Non-Equilibrium Fluctuations for a Spatial Logistic Branching Process with Weak Competition
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916386242035712 |
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| author | Tendron, Thomas |
| author_facet | Tendron, Thomas |
| contents | The spatial logistic branching process is a population dynamics model in which particles move on a lattice according to independent simple symmetric random walks, each particle splits into a random number of individuals at rate one, and pairs of particles at the same location compete at rate c. We consider the weak competition regime in which c tends to zero, corresponding to a local carrying capacity tending to infinity like 1/c. We show that the hydrodynamic limit of the spatial logistic branching process is given by the Fisher-Kolmogorov-Petrovsky-Piskunov equation. We then prove that its non-equilibrium fluctuations converge to a generalised Ornstein-Uhlenbeck process with deterministic but heterogeneous coefficients. The proofs rely on an adaptation of the method of v-functions developed in Boldrighini et al. 1992. An intermediate result of independent interest shows how the tail of the offspring distribution and the precise regime in which c tends to zero affect the convergence rate of the expected population size of the spatial logistic branching process to the hydrodynamic limit. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_05269 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Equilibrium Fluctuations for a Spatial Logistic Branching Process with Weak Competition Tendron, Thomas Probability 60F17, 60J80 (Primary), 92D25, 92D40 (Secondary) The spatial logistic branching process is a population dynamics model in which particles move on a lattice according to independent simple symmetric random walks, each particle splits into a random number of individuals at rate one, and pairs of particles at the same location compete at rate c. We consider the weak competition regime in which c tends to zero, corresponding to a local carrying capacity tending to infinity like 1/c. We show that the hydrodynamic limit of the spatial logistic branching process is given by the Fisher-Kolmogorov-Petrovsky-Piskunov equation. We then prove that its non-equilibrium fluctuations converge to a generalised Ornstein-Uhlenbeck process with deterministic but heterogeneous coefficients. The proofs rely on an adaptation of the method of v-functions developed in Boldrighini et al. 1992. An intermediate result of independent interest shows how the tail of the offspring distribution and the precise regime in which c tends to zero affect the convergence rate of the expected population size of the spatial logistic branching process to the hydrodynamic limit. |
| title | Non-Equilibrium Fluctuations for a Spatial Logistic Branching Process with Weak Competition |
| topic | Probability 60F17, 60J80 (Primary), 92D25, 92D40 (Secondary) |
| url | https://arxiv.org/abs/2409.05269 |