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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2020
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| Online Access: | https://arxiv.org/abs/2009.11820 |
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| _version_ | 1866911919348121600 |
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| author | Ojeda, Gabriel Berzunza Pardo, Juan Carlos |
| author_facet | Ojeda, Gabriel Berzunza Pardo, Juan Carlos |
| contents | In this manuscript, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. Here, we provide a series of integral tests that explain how competition and cooperation regulate the long-term behaviour of BPI-processes. In particular, such integral tests describe the events of explosion and extinction and provide conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use a random time change representation in terms of a modified branching process with immigration and moment duality. The moment dual of BPI-processes turns out to be a family of diffusions taking values on $[0,1]$ which are interesting in their own right and that we introduce as generalised Wright-Fisher diffusions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_11820 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Branching processes with pairwise interactions Ojeda, Gabriel Berzunza Pardo, Juan Carlos Probability 60J80, 60K35 In this manuscript, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. Here, we provide a series of integral tests that explain how competition and cooperation regulate the long-term behaviour of BPI-processes. In particular, such integral tests describe the events of explosion and extinction and provide conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use a random time change representation in terms of a modified branching process with immigration and moment duality. The moment dual of BPI-processes turns out to be a family of diffusions taking values on $[0,1]$ which are interesting in their own right and that we introduce as generalised Wright-Fisher diffusions. |
| title | Branching processes with pairwise interactions |
| topic | Probability 60J80, 60K35 |
| url | https://arxiv.org/abs/2009.11820 |