The current paper considers the dynamics of the following chemotaxis system of parabolicelliptic type with local as well as nonlocal time and space dependent logistic sourcewhere Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary ∂Ω and a i (t, x) (i = 0, 1, 2) are locally Hölder continuous in t ∈ R uniformly with respect to x ∈Ω and continuous in x ∈Ω. We first prove the local existence and uniqueness of classical solutions (u(x, t; t 0 , u 0 ), v(x, t; t 0 , u 0 )) with u(x, t 0 ; t 0 , u 0 ) = u 0 (x) for various initial functions u 0 (x).Next, under some conditions on the coefficients a 1 (t, x), a 2 (t, x), and χ and the dimension n, we prove the global existence and boundedness of classical solutions (u(x, t; t 0 , u 0 ), v(x, t; t 0 , u 0 )) with given nonnegative initial function u(x, t 0 ; t 0 , u 0 ) = u 0 (x). Then, under the same conditions for the global existence, we show that the system has an entire positive classical solution (u * (x, t), v * (x, t)). Moreover, if a i (t, x) (i = 0, 1, 2) are periodic in t with period T or are independent of t, then the system has a time periodic positive solution (u * (x, t), v * (x, t)) with periodic T or a steady state positive solution (u * (x), v * (x)). If a i (t, x) (i = 0, 1, 2) are independent of x (i.e. are spatially homogeneous), then the system has a spatially homogeneous entire positive solution (u * (t), v * (t)). Finally, under some further assumptions, we prove that the system has a unique entire positive solution (u * (x, t), v * (x, t)) which is globally stable in the sense that for any given t 0 ∈ R and u 0 ∈ C 0 (Ω) with u 0 (x) ≥ 0 and u 0 (x) ≡ 0,Moreover, if a i (t, x) (i = 0, 1, 2) are periodic or almost periodic in t, then the unique entire positive solution (u * (x, t), v * (x, t)) is also periodic or almost periodic in t.

In this paper, we consider two species chemotaxis systems with Lotka-Volterra type competition terms in heterogeneous media. We first find various conditions on the parameters which guarantee the global existence and boundedness of classical solutions with nonnegative initial functions. Next, we find further conditions on the parameters which establish the persistence of the two species. Then, under the same set of conditions for the persistence of two species, we prove the existence of coexistence states. Finally we prove the extinction phenomena in the sense that one of the species dies out asymptotically and the other reaches its carrying capacity as time goes to infinity. The persistence in general two species chemotaxis systems is studied for the first time. Several important techniques are developed to study the persistence and coexistence of two species chemotaxis systems. Many existing results on the persistence, coexistence, and extinction on two species competition systems without chemotaxis are recovered.

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The current paper is concerned with the asymptotic dynamics of two species competition systems with/without chemotaxis in heterogeneous media. In the previous work [15], we find conditions on the parameters in such systems for the persistence of the two species and the existence of positive coexistence states. In this paper, we find conditions on the parameters for the uniqueness and stability of positive coexistence states of such systems. The established results are new even for the two species competition systems without chemotaxis but with space dependent coefficients.Key words. Ultimates bounds of solutions, asymptotic stability and uniqueness of coexistence states.and b i,inf and b i,sup are defined similarly. Note that the occurrence of persistence in (1.2) implies the existence of coexistence states. However, there is little study on the uniqueness and stability of coexistence states of (1.2) in the case that a i and b i (i = 0, 1, 2) depend on x. The uniqueness and stability of coexistence states of (1.2) proved in Corollary 1.1 and in Theorem 1.5(3) are new. It should be pointed out that, in the study of (1.2), the so called competitive comparison principle plays an important role.Consider (1.1). It is important to investigate the role of chemotaxis in determining the dynamical behavior of the solutions with nonnegative initial functions. In particular, it is important to investigate the following questions: whether the presence of chemotaxis affects the global existence of solutions with nonnegative initial functions, or whether finite time blow-up occurs; how to identify the circumstances under which persistence or extinction occurs; in the case that persistence occurs, whether the system has coexistence states, and if so, whether the coexistence states are unique and stable; whether chemotaxis induces new solution patterns; etc. Note that chemotaxis induces several difficulties in the study of (1.1), including the lack of the so called competitive comparison principle.Several authors have studied the issues mentioned in the above for system (1.1) with constant coefficients (see [4,14,19,20,23,24,25]). For example, in [24], the authors studied the global stability of positive constant coexistence state under some assumption on the coefficients. In [23], the authors considered the competitive exclusion under some complicated smallness assumptions on the chemotaxis rates. In [25], the authors obtained nonconstant positive coexistence states induced by chemotaxis for parameters in certain region. In [14], the authors considered a more general competitive-cooperative chemotaxis system with nonlocal terms logistic sources and proved both the phenomena of coexistence and of exclusion for parameters in some natural range.However, there is little study on these asymptotic dynamical issues for (1.1) with general time and space dependent coefficients. Besides the difficulties induced by the chemotaxis, the time and space dependence of the coefficients induces additional difficulties in the study of (1.1). For e...

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