Abstract We study the following Neumann problem in one-dimension space arising from population genetics: { u t = d u x x + h ( x ) u 2 ( 1 − u ) in ( − 1 , 1 ) × ( 0 , ∞ ) , 0 ≤ u ≤ 1 in ( − 1 , 1 ) × ( 0 , ∞ ) , u ′ ( − 1 , t ) = u ′ ( 1 , t ) = 0 in ( 0 , ∞ ) , where h changes sign in ( − 1 , 1 ) and d is a positive parameter. Lou and Nagylaki (2002) [6] conjectured that if ∫ − 1 1 h ( x ) d x ≥ 0 , then this problem has at most one nontrivial steady state (i.e., a time-independent solution which is not identically equal to zero or one). Nakashima (2018) [15] proved this uniqueness under some additional conditions on h ( x ) . Unexpectedly, in this paper, we discover 3 nontrivial steady states for some h ( x ) satisfying ∫ − 1 1 h ( x ) d x ≥ 0 . Moreover, bi-stable phenomenon occurs in this scenario: one with two layers is stable; two with one layer each are ordered with the smaller one being stable and the larger one being unstable.

中文翻译：

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群体遗传学中一个不定非线性扩散问题的非唯一性

摘要 我们研究了由种群遗传学引起的一维空间中的以下诺依曼问题： { ut = duxx + h ( x ) u 2 ( 1 − u ) in ( − 1 , 1 ) × ( 0 , ∞ ) , 0 ≤ u ≤ 1 in ( − 1 , 1 ) × ( 0 , ∞ ) , u ′ ( − 1 , t ) = u ′ ( 1 , t ) = 0 in ( 0 , ∞ ) ，其中 h 改变登录 ( − 1 , 1 ) 和 d 是一个正参数。Lou 和 Nagylaki (2002) [6] 推测如果 ∫ − 1 1 h ( x ) dx ≥ 0 ，那么这个问题至多有一个非平凡的稳态（即，一个不完全等于零或一）。Nakashima (2018) [15] 在 h ( x ) 上的一些附加条件下证明了这种唯一性。出乎意料的是，在本文中，我们发现了一些 h ( x ) 满足 ∫ − 1 1 h ( x ) dx ≥ 0 的 3 个非平凡稳态。而且在这种情况下会出现双稳态现象：一层二层稳定；