Mathematical Modelling of Harmful Algal Blooms on West Coast of Sabah
Abstract
:1. Introduction
2. Materials and Methods
2.1. Nutrient-Phytoplankton-Zooplankton Interaction Model
- The linear mass action law is used for the maximal zooplankton predation rate for NTP and TPP [17].
- The model considered interspecies competition to obtain nutrients [17].
- TPP harm the zooplankton whenever they are consumed and the toxin content is produced at a high level [18].
- 1.
- 2.
- 3.
- 4.
2.2. Plankton–Zooplankton–Fish Interaction Model
- Let be the toxin production phytoplankton (TPP) which are being consumed by the zooplankton population which in turn serves as food for the fish population, .
- Let r be the intrinsic growth rate of phytoplankton; K be the environmental capacity of phytoplankton; and be the predation rate of zooplankton while is the predation rate of fish.
- is the birth rate of zooplankton while is the birth rate of fish, is the mortality rate of zooplankton and is mortality rate of fish, and f is coefficient of toxin substance from TPP.
- Let be the time delay for the fish to die when feeding on the infected zooplankton as this is not an instantaneous process. The infected zooplankton become harmful to the fish when eaten.
3. Results
3.1. Nutrient–Phytoplankton–Zooplankton Interaction Model
3.2. Plankton–Zooplankton–Fish Interaction Model
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
HAB | Harmful Algal Bloom |
TPP | Toxin-Producing Phytoplankton |
NTP | Non-Toxic Phytoplankton |
PSP | Paralytic Shellfish Poisoning |
PST | Paralytic Shellfish Toxin |
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Parameters | Symbols | Values |
---|---|---|
Dilution rate of nutrient | D | 0.3 (h) |
Constant input of nutrient concentration | 1.58 (h) | |
Nutrient uptake rate for the NTP | 0.03 (mL · h) | |
Nutrient uptake rate for the TPP | 0.022 (mL·h) | |
Conversion rate of NTP | 0.02 (mL·h) | |
Conversion rate of TPP | 0.02 (mL·h) | |
Natural death rate of NTP | 0.006 (h) | |
Natural death rate of TPP | 0.006 (h) | |
Natural death rate of zooplankton | 0.005 (h) | |
Competition coefficient | 0.02 (mL·h) | |
Competition coefficient | 0.02 (mL·h) | |
Predation rate of NTP | 0.02 (mL·h) | |
Predation rate of TPP | 0.01 (mL·h) | |
Conversion rate for NTP | 0.01 (mL·h) | |
Death rate due to consumption of TPP | 0.008 (mL·h) | |
Dilution rate of NTP | 0.0004 (h) | |
Dilution rate of TPP | 0.0004 (h) | |
Dilution rate of zooplankton | 0.0003 (h) |
Parameters | Symbols | Values |
---|---|---|
Intrinsic growth rate | r | 0.7 (mL·h) |
Constant input of nutrient concentration | K | 28 (h) |
Mortality rate of zooplankton | 0.23 (h) | |
Mortality rate of fish | 0.15 (h) | |
Predation rate of zooplankton | 0.65 (mL·h) | |
Predation rate of fish | 0.45 (mL·h) | |
Birth rate of zooplankton | 0.9 (mL·h) | |
Birth rate of fish | 0.99 (mL·h) | |
Coefficient of toxicity | f | 0.1 (h) |
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Yussof, F.N.; Maan, N.; Md Reba, M.N.; Khan, F.A. Mathematical Modelling of Harmful Algal Blooms on West Coast of Sabah. Mathematics 2022, 10, 2836. https://doi.org/10.3390/math10162836
Yussof FN, Maan N, Md Reba MN, Khan FA. Mathematical Modelling of Harmful Algal Blooms on West Coast of Sabah. Mathematics. 2022; 10(16):2836. https://doi.org/10.3390/math10162836
Chicago/Turabian StyleYussof, Fatin Nadiah, Normah Maan, Mohd Nadzri Md Reba, and Faisal Ahmed Khan. 2022. "Mathematical Modelling of Harmful Algal Blooms on West Coast of Sabah" Mathematics 10, no. 16: 2836. https://doi.org/10.3390/math10162836
APA StyleYussof, F. N., Maan, N., Md Reba, M. N., & Khan, F. A. (2022). Mathematical Modelling of Harmful Algal Blooms on West Coast of Sabah. Mathematics, 10(16), 2836. https://doi.org/10.3390/math10162836