A remark on the global dynamics of competitive systems on ordered Banach spaces
A remark on the global dynamics of competitive systems on ordered Banach spaces
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DOI：
10.1090/proc12768
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发表时间：
2015-03
期刊：
影响因子：
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通讯作者：
King-Yeung Lam;Daniel S. Munther
中科院分区：
文献类型：
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作者：
King-Yeung Lam;Daniel S. Munther
[Hsu-Smith-Waltman，Trans. AMS（1996）]中的一个著名结果指出，在定义在X+ = X1 ×X + 2上的竞争半流中，如果（u ∈，0）和（0，v ∈）分别是X1 × {0}和{0} ×X + 2中的全局吸引子，则对于两个竞争者，以下三个结果之一是可能的：或者存在至少一个共存稳态，或者（u_n，0），（0，v_n）中的一个吸引在顺序区间I = [0，u_n] × [0，v_n]中开始的所有轨迹。然而，通过一个例子证明了，在某些情况下，如果我们把我们的范围扩大到所有的X+，那么（u_∞，0）和（0，v_∞）都不是全局渐近稳定的.本文给出了在不存在共存平衡态的情况下，X+中所有轨迹中（u ∈，0）或（0，v ∈）之一全局渐近稳定的两个充分条件.也就是说，（u，0）或（0，v）中的一个是（i）线性不稳定的，或者（ii）是线性中性稳定的，但零是简单的本征值。我们的研究结果补充了上述论文中提到的反例以及实践中经常出现的应用。
A well-known result in [Hsu-Smith-Waltman, Trans. AMS (1996)] states that in a competitive semiflow defined on X+ = X 1 ×X + 2 , the product of two cones in respective Banach spaces, if (u∗, 0) and (0, v∗) are the global attractors in X 1 × {0} and {0} ×X + 2 respectively, then one of the following three outcomes is possible for the two competitors: either there is at least one coexistence steady state, or one of (u∗, 0), (0, v∗) attracts all trajectories initiating in the order interval I = [0, u∗] × [0, v∗]. However, it was demonstrated by an example that in some cases neither (u∗, 0) nor (0, v∗) is globally asymptotically stable if we broaden our scope to all of X+. In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of (u∗, 0) or (0, v∗) among all trajectories in X+. Namely, one of (u∗, 0) or (0, v∗) is (i) linearly unstable, or (ii) is linearly neutrally stable but zero is a simple eigenvalue. Our results complement the counter example mentioned in the above paper as well as applications that frequently arise in practice.