In a mathematical model of a system with two reaction-diffusion equations with Neumann-Dirichlet boundary conditions, we formulated zooplankton-phytoplankton in the aquatic environment on the circular domain. The attention has been focused on the toxin producing role of the space in explaining heterogeneity, the distribution of the species and the influence of the spatial structure on their abundance. The key idea of the model formulation is based on a nonlinear equations systems version with Holling II functional response. We base our mathematical analysis on the search for local and global solution with spatial diffusion. We present some mathematical results concerning the solution existence, the stability of the model equilibria. We have obtained important mathematical results for model equilibria stability at long time. Under certain mathematical conditions, the model without diffusion is locally asymptotically stable. Mathematical analysis also shows that the Hopf bifurcation breaks the time symmetry of the system and leads to uniform oscillations in space and periodic oscillations. The Turing bifurcation breaks the space symmetry and leads to the formation of stationary patterns in time and oscillatory patterns in space. A series of structured numerical simulations highlighted the formation of patterns and allowed to identify critical threshold of toxin released by phytoplankton leading to phytoplankton blooms.
| Published in | Mathematics Letters (Volume 11, Issue 1) |
| DOI | 10.11648/j.ml.20251101.12 |
| Page(s) | 10-31 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Cicular Domain, Phytoplankton-zooplankton, Toxin Parameter, Diffusion Coefficients, Global Stability, Dirichlet Boundary, Bifurcation Analysis, Pattern Formation
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APA Style
Ouedraogo, H., Ouedraogo, W., Ouedraogo, D., Sangare, B. (2025). A Circular Spatial-diffusion Mathematical Model to Analysis Hopf-Turing Bifurcation in Plankton Population Under the Toxin Control Variation in 2D. Mathematics Letters, 11(1), 10-31. https://doi.org/10.11648/j.ml.20251101.12
ACS Style
Ouedraogo, H.; Ouedraogo, W.; Ouedraogo, D.; Sangare, B. A Circular Spatial-diffusion Mathematical Model to Analysis Hopf-Turing Bifurcation in Plankton Population Under the Toxin Control Variation in 2D. Math. Lett. 2025, 11(1), 10-31. doi: 10.11648/j.ml.20251101.12
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author = {Hamidou Ouedraogo and Wendkouni Ouedraogo and Desire Ouedraogo and Boureima Sangare},
title = {A Circular Spatial-diffusion Mathematical Model to Analysis Hopf-Turing Bifurcation in Plankton Population Under the Toxin Control Variation in 2D},
journal = {Mathematics Letters},
volume = {11},
number = {1},
pages = {10-31},
doi = {10.11648/j.ml.20251101.12},
url = {https://doi.org/10.11648/j.ml.20251101.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20251101.12},
abstract = {In a mathematical model of a system with two reaction-diffusion equations with Neumann-Dirichlet boundary conditions, we formulated zooplankton-phytoplankton in the aquatic environment on the circular domain. The attention has been focused on the toxin producing role of the space in explaining heterogeneity, the distribution of the species and the influence of the spatial structure on their abundance. The key idea of the model formulation is based on a nonlinear equations systems version with Holling II functional response. We base our mathematical analysis on the search for local and global solution with spatial diffusion. We present some mathematical results concerning the solution existence, the stability of the model equilibria. We have obtained important mathematical results for model equilibria stability at long time. Under certain mathematical conditions, the model without diffusion is locally asymptotically stable. Mathematical analysis also shows that the Hopf bifurcation breaks the time symmetry of the system and leads to uniform oscillations in space and periodic oscillations. The Turing bifurcation breaks the space symmetry and leads to the formation of stationary patterns in time and oscillatory patterns in space. A series of structured numerical simulations highlighted the formation of patterns and allowed to identify critical threshold of toxin released by phytoplankton leading to phytoplankton blooms.},
year = {2025}
}
TY - JOUR T1 - A Circular Spatial-diffusion Mathematical Model to Analysis Hopf-Turing Bifurcation in Plankton Population Under the Toxin Control Variation in 2D AU - Hamidou Ouedraogo AU - Wendkouni Ouedraogo AU - Desire Ouedraogo AU - Boureima Sangare Y1 - 2025/03/24 PY - 2025 N1 - https://doi.org/10.11648/j.ml.20251101.12 DO - 10.11648/j.ml.20251101.12 T2 - Mathematics Letters JF - Mathematics Letters JO - Mathematics Letters SP - 10 EP - 31 PB - Science Publishing Group SN - 2575-5056 UR - https://doi.org/10.11648/j.ml.20251101.12 AB - In a mathematical model of a system with two reaction-diffusion equations with Neumann-Dirichlet boundary conditions, we formulated zooplankton-phytoplankton in the aquatic environment on the circular domain. The attention has been focused on the toxin producing role of the space in explaining heterogeneity, the distribution of the species and the influence of the spatial structure on their abundance. The key idea of the model formulation is based on a nonlinear equations systems version with Holling II functional response. We base our mathematical analysis on the search for local and global solution with spatial diffusion. We present some mathematical results concerning the solution existence, the stability of the model equilibria. We have obtained important mathematical results for model equilibria stability at long time. Under certain mathematical conditions, the model without diffusion is locally asymptotically stable. Mathematical analysis also shows that the Hopf bifurcation breaks the time symmetry of the system and leads to uniform oscillations in space and periodic oscillations. The Turing bifurcation breaks the space symmetry and leads to the formation of stationary patterns in time and oscillatory patterns in space. A series of structured numerical simulations highlighted the formation of patterns and allowed to identify critical threshold of toxin released by phytoplankton leading to phytoplankton blooms. VL - 11 IS - 1 ER -
Laboratory of Mathematics, Informatics (LaMIA), University Joseph Ki-Zerbo, Ouagadougou, Burkina Faso
Laboratory of Mathematics, Informatics and Applications (LaMIA), Nazi Boni University, Bobo Dioulasso, Burkina Faso
Laboratory of Mathematics, Informatics (LaMIA), University Joseph Ki-Zerbo, Ouagadougou, Burkina Faso
Laboratory of Mathematics, Informatics and Applications (LaMIA), Nazi Boni University, Bobo Dioulasso, Burkina Faso
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