By appealing to the theory of global attractors on complete metric spaces, we obtain weaker sufficient conditions for the existence of interior global attractors for uniformly persistent dynamical systems, and hence generalize the earlier results on coexistence steady states. We also provide examples to show applicability of our interior fixed point theorem in the case of convex κ-contracting maps, and to prove the existence of discrete-and continuous-time dynamical systems that admit global attractors, but no strong global attractors, which gives an affirmative answer to an open question presented by Sell and You [Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002] in the case of continuous-time semiflows.
Introduction.Uniform persistence is an important concept in population dynamics since it characterizes the long-term survival of some or all interacting species in an ecosystem. There have been extensive investigations on uniform persistence for discrete-and continuous-time dynamical systems. We refer to [13,27,30] for surveys and reviews. Looked at abstractly, uniform persistence is the notion that a closed subset of the state space (e.g., the set of extinction for one or more populations) is repelling for the dynamics on the complementary set. A natural question concerns the existence of "interior" global attractors and "coexistence" steady states for uniformly persistent dynamical systems. The existence of interior global attractors was addressed by Hale and Waltman [10], and the existence of coexistence steady states under a general setting was investigated by Zhao [29]. In [10, 29] the traditional concept of global attractors was employed: a global attractor is a compact, invariant set which attracts every bounded set in the phase space (see, e.g., Hale [7], Temam [24], and Raugel [20]).Recently, the following weaker concept of global attractors was introduced by Hirsch, Smith, and Zhao [11] and Sell and You [22]: a global attractor is a compact, invariant set which attracts some neighborhood of itself and every point in the phase space. For convenience, we refer to a traditional global attractor as a strong global attractor. With the concept of strong global attractor, Zhao [29, Theorem 2.3] assumed more conditions than necessary for the existence of a coexistence fixed point. However, the proof of [29, Theorem 2.3] needs only the property that the interior attractor attracts every compact set, and hence actually implies a general fixed point theorem that, if a continuous and κ-condensing map T has an interior global attractor, then it has a coexistence fixed point (see Theorem 4.1). So an important problem is to obtain sufficient conditions for the existence of interior global attractors for uniformly
