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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 948480, 13 pages
Dynamic Properties of a Forest Fire Model
Department of Mathematics, Northeast Forestry University, Harbin 150040, China
Received 11 June 2012; Revised 7 September 2012; Accepted 17 September 2012
Academic Editor: Bashir Ahmad
Copyright © 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.
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 .
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 . Lots of articles related to percolation theory  and self-organized criticality  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  in the 1930s. From then on, it had been applied in various fields including forest fire: where can be seen as the area of the burned forest. is a diffusion term of in space, and is the diffusion coefficient. is a nonlinear function. The equation can describe the speed of fire spreading. Zeldovich et al. gave the famous theory of combustion and explosions . 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, is the area where the fire has been put out. is a diffusion term of in space, and is the diffusion coefficient. is the resurgence probability of . 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, and influence each other. Thus, and must be functions of . We define by referring to the combustion model :
Since has opposite effect on the fire area (or ), we can also define by taking into account KPP model :
Then we get a new model: Define and an inner product is given by where . From the standpoint of biology, we are only interested in the dynamics of model (1.5) in the region:
2. Stability Analysis
Firstly, we consider the location stability [14, 15] and the number of the equilibria of model (1.5) in . We can also study autowave solutions  of the model. The interior equilibrium point is a root of the following equation: It is obvious that (2.1) has an only real solution , where and .
Now, we analyze the asymptotic stability of by Lyapunov function.
Lemma 2.1. For the model (1.5),
(1) if , is global asymptotic stability.
(2) if and also has global asymptotic stability.
Proof. Defining we can get where In what follows, we split it into two cases to prove. If , for all , since , we can get If and (equal to , 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 and . Introducing perturbations , and replace with , for which model (1.5) yields Now we can get the linearized system of parametric model (2.8) at , where The eigenvalues of are as follows: and the corresponding eigenvectors as follows: where Define for all It is easy to get, , if and only if the equation is held.
We obtain Rewrite it as where From (2.18), when is held, we can get . So the system's eigenvalues have negative real part, and has local asymptotic stability. Then, we can conclude that the system has Hopf bifurcation  in .
Define , are characteristic roots of , where Now we compute transversality condition: Now we consider and is positive in . So we can get the maximum value of (defined as ):
Define for all and are two roots of the equation It is easy to get Then we give the condition of especially .
As we know Then is held if and only if
Theorem 2.2. Assume that , and the equation is held:
then for all , existing or ; there are Hopf bifurcations at the real solution of model (1.5).
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 or . 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. .
Firstly, by the transformation , and replacing with , the parametric system (1.5) is equivalent to the following functional differential equation (FDE) system: where The adjoint operator of is defined as It is easy to get From the discussions in Section 2, define . We have Decompose as , where and . For all , existing and , we can obtain Rewrite (3.1) as where Using the same notations as in , where and are symmetrical multilinear functions. We can compute where Define where On the center manifold, we have We can obtain Comparing (3.9) and (3.13), we can get Similarly Then on the center manifold rewrite as where Using conclusions in  we can get then where
Now we give a conclusion.
Conclusion. (1) The sign of determines the direction of Hopf bifurcation. When , the Hopf bifurcation is supercritical; when , the Hopf bifurcation is subcritical.
(2) determines the stability of bifurcated periodic solutions. When , the periodic solutions are stable; when , the periodic solutions are unstable.
(3) determines the period of bifurcated periodic solutions. When , the period increases; when , the period decreases.
In this section, we use a numerical simulation to illustrate the analytical results we obtained in previous sections.
Let . The system (1.5) is Now we determine the direction of a Hopf bifurcation with and the other properties of bifurcating periodic solutions based on the theory of Hassard et al. , as discussed before. By means of software MATLAB 7.0, we can get some figures to illustrate the effectiveness of our results. , , , , and (2.27) is held. When , , , . We can get . The only positive equilibrium point of (4.1) is . When , we can compute The positive equilibrium point of (4.1) is unstable and the Hopf bifurcation is supercritical. The positive equilibrium point of system (4.1) is locally asymptotically stable when as is illustrated by computer simulations in Figure 1. And periodic solutions occur from when as is illustrated by computer simulations in Figure 2. When , we can easily show that the positive equilibrium point 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 .
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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