Dynamic Properties of a Forest Fire Model

Na Min,Zhilin Qu,Hongyang Zhang

doi:10.1155/2012/948480

Abstract

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.

Highlights

The forest fire is an important issue in the world
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}
The interior equilibrium point is a root of the following equation: u − au[2]

Summary

Introduction

The forest fire is an important issue in the world. 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. We analyze dynamic properties of the reaction-diffusion equations. Kolmogorov et al proposed the famous KPP model 11 in the 1930s. It had been applied in various fields including forest fire: ut d1uxx u f u , x ∈ R, t ≥ 0, 1.1 where u u x, t can be seen as the area of the burned forest. 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}

Stability Analysis

Hopf Bifurcation

Conclusion

Example