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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World Scientific Publishing
2024-12-01
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| Series: | International Journal of Mathematics for Industry |
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| Online Access: | https://www.worldscientific.com/doi/10.1142/S2661335224500254 |
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| author | Philip Broadbridge |
| author_facet | Philip Broadbridge |
| author_sort | Philip Broadbridge |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-678b33170933425bbb6e5c05deb39d5b |
| institution | Kabale University |
| issn | 2661-3352 2661-3344 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | World Scientific Publishing |
| record_format | Article |
| series | International Journal of Mathematics for Industry |
| spelling | doaj-art-678b33170933425bbb6e5c05deb39d5b2025-01-31T06:15:28ZengWorld Scientific PublishingInternational Journal of Mathematics for Industry2661-33522661-33442024-12-01160110.1142/S2661335224500254Reaction-diffusion for fish populations with realistic mobilityPhilip Broadbridge0Department of Mathematical and Physical Science, La Trobe University, Plenty Road Bundoora, VIC 3086, AustraliaNonlinear 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.https://www.worldscientific.com/doi/10.1142/S2661335224500254Nonclassical symmetryreaction-diffusion equationfisheriesLévy distributiondelayed reaction |
| spellingShingle | Philip Broadbridge Reaction-diffusion for fish populations with realistic mobility International Journal of Mathematics for Industry Nonclassical symmetry reaction-diffusion equation fisheries Lévy distribution delayed reaction |
| title | Reaction-diffusion for fish populations with realistic mobility |
| title_full | Reaction-diffusion for fish populations with realistic mobility |
| title_fullStr | Reaction-diffusion for fish populations with realistic mobility |
| title_full_unstemmed | Reaction-diffusion for fish populations with realistic mobility |
| title_short | Reaction-diffusion for fish populations with realistic mobility |
| title_sort | reaction diffusion for fish populations with realistic mobility |
| topic | Nonclassical symmetry reaction-diffusion equation fisheries Lévy distribution delayed reaction |
| url | https://www.worldscientific.com/doi/10.1142/S2661335224500254 |
| work_keys_str_mv | AT philipbroadbridge reactiondiffusionforfishpopulationswithrealisticmobility |