Impact of diffusion mechanisms on persistence and spreading
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| Format: | Preprint |
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2025
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| _version_ | 1866912619341807616 |
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| author | Boutillon, Nathanaël Kim, Yong-Jung Roques, Lionel |
| author_facet | Boutillon, Nathanaël Kim, Yong-Jung Roques, Lionel |
| contents | We examine a generalized KPP equation with a ``$q$-diffusion", which is a framework that unifies various standard linear diffusion regimes: Fickian diffusion ($q = 0$), Stratonovich diffusion ($q = 1/2$), Fokker-Planck diffusion ($q = 1$), and nonstandard diffusion regimes for general $q\in\mathbb{R}$. Using both analytical methods and numerical simulations, we explore how the ability of persistence (measured by some principal eigenvalue) and how the asymptotic spreading speed depend on the parameter $q$ and on the phase shift between the growth rate $r(x)$ and the diffusion coefficient $D(x)$.
Our results demonstrate that persistence and spreading properties generally depend on $q$: for example, appropriate configurations of $r(x)$ and $D(x)$ can be constructed such that $q$-diffusion either enhances or diminishes the ability of persistence and the spreading speed with respect to the traditional Fickian diffusion. We show that the spatial arrangement of $r(x)$ with respect to $D(x)$ has markedly different effects depending on whether $q > 0$, $q = 0$, or $q < 0$. The case where $r$ is constant is an exception: persistence becomes independent of $q$, while the spreading speed displays a symmetry around $q = 1/2$.
This work underscores the importance of carefully selecting diffusion models in ecological and epidemiological contexts, highlighting their potential implications for persistence, spreading, and control strategies. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_03816 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Impact of diffusion mechanisms on persistence and spreading Boutillon, Nathanaël Kim, Yong-Jung Roques, Lionel Analysis of PDEs 35K57, 35B40, 92D25, 60J70, 35Q92 We examine a generalized KPP equation with a ``$q$-diffusion", which is a framework that unifies various standard linear diffusion regimes: Fickian diffusion ($q = 0$), Stratonovich diffusion ($q = 1/2$), Fokker-Planck diffusion ($q = 1$), and nonstandard diffusion regimes for general $q\in\mathbb{R}$. Using both analytical methods and numerical simulations, we explore how the ability of persistence (measured by some principal eigenvalue) and how the asymptotic spreading speed depend on the parameter $q$ and on the phase shift between the growth rate $r(x)$ and the diffusion coefficient $D(x)$. Our results demonstrate that persistence and spreading properties generally depend on $q$: for example, appropriate configurations of $r(x)$ and $D(x)$ can be constructed such that $q$-diffusion either enhances or diminishes the ability of persistence and the spreading speed with respect to the traditional Fickian diffusion. We show that the spatial arrangement of $r(x)$ with respect to $D(x)$ has markedly different effects depending on whether $q > 0$, $q = 0$, or $q < 0$. The case where $r$ is constant is an exception: persistence becomes independent of $q$, while the spreading speed displays a symmetry around $q = 1/2$. This work underscores the importance of carefully selecting diffusion models in ecological and epidemiological contexts, highlighting their potential implications for persistence, spreading, and control strategies. |
| title | Impact of diffusion mechanisms on persistence and spreading |
| topic | Analysis of PDEs 35K57, 35B40, 92D25, 60J70, 35Q92 |
| url | https://arxiv.org/abs/2501.03816 |