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arXiv · 2503.20390

Periodic KPP equations: new insights into persistence, spreading, and the role of advection

Nathanaël Boutillon·François Hamel·Lionel Roques
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Abstract

We focus on the persistence and spreading properties for a heterogeneous Fisher-KPP equation with advection. After reviewing the different notions of persistence and spreading speeds, we focus on the effect of the direction of the advection term, denoted by $b$. First, we prove that changing $b$ to $-b$ can have an effect on the spreading speeds and the ability of persistence. Next, we provide a class of relationships between the intrinsic growth term $r$ and the advection term $b$ such that changing $b$ to $-b$ does not change the spreading speeds and the ability of persistence. We briefly mention the cases of slowly and rapidly varying environments, and bounded domains. Lastly, we show that in general, the spreading speeds are not controlled by the ability of persistence, and conversely. However, when there is no advection term, the spreading speeds are controlled by the ability of persistence, though the converse still does not hold.

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Nathanaël Boutillon, François Hamel, Lionel Roques. 2025-03-28. Periodic KPP equations: new insights into persistence, spreading, and the role of advection. https://arxiv.org/abs/2503.20390

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