The compact interface property for the stochastic heat equation with seed bank

TL;DR

This paper demonstrates that the inclusion of seed banks in a stochastic heat equation preserves the compactness of interfaces over time, despite the added complexity of dormancy and memory effects.

Contribution

It introduces a novel analysis showing that seed banks do not alter the compact interface property in the stochastic heat equation.

Findings

Interfaces remain compact over time with seed banks.

Duality and comparison techniques handle lack of martingale property.

Seed banks do not qualitatively change interface behavior.

Abstract

We investigate the compact interface property in a recently introduced variant of the stochastic heat equation that incorporates dormancy, or equivalently seed banks. There individuals can enter a dormant state during which they are no longer subject to spatial dispersal and genetic drift. This models a state of low metabolic activity as found in microbial species. Mathematically, one obtains a memory effect since mass accumulated by the active population will be retained for all times in the seed bank. This raises the question whether the introduction of a seed bank into the system leads to a qualitatively different behaviour of a possible interface. Here, we aim to show that nevertheless in the stochastic heat equation with seed bank compact interfaces are retained through all times in both the active and dormant population. We use duality and a comparison argument with partial functional differential equations to tackle technical difficulties that emerge due to the lack of the martingale property of our solutions which was crucial in the classical non seed bank case.