Global Solutions For The Tritrophic Food-Chain Model With Cross-Diffusion

Global behavior of solutions is investigated for the tritrophic food-chain model with Hollingâ…¡type functional response u_t=â–³(d₁u+α₁₁u²+α₁₂uv+α₁₃uw)+u[r(1-u/k)-(a₁v)/(1+b₁u)],x∈Ω,t＞0, v_t=â–³(d₂v+α₂₁uv+α₂₂v²+α₂₃vw)+v[(a₁u)/(1+b₁u)-(a₂w)/(1+b₂v)-D₁],x∈Ω,t＞0.(1) w_t=â–³(d₃w+α₃₁uw+α₃₂vw+α₃₃w²)+w[(a₂v)/(1+b₂v)-D₂],x∈Ω,t＞0, whereΩis a bounded smooth domain in Rⁿ.The organization of this paper is follows:In Section 1,the asymptotical stability of nonnegative equilibrium points for the system of ordinary differential equations of(1)is studied.In Section 2,the weakly coupled reaction-diffusion system of this model with homo-geneous Neumann boundary conditions is studied.First the existence and uniqueness of the global solutions for the system of(1)are discussed.Then stability of nonnegative equilibrium points is discussed.In Section 3,the existence and uniform boundedness of global solutions for tritrophic food-chain model with self and cross-diffusion are investigated.Firstly,using the method of energy estimates and Gagliardo-Nirenberg type inequalities,the existence and uniform boundedness of nonnegative global solutions for the system of(1)with homogeneous Neu-mann boundary conditions are established when the space dimension is one.Secondly, using Sobolev embedding theorems and bootstrap arguments,the existence and unique-ness of nonnegative global classical solutions for the system of(1)with homogeneous Neumann boundary conditions are investigated when the space dimension be at most 5.