ⎝ ⎠⎦, n 0.
⎪
0
⎪
⎩
c02
b02
b0
3.17
Then on the center manifold rewrite dU as
gij
dz
iω0 z q∗ , F0∗ iω0 z zi zj o |z|4 ,
dt
i!j!
2≤ij≤3
3.18
where
4cω0 a2 − a1 − a2 ω0 − 2ca2 3a − 1i
g20 q∗ , Q q, q ,
1 − a2 ω0
a a − q ω0 − 2ca2 i
,
g11 q∗ , Q q, q 1 aω0
a1 − a2 2cω0 a2 − 4ca2 i
,
g02 q∗ , Q q, q 1 − a2 ω0
g21 2 q∗ , Q w11 , q q∗ , Q w20 , q q∗ , C q, q, q
−12a3 1 − aω0 − 8ca3 ω0 4ca33−5ai
1 − a1 a2 ω0
3.19
.
Using conclusions in 14 we can get
2
2
g02 g11 g21
g20 g11 3αb iωb
,
C1 b 2
2
αb iωb 2αb 3iωb
2
2α b ω b
3.20
then
C1 b0 2 1 2
g21
i
,
g20 g11 − 2g11 − g02 2ω0
3
2
Re{c1 0}
,
β2 2 Re{c1 0},
Re{b τn }
2π 1 τ2 s2 o s4 ,
T2 ω0
μ2 −
3.21
Abstract and Applied Analysis
11
3
u(2)
u(1)
2
1
0
2
1
0
10
x
5
0
0
10
10
20
x
t
20
5
a
0 0
10
t
b
Figure 1: When b 0.1, the positive equilibrium point Y0 is asymptotically stable.
where
.
1
Rec1 b0 τ2 −
ω b0 .
Imc1 b0 −
ω0
α b0 3.22
Now we give a conclusion.
Conclusion. 1 The sign of μ2 determines the direction of Hopf bifurcation. When μ2 > 0, the
Hopf bifurcation is supercritical; when μ2 < 0, the Hopf bifurcation is subcritical.
2 β2 determines the stability of bifurcated periodic solutions. When β2 < 0, the
periodic solutions are stable; when β2 > 0, the periodic solutions are unstable.
3 T2 determines the period of bifurcated periodic solutions. When T2 > 0, the period
increases; when T2 < 0, the period decreases.
4. Example
In this section, we use a numerical simulation to illustrate the analytical results we obtained
in previous sections.
Let x ∈ 0, lπ, d1 1, d2 3, c 4, a 0.0588. The system 1.5 is
ut − uxx u − 0.0588u2 −
vt − 3vxx −4v uv
,
bu 1
uv
,
bu 1
ux 0, t vx 0, t 0,
ux lπ, t vx lπ, t 0,
ux, 0 ≥ 0,
x ∈ 0, lπ.
vx, 0 ≥ 0,
4.1
Now we determine the direction of a Hopf bifurcation with b ∈ B and the other properties of
bifurcating periodic solutions based on the theory of Hassard et al. 14, as discussed before.
By means of software MATLAB 7.0, we can get some figures to illustrate the effectiveness of
12
Abstract and Applied Analysis
15
5
u(2)
u(1)
10
10
0
20
15
20
x
10
0 0
10
t
10
x
a
5
0
0
5
10
15
t
b
Figure 2: When b 0.1253, periodic solutions occur from Y0 .
20
u(2)
u(1)
40
20
0
10
x
0 0
10
t
10
x
a
0 0
10
t
b
Figure 3: When b 1.3, the positive equilibrium point Y0 is unstable.
our results. b0B 0.2222, b∗ 0.1667, Ab∗ 0.4706, ln 2.9155n, and 2.27 is held. When n 2, l 3.1162, ln , ln1 2.9155, 5.8302, l ∈ ln , ln1 . We can get B 0.1253, 0.1944, 0.2222.
The only positive equilibrium point of 4.1 is Y0 4b/1−4b, b0.0588−u0 1u0 /0.0588.
When b 0.1253, we can compute
Re c1 0.1253 0.2856 > 0,
μ2 −0.3423 < 0,
T2 1.2346 > 0.
4.2
The positive equilibrium point of 4.1 is unstable and the Hopf bifurcation is supercritical.
The positive equilibrium point Y0 of system 4.1 is locally asymptotically stable when b 0.1
as is illustrated by computer simulations in Figure 1. And periodic solutions occur from Y0
when b 0.1253 as is illustrated by computer simulations in Figure 2. When b 1.3, we
can easily show that the positive equilibrium point Y0 is unstable as is illustrated in Figure 3.
From the above results, we can conclude that the stability properties of the system could
switch with parameter b.
Abstract and Applied Analysis
13
Acknowledgments
This research was supported by the National Natural Science Foundations of Heilongjiang
province C200941. The authors are grateful to the anonymous reviewers for their valuable
comments and suggestions which essentially improve the quality of the paper.
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