Extinction for small KPP patches with fixed bistable-patch length

Determine whether extinction occurs for the KPP-bistable periodic patch model when the bistable patch length \(l_2\) is fixed and the KPP patch length \(l_1\) is sufficiently small.

Background

The paper proves extinction for sufficiently small initial data when the principal eigenvalue is positive, and also establishes extinction for certain larger initial data supported inside sufficiently large unfavorable bistable patches. These results require either smallness of the initial datum or a large-bistable-patch regime.

The authors explicitly identify as unresolved the complementary situation in which $l_2$ is fixed while $l_1$ becomes sufficiently small, rather than allowing the bistable patch length to grow.