Reaction-diffusion for fish populations with realistic mobility
Nonlinear reaction-diffusion equations, with Fisher logistic growth and constant diffusion coefficient, have been used in fisheries research to estimate sustainable harvesting rates and critical domain sizes of no-take areas. However, constant diffusivity in a population density corresponds to stand...
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| Format: | Article |
| Language: | English |
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World Scientific Publishing
2024-12-01
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| Series: | International Journal of Mathematics for Industry |
| Subjects: | |
| Online Access: | https://www.worldscientific.com/doi/10.1142/S2661335224500254 |
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| Summary: | Nonlinear reaction-diffusion equations, with Fisher logistic growth and constant diffusion coefficient, have been used in fisheries research to estimate sustainable harvesting rates and critical domain sizes of no-take areas. However, constant diffusivity in a population density corresponds to standard Brownian motion of individuals, with a normal distribution for displacement over a fixed time interval. For available good data sets on mobile fish populations, the distribution is certainly not normal. The data can be fitted with a long-tailed stable Lévy distribution that results from diffusion by fractional Laplacian. Exact multidimensional solutions are developed here for realistic Fisher–Kolmogorov–Petrovski-Piscounov models with diffusion by fractional Laplacian. These can also account for a delay in the reaction term. It is then shown how to modify critical domain sizes of protected areas with Dirichlet and Robin boundary conditions for populations. |
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| ISSN: | 2661-3352 2661-3344 |