Global existence and stability in a two-species chemotaxis system

Abstract

This paper deals with the following two-species chemotaxis system w zu v t t tt= = == ∆ ∆ ∆∆ z v uw − − −− z v χχ+ + 12∇∇ h h (· ( · u w ((u) ) w∇∇vz))++µµ12uw(1(1−−uw−−a1aw2)u)xx ∈∈ΩΩtt>>00 x ∈ Ω, t > 0, x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n with smooth boundary. The parameters in the system are positive and the signal production function h is a prescribed C 1 -regular function. The main objectives of this paper are two-fold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium (u∗, v∗, w∗, z∗) may be globally attractive in the weak competition case (i.e., 0 < a1, a2 < 1), while the constant stationary solution (0, h(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e., a1 > 1 > a2 > 0). In the fully strong competition case (i.e. a1, a2 > 1), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous (u∗, v∗, w∗, z∗). The matter which species ultimately wins out depends crucially on the starting advantage each species has.