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