Dynamic Properties of a Forest Fire Model

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: R2 { u, v | u > 0, v > 0}. 1.8 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 − uv b u 1 0, −cv uv b u 1 0. 2.1 It is obvious that 2.1 has an only real solution Y0 u0, v0 , where u0 bc 1 − bc , v0 ab ( 1 a − u0 ) 1 u0 , 2.2 and b < 1/c a 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 ω u, v ∫ lπ 0 ∫u u0 r/b r 1 − c r/ r 1 dr dx 1 b ∫ lπ 0 ∫v v0 s − v0 s ds dx, 2.3


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: where u u x, t can be seen as the area of the burned forest.u xx is a diffusion term of u in space, and d 1 is the diffusion coefficient.f u is a nonlinear function.The equation can 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.
In this equation, v v x, t is the area where the fire has been put out.v xx is a diffusion term of v in space, and d 2 is the diffusion coefficient.c is the resurgence probability of v. g v 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 g u, v by referring to the combustion model 13 : Since g u, v has opposite effect on the fire area or u , we can also define f u, v by taking into account KPP model 8 : Then we get a new model: and an inner product is given by Abstract and Applied Analysis 3 where From the standpoint of biology, we are only interested in the dynamics of model 1.5 in the region: 1.8

Stability Analysis
Firstly, we consider the location stability 14, 15 and the number of the equilibria of model 1.5 in R 2 .We can also study autowave solutions 16 of the model.The interior equilibrium point is a root of the following equation:

2.1
It is obvious that 2.1 has an only real solution Y 0 u 0 , v 0 , where Proof.Defining where

2.5
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 1 2 > 0, we can get If a < 1 and 1 − a /c ≤ b ≤ 1/ ac c equal to v 0 ≤ b , we can still get 2.6 .That is to say We prove the conclusion.
Because of the conclusion of Lemma 2.1, we always assume a < 1 and 0 and replace u * , v * with u, v , for which model 1.5 yields

2.8
Now we can get the linearized system of parametric model 2.8 at 0, 0 , where The eigenvalues of Δ are as follows: and the corresponding eigenvectors as follows: where

2.13
Define for all y ∈ H It is easy to get, λ ∈ Δ L b , if and only if the equation Rewrite it as where

2.18
From 2.18 , when 1 − a / ac c < b < 1 − a /c is held, we can get T n b < 0, D n b > 0. So the system's eigenvalues have negative real part, and u 0 , v 0 has local asymptotic stability.Then, we can conclude that the system has Hopf bifurcation 14 in b ∈ 0, 1 − a / ac c .Define

2.20
Now we compute transversality condition:

2.22
for all l ∈ l n , l n 1 , 0 ≤ j ≤ n, b B j,− and b B j, are two roots of the equation

2.23
It is easy to get

2.24
Then we give the condition of D n b B j,± / 0 especially D n b B j,± > 0. As we know

2.25
Then D n b > 0 is held if and only if

2.28
then for all l ∈ l n , l n 1 , existing b b B j,± or b b B 0 ; there are Hopf bifurcations at the real solution of model 1.5 .Furthermore 2.29

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 b 0 or b j,± 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 − u 0 , v * v − v 0 , and replacing u * , v * with u, v , the parametric system 1.5 is equivalent to the following functional differential equation FDE system: The adjoint operator of L n b is defined as From the discussions in Section 2, define q * a * n , b * n T cos n/l x.We have where Using the same notations as in 11 , where U u, v and Q, C are symmetrical multilinear functions.We can compute

3.11
Define 12 where H 20 Q q, q − q * , Q q, q q − q * , Q q, q q, H 11 Q q, q − q * , Q q, q q − q * , Q q, q q.

3.13
On the center manifold, we have

3.14
We can obtain

3.15
Comparing 3.9 and 3.13 , we can get

3.19
Using conclusions in 14 we can get where Now we give a conclusion.
3 T 2 determines the period of bifurcated periodic solutions.When T 2 > 0, the period increases; when T 2 < 0, the period decreases.

Example
In this section, we use a numerical simulation to illustrate the analytical results we obtained in previous sections.
Let x ∈ 0, lπ , d 1 1, d 2 3, c 4, a 0.0588.The system 1.5 is The positive equilibrium point of 4.1 is unstable and the Hopf bifurcation is supercritical.
The positive equilibrium point Y 0 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 Y 0 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 Y 0 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.
A b is positive in 0, b B 0 .So we can get the maximum value of A b defined as A b * : Define

Figure 1 :
Figure 1: When b 0.1, the positive equilibrium point Y 0 is asymptotically stable.