## Abstract

We study the following Neumann problem in one-dimension space arising from population genetics: {u_{t}=du_{xx}+h(x)u^{2}(1−u)in(−1,1)×(0,∞),0≤u≤1in(−1,1)×(0,∞),u^{′}(−1,t)=u^{′}(1,t)=0in(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)dx≥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)dx≥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.

Original language | English |
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Pages (from-to) | 4643-4682 |

Number of pages | 40 |

Journal | Journal of Differential Equations |

Volume | 269 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2020 Sept 5 |

Externally published | Yes |

## Keywords

- Complete dominance
- Indefinite weight
- Layers
- Reaction-diffusion equation
- Singular perturbation

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics