Responsive dormancy of a spatial population among a moving trap
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918133647802368 |
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| author | Shafigh, Helia Tyrpak, Leo |
| author_facet | Shafigh, Helia Tyrpak, Leo |
| contents | 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 trap 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Responsive dormancy of a spatial population among a moving trap Shafigh, Helia Tyrpak, Leo Probability 60J28, 60K35, 60K37, 92D25 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 trap 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. |
| title | Responsive dormancy of a spatial population among a moving trap |
| topic | Probability 60J28, 60K35, 60K37, 92D25 |
| url | https://arxiv.org/abs/2509.01288 |