QUALITATIVE PROPERTIES FOR A THREE-SPECIES FOOD CHAIN MODEL WITH CROSS-DIFFUSION AND INTRA-SPECIFIC COMPETITION

摘要

In this paper, we consider a three-species spatial food chain system as follows @@@ {u(t) = d(1)Delta u + u(1-u) - b(1)uv; x is an element of Omega, t > 0; @@@ v(t) = del. (gamma(1)(u del)v) - del . (chi(1) (u)v del u) + uv - b(2)vw - theta(1)v - alpha(1)v(2), x is an element of Omega, t > 0, @@@ w(t) = del. (gamma(2)(v)del w) - del. (chi(2)(v)w del. v) + vw - theta(2)w - alpha(2)w(2), x is an element of Omega, t > 0, @@@ partial derivative u/partial derivative nu = partial derivative v/partial derivative nu = partial derivative w/partial derivative nu = 0, x is an element of partial derivative Omega, t > 0; @@@ (u, v, w) (x, 0) = (u0, v0, w0)(x), x is an element of Omega, @@@ where Omega subset of R-2 is a bounded domain with smooth boundary. For i = 1, 2, all the parameters b(i), alpha(i), theta(i) are positive and the functions gamma(i) > 0 and chi(i) > 0 satisfy (gamma(i), chi(i)) is an element of [C-2([0, infinity))](2): We first establish the global existence of classical solutions with uniform-in-time bound by using the coupling energy estimates and Moser iteration. Moreover, by constructing Lyapunov functionals, the global stability and convergence rate of steady states are established under certain conditions.

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