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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2105.06985 |
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| _version_ | 1866910757239652352 |
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| author | Maillard, Pascal Raoul, Gaël Tourniaire, Julie |
| author_facet | Maillard, Pascal Raoul, Gaël Tourniaire, Julie |
| contents | We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale $1/\varepsilon$ in space and time. This can be seen as a model for the spatial dynamics of a biological population in a habitat which is heterogeneous at a large scale (mountains, temperature or precipitation gradient\ldots). The model incorporates another parameter, $K$, which is a measure of the local population density. We study the model in the limit when first $\varepsilon\to 0$ and then $K\to\infty$. In this asymptotic regime, we show that the rescaled position of the front as a function of time converges to the solution of an explicit ODE. We further discuss the relation with another popular model of population dynamics, the Fisher-KPP equation, which arises in the limit $K\to\infty$. Combined with known results on the Fisher-KPP equation, our results show in particular that the limits $\varepsilon\to0$ and $K\to\infty$ do not commute in general. We conjecture that an interpolating regime appears when $\log K$ and $1/\varepsilon$ are of the same order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_06985 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Spreading speed of locally regulated population models in macroscopically heterogeneous environments Maillard, Pascal Raoul, Gaël Tourniaire, Julie Probability Analysis of PDEs 60J80 We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale $1/\varepsilon$ in space and time. This can be seen as a model for the spatial dynamics of a biological population in a habitat which is heterogeneous at a large scale (mountains, temperature or precipitation gradient\ldots). The model incorporates another parameter, $K$, which is a measure of the local population density. We study the model in the limit when first $\varepsilon\to 0$ and then $K\to\infty$. In this asymptotic regime, we show that the rescaled position of the front as a function of time converges to the solution of an explicit ODE. We further discuss the relation with another popular model of population dynamics, the Fisher-KPP equation, which arises in the limit $K\to\infty$. Combined with known results on the Fisher-KPP equation, our results show in particular that the limits $\varepsilon\to0$ and $K\to\infty$ do not commute in general. We conjecture that an interpolating regime appears when $\log K$ and $1/\varepsilon$ are of the same order. |
| title | Spreading speed of locally regulated population models in macroscopically heterogeneous environments |
| topic | Probability Analysis of PDEs 60J80 |
| url | https://arxiv.org/abs/2105.06985 |