Responsive dormancy of a spatial population among a moving trap
Authors
- Shafigh, Helia
ORCID: 0009-0003-8160-9204 - Tyrpak, Leo
2020 Mathematics Subject Classification
- 60J28 60K35 60K37 92D25
Keywords
- Parabolic Anderson model, dormancy, populations with seed-bank, branching random walk, Lyapunov exponents, switching diffusions, Feynman-Kac formula
DOI
Abstract
In this paper, we study a spatial model for dormancy in a random environment via a two-type branching random walk in continuous-time, where individuals switch between dormant and active states depending on the current state of a fluctuating environment (responsive switching). The branching mechanism is governed by the same random environment, which is here taken to be a simple symmetric random walk. We will interpret the presence of this random walk as a emphtrap which attempts to kill the individuals whenever it meets them. The responsive switching between the active and dormant state is defined so that active individuals become dormant only when a trap is present at their location and remain active otherwise. Conversely, dormant individuals can only wake up once the environment becomes trap-free again. We quantify the influence of dormancy on population survival by analyzing the long-time asymptotics of the expected population size. The starting point for our mathematical considerations and proofs is the parabolic Anderson model via the Feynman-Kac formula. Specifically, we investigate the quantitative role of dormancy by extending the Parabolic Anderson model to a two-type random walk framework.
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