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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. 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