With Cross-diffusion Model Of Forest. A Traveling Wave Solution Stability

we investigate a biological model of mono-species forest with two age classes which takes account of seed production and disperal was first presented in[1].where γ(v) = av — b² + c with postive a , b , c.By means of an asymototic procedure , (0.3) is then reduceed to the following lewer-dimensional reaction-cross-diffusion model[1]Due to the cross-diffusion term , (0.4) is no longer a parabolic system ,so the classical theory of stability of travelling waves for parabolic systems [3] [4] cannot be applied directly to (0.4).In this paper , combining the theory of the Co-semigroup with some basic ideas in [3] , by a series of detailed spectral analysis , subtle estimates , we show that the travelling waves obtained in [2] are exponentially stable with shift in a suitable space.when (ρ, h) ∈ regionl={(ρ, h)|sh ≤ ρ ≤ (s + 1)h} and the parameters satisfy some conditions , we can show that the travelling waves obtained in Theorem IV are exponentially stable with shift in a power space X_ω for the wave speed satisfied c^* < c < 1.when (ρ,h) ∈ region2= {(ρ,h)|(s + 1)h ≤ ρ} and the parameters satisfy some conditions , we can show that the travelling waves obtained in Theorem V are exponentially stable with shift in a power space X_ω|- for the wave speed satisfied c|-^* < c < 1.